Math Formulas: Complete List by Topic, With Examples

Grades 6 to 8

area and perimeter of rectangles, triangles and circles, simple interest, percent change, laws of exponents, $(a + b)^{2}$ and $(a - b)^{2}$

every later chapter reuses them

Grades 9 to 10

Pythagorean theorem, quadratic formula, distance and midpoint formulas, surface area and volume, AP $n$th term and sum, basic trig ratios, mean of grouped data

board exams test them directly

Grades 11 to 12

trig identities, binomial theorem, $^{n}C_{r}$, limits and derivatives, standard integrals, conic sections, probability rules

entrance exams such as JEE and the SAT build on them

Brackets or parentheses first

$2 \times (3 + 4) = 2 \times 7 = 14$

Simplify whatever is inside brackets before anything else, starting with the innermost pair and working outward.

Nested brackets

$\lbrace 2 + [3 \times (5 - 1)]\rbrace = \lbrace 2 + 12\rbrace = 14$

Clear round brackets ( ) first, then square brackets [ ], then curly braces { }, from the inside out.

Orders or exponents

$3 + 2^{3} = 3 + 8 = 11$

Powers and roots come next. BODMAS calls them Orders and PEMDAS calls them Exponents.

"Of" means multiply

$\frac{1}{2} \text{ of } 10 = \frac{1}{2} \times 10 = 5$

The word "of" between a fraction or percent and a number means multiplication, as in half of 10.

Division and multiplication, left to right

$12 \div 3 \times 2 = 4 \times 2 = 8$

These two share the same rank, so work them in the order they appear from left to right, not division first.

Addition and subtraction, left to right

$10 - 4 + 3 = 6 + 3 = 9$

These also share one rank and are done last, again strictly from left to right.

Fraction bar as a bracket

$\frac{6 + 4}{7 - 2} = \frac{10}{5} = 2$

A fraction bar groups its numerator and its denominator, so simplify each one fully before dividing.

Minus sign and powers

$-3^{2} = -9$

The exponent binds only to the 3, so square first and apply the minus after. Compare $(-3)^{2} = 9$.

Adding like fractions

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

When the denominators are equal, add the numerators and keep the common denominator $c$, where $c \neq 0$.

Subtracting like fractions

$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$

Subtract the numerators and keep the common denominator $c$, with $c \neq 0$.

Adding unlike fractions

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Rewrite both over the common denominator $bd$, with $b, d \neq 0$. Using the LCM of $b$ and $d$ keeps numbers smaller.

Subtracting unlike fractions

$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$

Same method as addition, subtract the cross products over $bd$, then reduce to lowest terms.

Multiplying fractions

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Multiply numerator by numerator and denominator by denominator, with $b, d \neq 0$. Cancel common factors first to save work.

Dividing fractions

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Keep the first fraction, flip the second and multiply. Needs $b, c, d \neq 0$ because you cannot divide by zero.

Reciprocal of a fraction

$\frac{a}{b} \to \frac{b}{a}$

Swap numerator and denominator, valid when $a \neq 0$. A number times its reciprocal always equals 1.

Mixed number to improper fraction

$a\frac{b}{c} = \frac{ac + b}{c}$

Multiply the whole number $a$ by the denominator $c$, add the numerator $b$ and keep $c$ as the denominator.

Improper fraction to mixed number

$\frac{p}{q} = Q\frac{R}{q}$

Divide $p$ by $q$. The quotient $Q$ is the whole part and the remainder $R$ is the new numerator over $q$.

Equivalent fractions

$\frac{a}{b} = \frac{a \times k}{b \times k}$

Multiplying or dividing top and bottom by the same nonzero number $k$ gives the same value in a different form.

Simplest form

$\frac{a}{b} = \frac{a \div h}{b \div h}$

Divide numerator and denominator by $h$, the HCF of $a$ and $b$, to reduce the fraction to lowest terms.

Comparing fractions

$\frac{a}{b} > \frac{c}{d} \iff ad > bc$

Cross-multiply to compare two fractions without finding a common denominator. Valid when $b$ and $d$ are both positive.

Fraction of a quantity

$\frac{a}{b} \text{ of } N = \frac{a \times N}{b}$

To find a fraction of an amount $N$, multiply $N$ by the numerator and divide by the denominator.

Percent as a fraction

$p% = \frac{p}{100}$

Percent means "per hundred", so $p$ percent is $p$ parts out of 100. For example $35% = 0.35$.

Fraction to percent

$\frac{a}{b} = \left(\frac{a}{b} \times 100\right)%$

Multiply a fraction by 100 to express it as a percent, with $b \neq 0$.

Decimal to percent

$d = (100d)%$

Move the decimal point two places to the right and add the percent sign, so $0.075 = 7.5%$.

Terminating decimal to fraction

$0.d_{1}d_{2}\ldots d_{k} = \frac{d_{1}d_{2}\ldots d_{k}}{10^{k}}$

Write the $k$ digits after the point over $10^{k}$, then simplify. For example $0.125 = \frac{125}{1000} = \frac{1}{8}$.

Pure recurring decimal to fraction

$0.\overline{d_{1}d_{2}\ldots d_{k}} = \frac{d_{1}d_{2}\ldots d_{k}}{10^{k} - 1}$

A block of $k$ digits that repeats forever equals that block over $k$ nines, so $0.\overline{7} = \frac{7}{9}$.

Mixed recurring decimal to fraction

$0.a\overline{b} = \frac{ab - a}{90}$

Here $ab$ is the two-digit number formed by the digits. One non-repeating digit $a$ is followed by a repeating digit $b$, so $0.1\overline{6} = \frac{15}{90} = \frac{1}{6}$.

Multiplying by a power of 10

$x \times 10^{n}$

Shift the decimal point of $x$ by $n$ places to the right. Dividing by $10^{n}$ shifts it $n$ places to the left.

Decimal places in a product

$\text{places in } (x \times y) = \text{places in } x + \text{places in } y$

Multiply as whole numbers, then place the point so the answer has as many decimal places as both factors together.

Scientific notation

$N = a \times 10^{n}$

Write any number with $1 \leq a < 10$ and integer $n$. Large numbers get positive $n$ and small numbers negative $n$.

Percent of a number

$p% \text{ of } N = \frac{p}{100} \times N$

Convert the percent to a fraction of 100 and multiply by the amount $N$.

What percent A is of B

$\frac{A}{B} \times 100%$

Gives $A$ as a percent of the whole $B$, with $B \neq 0$. Used for marks, shares and proportions of a total.

Percentage change

$\frac{\text{new} - \text{old}}{\text{old}} \times 100%$

A positive result is an increase and a negative result is a decrease. Always divide by the original (old) value.

Percentage increase

$\frac{\text{increase}}{\text{original}} \times 100%$

The rise in value as a percent of the starting value, where increase equals new minus original.

Percentage decrease

$\frac{\text{decrease}}{\text{original}} \times 100%$

The fall in value as a percent of the starting value, where decrease equals original minus new.

Increasing by p%

$N \times \left(1 + \frac{p}{100}\right)$

Multiplies $N$ by a single factor to give the value after a $p$ percent rise, so a 20 percent rise means times 1.2.

Decreasing by p%

$N \times \left(1 - \frac{p}{100}\right)$

Gives the value after a $p$ percent fall, so a 15 percent fall means multiplying by 0.85.

Reverse percentage after an increase

$\text{original} = \frac{\text{final}}{1 + \frac{p}{100}}$

Finds the starting value when you know the value after a $p$ percent increase. Divide, do not subtract $p$ percent of the final value.

Reverse percentage after a decrease

$\text{original} = \frac{\text{final}}{1 - \frac{p}{100}}$

Finds the starting value from the value after a $p$ percent decrease, with $p < 100$.

Two successive percentage changes

$\text{net change} = a + b + \frac{ab}{100}$

Net percent change after changes of $a$ percent then $b$ percent. Write a decrease as a negative number.

Percent error

$\frac{\lvert \text{measured} - \text{actual} \rvert}{\text{actual}} \times 100%$

How far a measured or estimated value is from the true value, as a percent of the true value.

Ratio formula

$a : b = \frac{a}{b}$

Compares two quantities in the same unit by division, read "a to b", with $b \neq 0$.

Simplest form of a ratio

$a : b = \frac{a}{h} : \frac{b}{h}$

Divide both terms by $h$, their HCF, so the terms have no common factor other than 1.

Proportion and cross-multiplication

$a : b = c : d \iff ad = bc$

Two ratios are equal when the product of the extremes $a, d$ equals the product of the means $b, c$.

Fourth proportional

$x = \frac{bc}{a}$

The value $x$ that completes $a : b = c : x$, found by cross-multiplication with $a \neq 0$.

Third proportional

$x = \frac{b^{2}}{a}$

The value $x$ in $a : b = b : x$, where the middle term is repeated.

Mean proportional

$x = \sqrt{ab}$

The value $x$ in $a : x = x : b$ for positive $a$ and $b$. It is the geometric mean of the two numbers.

Direct proportion

$\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}$

When $y = kx$, both quantities rise or fall together and their ratio stays fixed at the constant $k$.

Inverse proportion

$x_{1}y_{1} = x_{2}y_{2}$

When $y = \frac{k}{x}$, one quantity rises as the other falls and their product stays fixed. Typical case: workers and days.

Dividing N in the ratio a : b

$\frac{a}{a + b} \times N$ and $\frac{b}{a + b} \times N$

Each share is its ratio part over the total number of parts, multiplied by the amount $N$ being shared.

Dividing N in the ratio a : b : c

$\frac{a}{a + b + c} \times N$

The first share of three. Replace the numerator by $b$ or $c$ for the other two shares.

Combining two ratios

$a : b : c = pr : qr : qs$

Joins $a : b = p : q$ and $b : c = r : s$ by making the shared term $b$ equal in both.

Compound ratio

$(a : b) \times (c : d) = ac : bd$

Multiply the first terms together and the second terms together to combine two ratios into one.

Componendo and dividendo

$\frac{a + b}{a - b} = \frac{c + d}{c - d}$

Follows from $\frac{a}{b} = \frac{c}{d}$ when $a \neq b$ and $c \neq d$. Useful for solving ratio equations quickly.

Average formula (arithmetic mean)

$\bar{x} = \frac{x_{1} + x_{2} + \cdots + x_{n}}{n}$

Add all $n$ values and divide by how many there are. This is the everyday meaning of "average".

Total from the average

$\text{sum} = \bar{x} \times n$

Rearranged mean formula. Most average word problems are solved by converting averages back to totals first.

Weighted average

$\bar{x}{w} = \frac{\sum{i=1}^{n} w_{i}x_{i}}{\sum_{i=1}^{n} w_{i}}$

Each value $x_{i}$ counts $w_{i}$ times, for example credits or marks weightage. Weights must not sum to zero.

Combined average of two groups

$\bar{x} = \frac{n_{1}\bar{x}{1} + n{2}\bar{x}{2}}{n{1} + n_{2}}$

Merges groups of sizes $n_{1}$ and $n_{2}$ with means $\bar{x}{1}$ and $\bar{x}{2}$. Do not simply average the two means.

Average of equally spaced numbers

$\bar{x} = \frac{\text{first} + \text{last}}{2}$

For any list with a constant gap, such as consecutive integers or an AP, the mean is the midpoint of the ends.

Average of the first n natural numbers

$\frac{n + 1}{2}$

The mean of $1, 2, \ldots, n$. For 1 to 100 it is 50.5.

Average of the first n even numbers

$n + 1$

The mean of $2, 4, \ldots, 2n$. For the first 10 even numbers it is 11.

Average of the first n odd numbers

$n$

The mean of the odd numbers $1, 3, \ldots, 2n - 1$ is simply $n$, so the first 8 odd numbers average 8.

New average after adding one value

$\bar{x}_{\text{new}} = \frac{n\bar{x} + a}{n + 1}$

When a value $a$ joins $n$ values with mean $\bar{x}$, rebuild the total and divide by the new count.

New average after replacing one value

$\bar{x}_{\text{new}} = \bar{x} + \frac{\text{new} - \text{old}}{n}$

Swapping one of $n$ values shifts the mean by the difference spread over all $n$ values.

HCF by prime factorisation

$\text{HCF} = \prod p_{i}^{\min(a_{i}, b_{i})}$

Write both numbers as products of primes $p_{i}$ and multiply each common prime raised to its lowest power.

LCM by prime factorisation

$\text{LCM} = \prod p_{i}^{\max(a_{i}, b_{i})}$

Multiply every prime that appears in either number, each raised to its highest power.

HCF times LCM

$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

True for any pair of positive integers $a$ and $b$. It does not hold for three or more numbers in general.

LCM from the HCF

$\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$

The quickest way to find an LCM when the HCF is known or easy to find.

Co-prime numbers

$\text{HCF}(a, b) = 1 \Rightarrow \text{LCM}(a, b) = ab$

Two numbers with no common factor except 1 have their product as their LCM, for example 8 and 15.

Euclid's division algorithm

$\text{HCF}(a, b) = \text{HCF}(b, r)$

Here $r$ is the remainder when $a$ is divided by $b$. Repeat until the remainder is 0, and the last divisor is the HCF.

HCF of fractions

$\text{HCF}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}$

For fractions in lowest terms, take the HCF of the numerators over the LCM of the denominators.

LCM of fractions

$\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$

For fractions in lowest terms, take the LCM of the numerators over the HCF of the denominators.

Greatest common factor divides the difference

$\text{HCF}(a, b) \mid (a - b)$

Any common factor of two numbers also divides their difference, which helps narrow down the HCF quickly.

Prime factorisation

$N = p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}}$

Every integer greater than 1 can be written as a product of prime powers in exactly one way, apart from order.

Number of factors

$d(N) = (a_{1} + 1)(a_{2} + 1)\cdots(a_{k} + 1)$

Add 1 to each exponent and multiply. The count includes 1 and $N$ itself.

Sum of factors

$\sigma(N) = \frac{p_{1}^{a_{1}+1} - 1}{p_{1} - 1} \times \cdots \times \frac{p_{k}^{a_{k}+1} - 1}{p_{k} - 1}$

Multiply one geometric-series sum per prime. The total includes 1 and $N$.

Product of all factors

$N^{d(N)/2}$

Factors pair up to multiply to $N$, so the product of all $d(N)$ factors is $N$ raised to half that count.

Number of odd factors

$(a_{2} + 1)(a_{3} + 1)\cdots(a_{k} + 1)$

For even $N$, take $p_{1} = 2$ and ignore its exponent, since odd factors cannot contain the prime 2. If $N$ is odd, every factor is odd.

Number of even factors

$d(N) - \text{odd factors}$

All factors minus the odd ones. Equivalently $a_{1}(a_{2} + 1)\cdots(a_{k} + 1)$ when $p_{1} = 2$.

Factor count of a perfect square

$d(N)$ is odd

A number has an odd number of factors exactly when it is a perfect square, because its square root pairs with itself.

Divisibility by 2

$2 \mid N \iff d_{0} \in \lbrace 0, 2, 4, 6, 8\rbrace$

A number is divisible by 2 when its last digit is even.

Divisibility by 3

$3 \mid N \iff 3 \mid (\text{sum of digits})$

Add all the digits. If that sum is a multiple of 3, so is the number.

Divisibility by 4

$4 \mid N \iff 4 \mid (\text{last two digits})$

Only the number formed by the tens and units digits matters, because 100 is a multiple of 4.

Divisibility by 5

$5 \mid N \iff d_{0} \in \lbrace 0, 5\rbrace$

A number is divisible by 5 when it ends in 0 or 5.

Divisibility by 6

$6 \mid N \iff 2 \mid N \text{ and } 3 \mid N$

The number must pass both the test for 2 and the test for 3, since 2 and 3 are co-prime.

Divisibility by 7

$7 \mid N \iff 7 \mid (\text{rest} - 2d_{0})$

Remove the units digit, double it and subtract from the remaining number. Repeat until the result is small.

Divisibility by 8

$8 \mid N \iff 8 \mid (\text{last three digits})$

Only the last three digits matter, because 1000 is a multiple of 8.

Divisibility by 9

$9 \mid N \iff 9 \mid (\text{sum of digits})$

If the digit sum is a multiple of 9, the number is too. The remainder on division by 9 equals the digit-sum remainder.

Divisibility by 10

$10 \mid N \iff d_{0} = 0$

A number is divisible by 10 when its last digit is 0.

Divisibility by 11

$11 \mid N \iff 11 \mid (S_{\text{odd}} - S_{\text{even}})$

Sum the digits in odd places and in even places counting from the right. A difference of 0 or a multiple of 11 passes.

Divisibility by 12

$12 \mid N \iff 3 \mid N \text{ and } 4 \mid N$

The number must pass the tests for both 3 and 4.

Divisibility by 25

$25 \mid N \iff \text{last two digits} \in \lbrace 00, 25, 50, 75\rbrace$

Only the last two digits matter, because 100 is a multiple of 25.

Co-prime divisors rule

$a \mid N \text{ and } b \mid N \Rightarrow ab \mid N$

Holds when $\text{HCF}(a, b) = 1$. This is why tests for 6, 12, 15 and 18 combine two simpler tests.

Square of a number

$n^{2} = n \times n$

A number multiplied by itself. The square of any real number is never negative.

Perfect squares

$N = k^{2}$

$N$ is a perfect square when $k$ is an integer. Perfect squares end in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8.

Cube of a number

$n^{3} = n \times n \times n$

A number used three times as a factor. The cube keeps the sign of the number, so $(-2)^{3} = -8$.

Perfect cube

$N = k^{3}$

$N$ is a perfect cube when $k$ is an integer, as in 1, 8, 27, 64 and 125.

Square root

$\sqrt{N} = k \iff k^{2} = N$

The principal square root is the non-negative value $k$, defined for real numbers only when $N \geq 0$.

Cube root

$\sqrt[3]{N} = k \iff k^{3} = N$

Defined for every real $N$, including negatives, so $\sqrt[3]{-27} = -3$.

Square root of a square

$\sqrt{x^{2}} = \lvert x \rvert$

The result is the absolute value of $x$, since a principal square root is never negative.

Product rule for roots

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

Valid for $a \geq 0$ and $b \geq 0$. Used to simplify roots by pulling out perfect-square factors.

Quotient rule for roots

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Valid for $a \geq 0$ and $b > 0$. The root of a fraction is the root of the top over the root of the bottom.

Simplifying a square root

$\sqrt{a^{2}b} = a\sqrt{b}$

For $a \geq 0$ and $b \geq 0$, move a perfect-square factor $a^{2}$ outside the root as $a$.

Squaring a number ending in 5

$(10a + 5)^{2} = 100a(a + 1) + 25$

Multiply the leading part $a$ by the next integer and write 25 after it, so $35^{2} = 1225$.

Gap between consecutive squares

$(n + 1)^{2} - n^{2} = 2n + 1$

Consecutive squares differ by consecutive odd numbers, so $n^{2}$ to $(n + 1)^{2}$ always steps by $2n + 1$.

Pythagorean triples

$(m^{2} - n^{2}, \ 2mn, \ m^{2} + n^{2})$

For integers $m > n > 0$ these three numbers always satisfy $a^{2} + b^{2} = c^{2}$. $m = 2, n = 1$ gives 3, 4, 5.

Sum of natural numbers formula

$1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}$

Adds the first $n$ counting numbers. Pairing the first and last terms gives $n$ halves of $n + 1$.

Sum of even numbers formula

$2 + 4 + 6 + \cdots + 2n = n(n + 1)$

Adds the first $n$ even numbers. It is twice the sum of the first $n$ natural numbers.

Sum of odd numbers

$1 + 3 + 5 + \cdots + (2n - 1) = n^{2}$

The first $n$ odd numbers always add to a perfect square, for example $1 + 3 + 5 = 9$.

Sum of squares

$1^{2} + 2^{2} + \cdots + n^{2} = \frac{n(n + 1)(2n + 1)}{6}$

Adds the squares of the first $n$ natural numbers. The answer is always a whole number.

Sum of cubes of n natural numbers

$1^{3} + 2^{3} + \cdots + n^{3} = \left[\frac{n(n + 1)}{2}\right]^{2}$

The sum of the first $n$ cubes equals the square of the sum of the first $n$ natural numbers.

Sum of squares of the first n even numbers

$2^{2} + 4^{2} + \cdots + (2n)^{2} = \frac{2n(n + 1)(2n + 1)}{3}$

Four times the sum of the first $n$ squares, because each term is $4k^{2}$.

Sum of squares of the first n odd numbers

$1^{2} + 3^{2} + \cdots + (2n - 1)^{2} = \frac{n(2n - 1)(2n + 1)}{3}$

Adds the squares of the first $n$ odd numbers, for example $1 + 9 + 25 = 35$ when $n = 3$.

Sum of consecutive integers from a to b

$a + (a + 1) + \cdots + b = \frac{(b - a + 1)(a + b)}{2}$

Number of terms times the average of the first and last terms, for integers $a \leq b$.

Sum of multiples of k up to N

$k \times \frac{m(m + 1)}{2}$

Here $m = \left\lfloor \frac{N}{k} \right\rfloor$ is the number of multiples of $k$ from 1 to $N$.

Dividend, divisor, quotient and remainder

$a = bq + r, \quad 0 \leq r < b$

Dividend $a$ equals divisor $b$ times quotient $q$ plus remainder $r$, for integers with $b > 0$. The remainder is always smaller than the divisor.

Finding the divisor

$b = \frac{a - r}{q}$

Rearranges the division algorithm when the dividend, quotient and remainder are known, with $q \neq 0$.

Finding the quotient

$q = \left\lfloor \frac{a}{b} \right\rfloor$

The quotient is the whole-number part of $a \div b$, found by rounding down, for positive $a$ and $b$.

Exact division

$r = 0 \iff b \mid a$

A zero remainder means $b$ is a factor of $a$ and $a$ is a multiple of $b$.

Same remainder test

$a \equiv c \pmod{m} \iff m \mid (a - c)$

Two integers leave the same remainder on division by $m$ exactly when $m$ divides their difference.

Remainder of a sum

$(a + c) \bmod m = [(a \bmod m) + (c \bmod m)] \bmod m$

Find each remainder first, add them, then take the remainder again. Saves work with large numbers.

Remainder of a product

$(a \times c) \bmod m = [(a \bmod m) \times (c \bmod m)] \bmod m$

Multiply the separate remainders and reduce again. Used to find remainders of large products and powers.

Division involving zero

$\frac{0}{a} = 0$

Zero divided by any nonzero $a$ is 0, but $\frac{a}{0}$ is undefined for every number $a$.

Integers from a to b, both included

$b - a + 1$

Counts every integer in the closed range from $a$ to $b$, where $a \leq b$. From 15 to 40 there are 26.

Integers from a to b, both excluded

$b - a - 1$

Counts only the integers strictly between $a$ and $b$, where $a < b$.

Integers from a to b, one end included

$b - a$

Counts the integers when exactly one of the two endpoints is part of the range.

Multiples of k from 1 to N

$\left\lfloor \frac{N}{k} \right\rfloor$

Divide $N$ by $k$ and drop the decimal part, for positive integers $N$ and $k$.

Multiples of k from a to b

$\left\lfloor \frac{b}{k} \right\rfloor - \left\lfloor \frac{a - 1}{k} \right\rfloor$

Multiples up to $b$ minus multiples below $a$, for positive integers $a \leq b$.

Terms in an equally spaced list

$\frac{\text{last} - \text{first}}{d} + 1$

Counts the numbers in a list with constant gap $d$, such as $7, 11, 15, \ldots, 99$.

Even number

$2k$

Any integer of the form $2k$, where $k$ is an integer, is even. Zero is even.

Odd numbers

$2k + 1$

Any integer of the form $2k + 1$, where $k$ is an integer, is odd. It leaves remainder 1 when divided by 2.

Sum of two evens or two odds

$2m + 2n = 2(m + n)$

Even plus even is even, and odd plus odd is also even, since $(2m + 1) + (2n + 1) = 2(m + n + 1)$.

Sum of an even and an odd

$2m + (2n + 1) = 2(m + n) + 1$

Adding or subtracting an even and an odd number always gives an odd number.

Products of evens and odds

$(2m + 1)(2n + 1) = 2(2mn + m + n) + 1$

Odd times odd is odd. If at least one factor is even, the product is even.

Consecutive integers

$n, \ n + 1, \ n + 2, \ \ldots$

Integers one apart. Consecutive even or odd integers step by 2, as in $n, \ n + 2, \ n + 4$.

Product of consecutive integers

$n(n + 1)$ is even

Any two consecutive integers include an even number, and any three consecutive integers have a product divisible by 6.

Simple interest

$SI = \frac{P \times R \times T}{100}$

$P$ is the principal, $R$ the rate in percent per year and $T$ the time in years. Interest is charged only on the original principal.

Amount

$A = P + SI$

The total repaid or received at the end of the period is the principal plus the interest.

Amount in one step

$A = P\left(1 + \frac{RT}{100}\right)$

Gives the final amount directly without finding the interest first. $P$, $R$ and $T$ mean the same as above.

Principal

$P = \frac{100 \times SI}{R \times T}$

Rearranged form used when the interest, rate and time are known and the sum lent is unknown.

Rate

$R = \frac{100 \times SI}{P \times T}$

Finds the yearly rate in percent from the interest, the principal and the time in years.

Time period

$T = \frac{100 \times SI}{P \times R}$

Finds the number of years needed to earn a given interest on principal $P$ at $R%$ per year.

Interest with a decimal rate

$I = Prt$

The same formula with $r$ written as a decimal (8% becomes 0.08) and $t$ in years, as in US textbooks.

Time in months or days

$T = \frac{m}{12}$ or $T = \frac{d}{365}$

Change $m$ months or $d$ days into years before putting $T$ into the simple interest formula.

Time for a sum to become k times

$T = \frac{100(k - 1)}{R}$

Years needed at simple interest for the amount to reach $k$ times the principal at a rate of $R%$ per year.

Amount, compounded annually

$A = P\left(1 + \frac{R}{100}\right)^{n}$

$P$ is the principal. Interest is added once a year, and each year's interest also earns interest after that.

Compound interest

$CI = A - P$

The compound interest earned is the final amount minus the principal that was invested.

Compounded half-yearly

$A = P\left(1 + \frac{R}{200}\right)^{2n}$

Interest is added every six months, so the rate per period is halved and the number of periods doubles.

Compounded quarterly

$A = P\left(1 + \frac{R}{400}\right)^{4n}$

Interest is added four times a year, so each quarter uses $\frac{R}{4}%$ and there are $4n$ periods.

Compounded n times a year

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

General form. Here $r$ is the annual rate as a decimal, $n$ the compounding periods per year and $t$ the years. Monthly uses $n = 12$, daily $n = 365$.

Compounded continuously

$A = Pe^{rt}$

$e \approx 2.71828$, $r$ is the annual rate as a decimal and $t$ the time in years. It is the limit of compounding ever more often.

Different rates in successive years

$A = P\left(1 + \frac{R_{1}}{100}\right)\left(1 + \frac{R_{2}}{100}\right)\left(1 + \frac{R_{3}}{100}\right)$

Used when the rate changes each year, with $R_{1}%$ in the first year, $R_{2}%$ in the second and so on.

Fractional number of years

$A = P\left(1 + \frac{R}{100}\right)^{n}\left(1 + \frac{fR}{100}\right)$

For $n$ whole years plus a fraction $f$ of a year, with simple interest on the part year. For 2.5 years, $n = 2$ and $f = \frac{1}{2}$.

CI minus SI for 2 years

$CI - SI = P\left(\frac{R}{100}\right)^{2}$

The extra interest over two years caused by the first year's interest earning interest. Assumes annual compounding.

CI minus SI for 3 years

$CI - SI = P\left(\frac{R}{100}\right)^{2}\left(3 + \frac{R}{100}\right)$

The gap between compound and simple interest over three years on the same principal at the same annual rate.

Rule of 72

$t \approx \frac{72}{R}$

A quick estimate of the years needed for money to double at $R%$ compounded yearly. It is most accurate near 8%.

Effective annual rate

$EAR = \left(1 + \frac{r}{n}\right)^{n} - 1$

The true yearly growth when a nominal decimal rate $r$ is compounded $n$ times a year. Used to compare savings or loan offers.

EAR with continuous compounding

$EAR = e^{r} - 1$

The effective yearly rate when the nominal decimal rate $r$ is compounded continuously.

Nominal rate from EAR

$r = n\left[(1 + EAR)^{\frac{1}{n}} - 1\right]$

Turns an effective annual rate back into the nominal rate that is compounded $n$ times per year.

Present value

$PV = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$

The sum to invest today to reach amount $A$ after $t$ years at nominal rate $r$ compounded $n$ times a year.

Compound annual growth rate

$CAGR = \left(\frac{V_{f}}{V_{i}}\right)^{\frac{1}{t}} - 1$

The steady yearly growth rate, as a decimal, that takes a starting value $V_{i}$ to a final value $V_{f}$ in $t$ years.

Loan EMI

$EMI = \frac{P \cdot i \cdot (1 + i)^{N}}{(1 + i)^{N} - 1}$

Equal monthly instalment on a loan $P$, where $i$ is the monthly rate as a decimal (annual rate divided by 12) and $N$ the number of months.

Appreciation

$V = V_{0}\left(1 + \frac{R}{100}\right)^{n}$

Value after $n$ years when an asset worth $V_{0}$ gains $R%$ of its current value every year, such as land.

Depreciation

$V = V_{0}\left(1 - \frac{R}{100}\right)^{n}$

Value after $n$ years when an item worth $V_{0}$ loses $R%$ of its current value every year, such as a car.

Original value from depreciated value

$V_{0} = \frac{V}{\left(1 - \frac{R}{100}\right)^{n}}$

Works backward from today's value $V$ to the value $n$ years ago under yearly depreciation at $R%$.

Population growth

$P_{n} = P_{0}\left(1 + \frac{R}{100}\right)^{n}$

Population after $n$ years when it grows by $R%$ a year from $P_{0}$. Use a minus sign for a yearly decline.

Population n years ago

$P_{0} = \frac{P_{n}}{\left(1 + \frac{R}{100}\right)^{n}}$

Finds the earlier population from the present one $P_{n}$ when the growth rate $R%$ has stayed constant.

Growth then decline

$V = V_{0}\left(1 + \frac{R_{1}}{100}\right)\left(1 - \frac{R_{2}}{100}\right)$

A rise of $R_{1}%$ in one year followed by a fall of $R_{2}%$ the next. Add one factor for each extra year.

Straight-line depreciation

$D = \frac{C - S}{n}$

Equal yearly depreciation $D$ for an asset that costs $C$ and has scrap value $S$ after a useful life of $n$ years.

Profit

$\text{Profit} = SP - CP$

A profit is made when the selling price $SP$ is more than the cost price $CP$.

Loss

$\text{Loss} = CP - SP$

A loss is made when the cost price $CP$ is more than the selling price $SP$.

Profit percentage

$\text{Profit},% = \frac{\text{Profit}}{CP} \times 100$

Profit written as a percent of the cost price. Profit and loss percents are based on $CP$ unless a question says otherwise.

Loss percentage

$\text{Loss},% = \frac{\text{Loss}}{CP} \times 100$

Loss written as a percent of the cost price $CP$, used to compare losses on items of different prices.

Selling price at a profit

$SP = CP \times \frac{100 + P%}{100}$

Finds the selling price when the cost price and the wanted profit percent $P%$ are known.

Selling price at a loss

$SP = CP \times \frac{100 - L%}{100}$

Finds the selling price when an item is sold at a loss of $L%$ on its cost price.

Cost price from SP and profit

$CP = \frac{100 \times SP}{100 + P%}$

Works back to the cost price when the selling price and the profit percent are given.

Cost price from SP and loss

$CP = \frac{100 \times SP}{100 - L%}$

Works back to the cost price when the selling price and the loss percent are given.

Profit margin

$\text{Margin},% = \frac{\text{Profit}}{SP} \times 100$

Profit as a percent of the selling price, common in business. It is smaller than the profit percent on cost whenever there is a profit.

Same SP with equal gain and loss

$\text{Net loss},% = \frac{x^{2}}{100}$

Two items sold at the same price, one at an $x%$ gain and one at an $x%$ loss, always give an overall loss of this percent.

Gain with a false weight

$\text{Gain},% = \frac{\text{True weight} - \text{False weight}}{\text{False weight}} \times 100$

A seller who charges the cost price but hands over less than the true weight makes this percent profit.

Partnership profit share

$\text{Share}{A} : \text{Share}{B} = C_{A}T_{A} : C_{B}T_{B}$

Partners divide profit in the ratio of each capital $C$ multiplied by the time $T$ for which it was invested.

Discount

$D = MP - SP$

Discount is the reduction from the marked price $MP$ to the selling price $SP$ that the buyer actually pays.

Discount percentage

$D% = \frac{D}{MP} \times 100$

Discount written as a percent of the marked price, never of the cost price.

Selling price after discount

$SP = MP\left(1 - \frac{d}{100}\right)$

The price paid after a discount of $d%$ is taken off the marked price $MP$.

Marked price from SP

$MP = \frac{100 \times SP}{100 - d}$

Finds the list price when the selling price and the discount rate $d%$ are known.

Markup

$MP = CP\left(1 + \frac{m}{100}\right)$

A seller who marks goods $m%$ above the cost price $CP$ sets the marked price this way.

Marked price for a target profit

$MP = CP \times \frac{100 + P%}{100 - d%}$

The marked price that still leaves a profit of $P%$ on cost after a discount of $d%$ is given.

Profit after markup and discount

$\text{Profit},% = m - d - \frac{md}{100}$

Net profit percent on cost when goods are marked $m%$ above cost and then sold at a $d%$ discount.

Buy x get y free

$\text{Discount},% = \frac{y}{x + y} \times 100$

The effective discount when a customer pays for $x$ items and takes home $x + y$ items.

Two successive discounts

$d = d_{1} + d_{2} - \frac{d_{1}d_{2}}{100}$

The single discount equal to $d_{1}%$ followed by $d_{2}%$. The second discount applies to the already reduced price.

Price after successive discounts

$SP = MP\left(1 - \frac{d_{1}}{100}\right)\left(1 - \frac{d_{2}}{100}\right)\left(1 - \frac{d_{3}}{100}\right)$

Multiply one factor for each discount. The order does not matter, but the discounts cannot simply be added.

Two successive changes

$\text{Net change},% = a + b + \frac{ab}{100}$

Net percent change after a change of $a%$ and then $b%$. Use positive values for increases and negative values for decreases.

Rise and fall by the same percent

$\text{Net change},% = -\frac{x^{2}}{100}$

An increase of $x%$ and a decrease of $x%$, in either order, always end in a net decrease.

Fixed spending after a price rise

$\text{Cut},% = \frac{r}{100 + r} \times 100$

How much consumption must drop when the price rises by $r%$ so that total spending stays the same.

Fixed spending after a price fall

$\text{Increase},% = \frac{r}{100 - r} \times 100$

How much consumption can grow when the price falls by $r%$ while total spending stays the same.

Tax amount

$\text{Tax} = \frac{t}{100} \times P$

Tax on a pre-tax price $P$ at a rate of $t%$. The same formula works for GST, VAT and US sales tax.

Price including tax

$P_{\text{incl}} = P\left(1 + \frac{t}{100}\right)$

The final bill when tax at $t%$ is added to the pre-tax price $P$.

Pre-tax price from inclusive price

$P = \frac{100 \times P_{\text{incl}}}{100 + t}$

Removes the tax from a tax-inclusive price. Subtracting $t%$ of the inclusive price gives the wrong answer.

Tax inside an inclusive price

$\text{Tax} = P_{\text{incl}} \times \frac{t}{100 + t}$

The tax portion contained in a price that already includes tax at $t%$.

CGST and SGST split

$\text{CGST} = \text{SGST} = \frac{\text{GST}}{2}$

For a sale within one Indian state, the GST is split equally between the central and the state government.

IGST

$\text{IGST} = \text{GST}$

For a sale between two Indian states, the full GST is charged as one integrated tax.

Net GST payable

$\text{GST payable} = \text{Output GST} - \text{Input tax credit}$

A business pays the tax collected on its sales minus the tax it already paid on its purchases.

Discount then tax

$P_{\text{final}} = MP\left(1 - \frac{d}{100}\right)\left(1 + \frac{t}{100}\right)$

Tax is charged on the price after discount, so apply the discount factor first and then the tax factor.

Speed

$S = \frac{D}{T}$

Speed $S$ is the distance $D$ covered per unit of time $T$. Keep units matched, such as km with hours.

Distance

$D = S \times T$

Distance travelled when moving at a constant speed $S$ for a time $T$.

Time

$T = \frac{D}{S}$

Time taken to cover a distance $D$ at a constant speed $S$.

km/h to m/s

$v_{\text{m/s}} = v_{\text{km/h}} \times \frac{5}{18}$

Multiply by $\frac{5}{18}$ because 1 km is 1000 m and 1 hour is 3600 seconds.

m/s to km/h

$v_{\text{km/h}} = v_{\text{m/s}} \times \frac{18}{5}$

The reverse conversion. For example, a speed of 10 m/s equals 36 km/h.

Average speed

$\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

Always divide total distance by total time. Taking the plain mean of the speeds is wrong when the times differ.

Average speed over two equal distances

$\bar{v} = \frac{2xy}{x + y}$

Average speed when two equal distances are covered at speeds $x$ and $y$. It is the harmonic mean of the two speeds.

Average speed over three equal distances

$\bar{v} = \frac{3xyz}{xy + yz + zx}$

Average speed when three equal distances are covered at speeds $x$, $y$ and $z$.

Average speed over equal times

$\bar{v} = \frac{x + y}{2}$

When the same amount of time is spent at speeds $x$ and $y$, the ordinary mean of the two speeds is correct.

Speed and time for a fixed distance

$\frac{S_{1}}{S_{2}} = \frac{T_{2}}{T_{1}}$

Over the same distance, speed and time are inversely proportional, so doubling the speed halves the time.

Distance from late and early arrival

$D = \frac{S_{1}S_{2}(t_{1} + t_{2})}{S_{2} - S_{1}}$

At the slower speed $S_{1}$ a traveller arrives $t_{1}$ late and at the faster speed $S_{2}$ arrives $t_{2}$ early. Times in hours.

Relative speed, same direction

$S_{r} = \lvert u - v \rvert$

When two objects move the same way at speeds $u$ and $v$, the gap between them changes at the difference.

Relative speed, opposite directions

$S_{r} = u + v$

When two objects move towards or away from each other at speeds $u$ and $v$, the speeds add.

Time to meet

$T = \frac{d}{u + v}$

Time for two objects $d$ apart, moving towards each other at speeds $u$ and $v$, to meet.

Time to catch up

$T = \frac{d}{u - v}$

Time for a faster object at speed $u$ to close a gap $d$ on a slower one at $v$ going the same way, with $u > v$.

Train passing a pole or person

$T = \frac{L}{S}$

A train of length $L$ moving at speed $S$ must cover its own length to pass a point object.

Train passing a platform or bridge

$T = \frac{L + P}{S}$

The train must cover its own length $L$ plus the length $P$ of the platform or bridge.

Two trains crossing, opposite directions

$T = \frac{L_{1} + L_{2}}{u + v}$

Time for trains of lengths $L_{1}$ and $L_{2}$, moving towards each other at speeds $u$ and $v$, to pass completely.

Two trains crossing, same direction

$T = \frac{L_{1} + L_{2}}{u - v}$

Time for the faster train at speed $u$ to fully overtake the slower train at speed $v$, with $u > v$.

Downstream speed

$v_{d} = b + s$

A boat with speed $b$ in still water moves along with a current of speed $s$, so the speeds add.

Upstream speed

$v_{u} = b - s$

Moving against the current, the boat's speed is its still-water speed $b$ minus the stream speed $s$.

Speed in still water

$b = \frac{v_{d} + v_{u}}{2}$

The boat's still-water speed is the mean of its downstream speed $v_{d}$ and upstream speed $v_{u}$.

Speed of the stream

$s = \frac{v_{d} - v_{u}}{2}$

The speed of the current is half the difference between the downstream and upstream speeds.

Work rate

$\text{Rate} = \frac{1}{n}$

A worker who finishes a whole job in $n$ days completes $\frac{1}{n}$ of the job each day.

Work done

$W = \text{Rate} \times T$

The fraction of the job finished equals the work rate multiplied by the time $T$ spent working.

Combined rate

$\frac{1}{T} = \frac{1}{a} + \frac{1}{b}$

When workers who take $a$ and $b$ days alone work together, their daily rates add. $T$ is the joint time.

Two workers together

$T = \frac{ab}{a + b}$

Time for two workers, who need $a$ and $b$ days alone, to finish the job together.

Three workers together

$T = \frac{abc}{ab + bc + ca}$

Joint time for three workers who need $a$, $b$ and $c$ days alone.

One worker alone from the joint time

$T_{B} = \frac{aT}{a - T}$

If A alone takes $a$ days and A and B together take $T$ days, B alone takes this long. Needs $a > T$.

Work equivalence

$\frac{M_{1}D_{1}H_{1}}{W_{1}} = \frac{M_{2}D_{2}H_{2}}{W_{2}}$

$M$ is workers, $D$ days, $H$ hours per day and $W$ the amount of work. Used to rescale a job to a new team.

Efficiency and time

$\frac{E_{A}}{E_{B}} = \frac{T_{B}}{T_{A}}$

Efficiency and time for the same job are inversely proportional, so a worker twice as efficient takes half the time.

Pipe filling with a leak

$T = \frac{ab}{b - a}$

A pipe fills a tank in $a$ hours and a leak empties it in $b$ hours, with $b > a$. $T$ is the net filling time.

Net rate of several pipes

$\frac{1}{T} = \sum \frac{1}{f_{i}} - \sum \frac{1}{e_{j}}$

Add the rates of the filling pipes (times $f_{i}$) and subtract the rates of the emptying pipes (times $e_{j}$).

Sharing wages

$\text{Share}{A} : \text{Share}{B} = W_{A} : W_{B}$

Pay for a joint job is divided in the ratio of the work each person did, which is rate times time.

Kilometre to metre

$1,\text{km} = 1000,\text{m}$

Multiply kilometres by 1000 to get metres. Metric length units change by powers of 10.

Metre to centimetre

$1,\text{m} = 100,\text{cm}$

Multiply metres by 100 to get centimetres, and divide centimetres by 100 to get metres.

Centimetre to millimetre

$1,\text{cm} = 10,\text{mm}$

Multiply centimetres by 10 to get millimetres. One metre is 1000 millimetres.

Inch to centimetre

$1,\text{in} = 2.54,\text{cm}$

Exact by definition. Multiply inches by 2.54 for centimetres, or divide centimetres by 2.54 for inches.

Foot to inches

$1,\text{ft} = 12,\text{in}$

One foot is 12 inches, so multiply feet by 12 to get inches.

Foot to meters

$1,\text{ft} = 0.3048,\text{m}$

Exact value. Multiply feet by 0.3048 for metres. One metre is about 3.2808 feet.

Yard to metre

$1,\text{yd} = 3,\text{ft} = 0.9144,\text{m}$

One yard is three feet or 36 inches, which is exactly 0.9144 metres.

Miles to km

$1,\text{mile} = 1.609344,\text{km}$

Exact value. Multiply miles by 1.609344 (often rounded to 1.61) to get kilometres. One mile is 1760 yards.

Kilometres to miles

$1,\text{km} \approx 0.621371,\text{mile}$

Multiply kilometres by about 0.6214 to get miles, the reverse of the row above.

Nautical mile

$1,\text{nautical mile} = 1852,\text{m}$

Used at sea and in the air. A speed of one nautical mile per hour is called one knot.

Kilogram to gram

$1,\text{kg} = 1000,\text{g}$

Multiply kilograms by 1000 to get grams. One gram is 1000 milligrams.

Tonne to kilogram

$1,\text{t} = 1000,\text{kg}$

A metric tonne is 1000 kilograms. It differs from the US short ton of 2000 pounds.

Pound to kilogram

$1,\text{lb} = 0.45359237,\text{kg}$

Exact by definition. Multiply pounds by about 0.4536 to get kilograms.

Kilogram to pound

$1,\text{kg} \approx 2.20462,\text{lb}$

Multiply kilograms by about 2.2046 to get pounds.

Pound to ounce

$1,\text{lb} = 16,\text{oz}$

One pound is 16 ounces, and one ounce is about 28.35 grams.

Litre to millilitre

$1,\text{L} = 1000,\text{mL}$

Multiply litres by 1000 to get millilitres. One millilitre is the same volume as one cubic centimetre.

US gallon to litre

$1,\text{US gal} = 3.785411784,\text{L}$

Exact value, about 3.785 litres. This is the gallon used for fuel and milk in the United States.

UK gallon to litre

$1,\text{UK gal} = 4.54609,\text{L}$

The imperial gallon, exactly 4.54609 litres, is larger than the US gallon.

Area conversion rule

$\text{Area factor} = k^{2}$

If one large unit of length equals $k$ small units, one square large unit equals $k^{2}$ square small units.

Volume conversion rule

$\text{Volume factor} = k^{3}$

If one large unit of length equals $k$ small units, one cubic large unit equals $k^{3}$ cubic small units.

Square centimetres to square metres

$1,\text{m}^{2} = 10000,\text{cm}^{2}$

Divide square centimetres by 10,000 to get square metres, because $100^{2} = 10000$.

Square kilometre to square metre

$1,\text{km}^{2} = 10^{6},\text{m}^{2}$

One square kilometre is one million square metres, since $1000^{2} = 10^{6}$.

Hectare

$1,\text{ha} = 10000,\text{m}^{2}$

A hectare is the area of a square 100 m on each side. One square kilometre is 100 hectares.

Acre

$1,\text{acre} = 4046.8564224,\text{m}^{2}$

Exact value, about 0.4047 hectare. One acre is also 43,560 square feet.

Square foot to square inches

$1,\text{ft}^{2} = 144,\text{in}^{2}$

Since one foot is 12 inches, one square foot is $12^{2} = 144$ square inches.

Square metre to square feet

$1,\text{m}^{2} \approx 10.7639,\text{ft}^{2}$

Multiply square metres by about 10.764 to get square feet, often used for room and flat sizes.

Cubic metre to cubic centimetre

$1,\text{m}^{3} = 10^{6},\text{cm}^{3}$

One cubic metre is one million cubic centimetres, since $100^{3} = 10^{6}$.

Cubic centimetre to millilitre

$1,\text{cm}^{3} = 1,\text{mL}$

Links volume to capacity. A 1 cm cube holds one millilitre, so 1000 cubic centimetres make one litre.

Cubic metre to litre

$1,\text{m}^{3} = 1000,\text{L}$

Used in tank and water-storage problems. Multiply cubic metres by 1000 to get litres.

Cubic foot to litre

$1,\text{ft}^{3} \approx 28.3168,\text{L}$

One cubic foot holds about 28.32 litres, found from $0.3048^{3},\text{m}^{3}$.

Celsius to Fahrenheit

$F = \frac{9}{5}C + 32$

Multiply the Celsius reading $C$ by 1.8 and then add 32 to get the Fahrenheit reading $F$.

Fahrenheit to Celsius

$C = \frac{5}{9}(F - 32)$

Subtract 32 from the Fahrenheit reading first, then multiply the result by $\frac{5}{9}$.

Celsius to Kelvin

$K = C + 273.15$

Add 273.15 to the Celsius reading. Kelvin is written without a degree sign, and 0 K is absolute zero.

Kelvin to Celsius

$C = K - 273.15$

Subtract 273.15 from the kelvin value. Many school problems round this constant to 273.

Kelvin to Fahrenheit

$F = \frac{9}{5}(K - 273.15) + 32$

Changes kelvin to Celsius and then Celsius to Fahrenheit in a single step.

Fahrenheit to Kelvin

$K = \frac{5}{9}(F - 32) + 273.15$

Changes Fahrenheit to Celsius and then adds 273.15 to reach the kelvin value.

Temperature change

$\Delta F = \frac{9}{5}\Delta C$ and $\Delta K = \Delta C$

For a difference in temperature, do not add 32 or 273.15. A rise of 10 degrees Celsius is a rise of 18 degrees Fahrenheit.

Equal reading

$-40^{\circ}\text{C} = -40^{\circ}\text{F}$

The only temperature at which the Celsius and Fahrenheit scales show the same number.

Minute to seconds

$1,\text{min} = 60,\text{s}$

Multiply minutes by 60 to get seconds, and divide seconds by 60 to get minutes.

Hour to seconds

$1,\text{h} = 60,\text{min} = 3600,\text{s}$

Multiply hours by 3600 to get seconds. This factor is used when changing km/h into m/s.

Day

$1,\text{day} = 24,\text{h} = 1440,\text{min} = 86400,\text{s}$

The standard lengths used to change days into hours, minutes or seconds.

Week

$1,\text{week} = 7,\text{days} = 168,\text{h}$

Multiply weeks by 7 to get days, or by 168 to get hours.

Year

$1,\text{year} = 365,\text{days} = 52,\text{weeks} + 1,\text{day}$

A common year. A leap year has 366 days, which is 52 weeks and 2 days.

Leap year rule

$4 \mid y$, except when $100 \mid y$ and $400 \nmid y$

Year $y$ is a leap year if divisible by 4, but a century year must also be divisible by 400. So 2000 was a leap year and 1900 was not.

Decimal hours to minutes

$\text{Minutes} = \text{decimal part} \times 60$

Changes a time such as 2.75 hours into hours and minutes. The fractional part times 60 gives the minutes.

12-hour to 24-hour clock

$H_{24} = H_{12} + 12$

Add 12 to the hour for times from 1 pm to 11 pm. 12 am becomes 00 and 12 pm stays 12.

Angle between clock hands

$\theta = \lvert 30H - 5.5M \rvert$

Angle in degrees at $H$ hours and $M$ minutes, with $H$ read on a 12-hour clock. If the result is more than $180^{\circ}$, the smaller angle is $360^{\circ} - \theta$.

Square of a sum (a + b whole square)

$(a+b)^{2} = a^{2} + 2ab + b^{2}$

The square of a sum equals both squares plus twice their product. Use it to expand brackets or square numbers like 103 mentally.

Square of a difference

$(a-b)^{2} = a^{2} - 2ab + b^{2}$

The square of a difference equals both squares minus twice their product. Handy for squares such as 98 or $(x-5)^{2}$.

Difference of squares

$a^{2} - b^{2} = (a+b)(a-b)$

A difference of two perfect squares always factors into their sum times their difference. Used in factoring and quick products like $51 \times 49$.

a square plus b square

$a^{2} + b^{2} = (a+b)^{2} - 2ab = (a-b)^{2} + 2ab$

Rewrites a sum of two squares using the sum or difference and the product. Use it when $a + b$ or $a - b$ and $ab$ are given.

Sum of the two binomial squares

$(a+b)^{2} + (a-b)^{2} = 2(a^{2} + b^{2})$

Adding the square of the sum and the square of the difference cancels the $2ab$ terms and doubles the squares.

Difference of the two binomial squares

$(a+b)^{2} - (a-b)^{2} = 4ab$

Subtracting the two squares leaves four times the product. Use it to find $ab$ from a known sum and difference.

Product with a common first term

$(x+a)(x+b) = x^{2} + (a+b)x + ab$

Multiplies two binomials that share $x$. The middle coefficient is the sum of $a$ and $b$ and the constant is their product.

Product of three binomials with a common term

$(x+a)(x+b)(x+c) = x^{3} + (a+b+c)x^{2} + (ab+bc+ca)x + abc$

Expands three brackets that share $x$. The coefficients are the sum, the sum of pairwise products and the product of $a$, $b$, $c$.

Cube of a sum

$(a+b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3} = a^{3} + b^{3} + 3ab(a+b)$

Cubes a binomial with coefficients 1, 3, 3, 1. The shorter second form is faster when $a + b$ and $ab$ are known.

Cube of a difference

$(a-b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3} = a^{3} - b^{3} - 3ab(a-b)$

Cubes a difference. The signs alternate, starting with plus, and the coefficients stay 1, 3, 3, 1.

a cube plus b cube

$a^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2})$

Factors a sum of two cubes. The linear factor keeps the plus sign and the middle term of the quadratic factor is negative.

a cube minus b cube

$a^{3} - b^{3} = (a-b)(a^{2} + ab + b^{2})$

Factors a difference of two cubes. The linear factor has the minus sign and every term of the quadratic factor is positive.

Square of a trinomial

$(a+b+c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca$

The square of three terms is the sum of their squares plus twice each pairwise product.

a square plus b square plus c square

$a^{2} + b^{2} + c^{2} = (a+b+c)^{2} - 2(ab + bc + ca)$

Finds the sum of three squares when the total $a + b + c$ and the sum of pairwise products are known.

Sum of three cubes identity

$a^{3} + b^{3} + c^{3} - 3abc = (a+b+c)(a^{2} + b^{2} + c^{2} - ab - bc - ca)$

Factors this symmetric cubic expression into the sum of the terms times a quadratic factor. Common in Class 9 and JEE problems.

Three cubes with zero sum

$a^{3} + b^{3} + c^{3} = 3abc$ when $a + b + c = 0$

A special case of the previous identity. When the three numbers add to zero, the left factor vanishes.

Product rule

$a^{m} \cdot a^{n} = a^{m+n}$

When powers with the same base $a$ are multiplied, keep the base and add the exponents $m$ and $n$.

Quotient rule

$\frac{a^{m}}{a^{n}} = a^{m-n}$

When powers with the same nonzero base are divided, keep the base and subtract the bottom exponent from the top one. Needs $a \neq 0$.

Power of a power rule

$(a^{m})^{n} = a^{mn}$

Raising a power to another power multiplies the exponents. Do not confuse it with the product rule, which adds them.

Power of a product

$(ab)^{n} = a^{n}b^{n}$

An exponent outside brackets applies to every factor inside. It does not spread over addition, since $(a+b)^{2} \neq a^{2} + b^{2}$.

Power of a quotient

$\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}$

The exponent applies to both the numerator and the denominator. The denominator $b$ must be nonzero.

Zero exponent

$a^{0} = 1$

Any nonzero base raised to zero equals 1. The expression $0^{0}$ is left undefined in school algebra.

Negative exponent

$a^{-n} = \frac{1}{a^{n}}$

A negative exponent means the reciprocal of the positive power. The base $a$ must be nonzero.

Negative exponent of a fraction

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$

Flip the fraction and make the exponent positive. Both $a$ and $b$ must be nonzero.

Fractional exponent

$a^{\frac{1}{n}} = \sqrt[n]{a}$

An exponent of $\frac{1}{n}$ means the $n$th root. For even $n$ the base must satisfy $a \geq 0$ to give a real answer.

Rational exponent

$a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}$

The denominator $n$ is the root and the numerator $m$ is the power. Taking the root first keeps the numbers small. Use $a > 0$.

Equal powers of the same base

$a^{m} = a^{n}$ implies $m = n$

For a base with $a > 0$ and $a \neq 1$, equal powers force equal exponents. This is how simple exponential equations are solved.

Definition of the nth root

$\sqrt[n]{a} = b$ means $b^{n} = a$

The $n$th root of $a$ is the number whose $n$th power is $a$. For even $n$ it is the non-negative root, with $a \geq 0$.

Square root of a square

$\sqrt{a^{2}} = \lvert a \rvert$

The square root of a square is the absolute value, not $a$ itself. For example, $\sqrt{(-3)^{2}} = 3$.

Product rule for roots

$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

Multiply the numbers under two square roots. Valid for $a \geq 0$ and $b \geq 0$.

Quotient rule for roots

$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Divide the numbers under two square roots. Valid for $a \geq 0$ and $b > 0$.

Product rule for nth roots

$\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$

Roots of the same order combine under one root sign. For even $n$, both $a$ and $b$ must be non-negative.

Root of a root

$\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$

Taking an $n$th root and then an $m$th root equals one root of order $mn$. For example, $\sqrt{\sqrt{16}} = \sqrt[4]{16} = 2$.

Simplifying a surd

$\sqrt{a^{2}b} = a\sqrt{b}$

Take a perfect-square factor $a^{2}$ out of the root, as in $\sqrt{72} = 6\sqrt{2}$. Needs $a \geq 0$ and $b \geq 0$.

Adding like surds

$p\sqrt{a} + q\sqrt{a} = (p+q)\sqrt{a}$

Only surds with the same number under the root combine. In general $\sqrt{a} + \sqrt{b} \neq \sqrt{a+b}$.

Rationalising a single surd

$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$

Multiply top and bottom by $\sqrt{a}$ to clear the root from the denominator. Needs $a > 0$.

Rationalising with the conjugate

$\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b}$

Multiply by the conjugate $\sqrt{a} - \sqrt{b}$ so the denominator becomes a difference of squares. Needs $a, b \geq 0$ and $a \neq b$.

Rationalising a binomial surd

$\frac{1}{p + \sqrt{q}} = \frac{p - \sqrt{q}}{p^{2} - q}$

Same conjugate method when one term is rational. Here $q > 0$ and $p^{2} \neq q$.

Square root of a binomial surd

$\sqrt{a + b + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$

Recognises a perfect square hidden under a root, as in $\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}$. Needs $a, b \geq 0$.

Log to exponential form

$\log_{b} x = y$ means $b^{y} = x$

A logarithm is the exponent $y$ that turns base $b$ into $x$. Needs $b > 0$, $b \neq 1$ and argument $x > 0$.

Product rule

$\log_{b}(MN) = \log_{b} M + \log_{b} N$

The log of a product is the sum of the logs. Valid for $M > 0$, $N > 0$, $b > 0$ and $b \neq 1$.

Quotient rule

$\log_{b}\left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N$

The log of a quotient is the log of the top minus the log of the bottom. Needs $M > 0$ and $N > 0$.

Power rule

$\log_{b}(M^{p}) = p\log_{b} M$

An exponent inside a log comes out as a multiplier. Valid for $M > 0$ and any real $p$.

Change of base

$\log_{b} M = \frac{\log_{k} M}{\log_{k} b}$

Converts a log to any new base $k$ with $k > 0$ and $k \neq 1$. Use base 10 or $e$ to evaluate on a calculator.

Reciprocal rule

$\log_{b} a = \frac{1}{\log_{a} b}$

Swapping the base and the argument gives the reciprocal. Needs $a, b > 0$ with $a \neq 1$ and $b \neq 1$.

Chain rule for bases

$\log_{a} b \cdot \log_{b} c = \log_{a} c$

The middle base cancels, like a chain of unit conversions. All bases must be positive and not equal to 1, and $c > 0$.

Power of the base

$\log_{b^{q}} M = \frac{1}{q}\log_{b} M$

A power on the base divides the log. Here $q \neq 0$ and $b^{q} \neq 1$. For example, $\log_{4} M = \frac{1}{2}\log_{2} M$.

Log of the base

$\log_{b} b = 1$

Any valid base raised to the power 1 is itself, so the log of the base is 1. For example, $\log_{10} 10 = 1$.

Log of one

$\log_{b} 1 = 0$

Any valid base raised to zero is 1, so the log of 1 is zero in every base.

Inverse properties

$b^{\log_{b} x} = x$ and $\log_{b}(b^{x}) = x$

Exponentials and logs of the same base undo each other. The first form needs $x > 0$, the second holds for all real $x$.

Exchange rule

$a^{\log_{b} c} = c^{\log_{b} a}$

The base-$a$ and the argument-$c$ positions can be swapped. Needs $a > 0$, $c > 0$, $b > 0$ and $b \neq 1$.

Common and natural logs

$\log x = \log_{10} x$ and $\ln x = \log_{e} x$

In school work, $\log$ with no base means base 10 and $\ln$ means base $e \approx 2.71828$. Both need $x > 0$.

Converting ln to log

$\ln x = \ln 10 \cdot \log_{10} x \approx 2.3026\log_{10} x$

A special case of change of base. Useful values are $\log_{10} 2 \approx 0.3010$ and $\log_{10} 3 \approx 0.4771$.

Linear equation in one variable

$ax + b = 0$ gives $x = -\frac{b}{a}$

A first-degree equation in one unknown has exactly one solution when the coefficient $a \neq 0$.

Slope formula

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Slope $m$ is rise over run between points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$. Needs $x_{1} \neq x_{2}$, since a vertical line has undefined slope.

Slope from an angle

$m = \tan\theta$

$\theta$ is the angle the line makes with the positive $x$-axis, measured anticlockwise. Not defined when $\theta = 90^{\circ}$.

Slope intercept form

$y = mx + c$

$m$ is the slope and $c$ is the $y$-intercept. US textbooks write the same form as $y = mx + b$.

Point slope form

$y - y_{1} = m(x - x_{1})$

The line through the point $(x_{1}, y_{1})$ with slope $m$. Use it when one point and the slope are known.

Two-point form

$y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}(x - x_{1})$

The line through two known points. It is point-slope form with the slope formula substituted. Needs $x_{1} \neq x_{2}$.

Intercept form

$\frac{x}{a} + \frac{y}{b} = 1$

The line cutting the $x$-axis at $(a, 0)$ and the $y$-axis at $(0, b)$. Needs $a \neq 0$ and $b \neq 0$.

Standard form of linear equations

$Ax + By = C$

$A$, $B$, $C$ are constants with $A$ and $B$ not both zero. The slope is $-\frac{A}{B}$ when $B \neq 0$.

General form

$ax + by + c = 0$

The form used in Indian textbooks. Its slope is $-\frac{a}{b}$ and its $y$-intercept is $-\frac{c}{b}$, both needing $b \neq 0$.

X and y intercepts

$x = \frac{C}{A}$ and $y = \frac{C}{B}$ for $Ax + By = C$

Set $y = 0$ to find where the line meets the $x$-axis and $x = 0$ for the $y$-axis. Needs $A \neq 0$ and $B \neq 0$.

Slope of parallel lines

$m_{1} = m_{2}$

Two non-vertical lines are parallel exactly when their slopes are equal. All vertical lines are parallel to each other.

Slope of perpendicular lines

$m_{1} \cdot m_{2} = -1$

Two lines, neither vertical, are perpendicular when the product of slopes is $-1$, so $m_{2} = -\frac{1}{m_{1}}$.

Horizontal and vertical lines

$y = k$ and $x = h$

$y = k$ is parallel to the $x$-axis with slope 0. $x = h$ is parallel to the $y$-axis with undefined slope.

Unique solution test for two lines

$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

For $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$, the lines meet at exactly one point.

Parallel or coincident lines test

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$

If this common ratio differs from $\frac{c_{1}}{c_{2}}$ the lines are parallel with no solution. If it equals $\frac{c_{1}}{c_{2}}$ there are infinitely many solutions.

Addition property

If $a < b$, then $a + c < b + c$

Adding or subtracting the same real number $c$ on both sides keeps the direction of the inequality.

Multiplying by a positive number

If $a < b$ and $c > 0$, then $ac < bc$

Multiplying or dividing both sides by a positive number keeps the direction of the inequality.

Multiplying by a negative number

If $a < b$ and $c < 0$, then $ac > bc$

Multiplying or dividing both sides by a negative number flips the inequality sign.

Reciprocal property

If $0 < a < b$, then $\frac{1}{a} > \frac{1}{b}$

Taking reciprocals of two positive numbers reverses the order. It also holds when both are negative, but not across zero.

Transitive property

If $a < b$ and $b < c$, then $a < c$

Inequalities in the same direction chain together. Used to join steps and to write compound inequalities.

Absolute value

$\lvert x \rvert = x$ if $x \geq 0$, and $\lvert x \rvert = -x$ if $x < 0$

The absolute value is the distance of $x$ from zero on the number line, so it is never negative.

Product and quotient of absolute values

$\lvert ab \rvert = \lvert a \rvert \lvert b \rvert$ and $\left\lvert \frac{a}{b} \right\rvert = \frac{\lvert a \rvert}{\lvert b \rvert}$

Absolute value splits over multiplication and division. The quotient form needs $b \neq 0$.

Absolute value equation

$\lvert x \rvert = a$ gives $x = \pm a$

For $a > 0$ there are two solutions, for $a = 0$ only $x = 0$, and for $a < 0$ there is no solution.

Less-than absolute inequality

$\lvert x \rvert < a$ is equivalent to $-a < x < a$

For $a > 0$, the solutions lie within $a$ units of zero, giving one bounded interval. No solution if $a \leq 0$.

Greater-than absolute inequality

$\lvert x \rvert > a$ is equivalent to $x < -a$ or $x > a$

For $a \geq 0$, the solutions lie more than $a$ units from zero, giving two separate rays.

Shifted absolute inequality

$\lvert x - h \rvert < a$ is equivalent to $h - a < x < h + a$

For $a > 0$, $x$ is within distance $a$ of the centre $h$. Use it for tolerance and error-bound problems.

Triangle inequality

$\lvert a + b \rvert \leq \lvert a \rvert + \lvert b \rvert$

The size of a sum never exceeds the sum of the sizes. Equality holds when $a$ and $b$ have the same sign or one is zero.

Reverse triangle inequality

$\lvert \lvert a \rvert - \lvert b \rvert \rvert \leq \lvert a - b \rvert$

The gap between two sizes is at most the distance between the numbers. Used to find lower bounds.

Standard form of a quadratic equation

$ax^{2} + bx + c = 0$

A second-degree equation in $x$ with constants $a$, $b$, $c$. If $a = 0$ it becomes linear, so $a \neq 0$ is required.

Quadratic formula

$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

Gives both roots of any quadratic directly. Use it when the quadratic does not factor easily.

Discriminant

$D = b^{2} - 4ac$

The expression under the square root in the quadratic formula. Its sign tells you the nature of the roots before solving. Also written $\Delta$.

Two distinct real roots

$D > 0$

The parabola crosses the $x$-axis twice. If $a$, $b$, $c$ are rational and $D$ is a perfect square, both roots are rational.

Two equal real roots

$D = 0$ gives $x = -\frac{b}{2a}$

The quadratic is a perfect square and the parabola touches the $x$-axis at its vertex.

No real roots

$D < 0$

The parabola does not meet the $x$-axis. The two roots are complex conjugates $\frac{-b \pm i\sqrt{-D}}{2a}$.

Sum of roots

$\alpha + \beta = -\frac{b}{a}$

The two roots add to minus the $x$ coefficient divided by the $x^{2}$ coefficient. Holds for real and complex roots.

Product of roots

$\alpha\beta = \frac{c}{a}$

The two roots multiply to the constant term divided by the $x^{2}$ coefficient.

Difference of roots

$\lvert \alpha - \beta \rvert = \frac{\sqrt{D}}{\lvert a \rvert}$

The distance between two real roots, found without solving. Needs $D \geq 0$.

Quadratic from its roots

$x^{2} - (\alpha + \beta)x + \alpha\beta = 0$

Builds a quadratic with given roots from their sum and product. Multiply through by any nonzero constant to clear fractions.

Factored form

$ax^{2} + bx + c = a(x - \alpha)(x - \beta)$

Any quadratic can be written using its leading coefficient and its two roots. Links roots to $x$-intercepts.

Vertex of a parabola

$\left(-\frac{b}{2a}, -\frac{D}{4a}\right)$

The turning point of $y = ax^{2} + bx + c$. The $y$-value equals $c - \frac{b^{2}}{4a}$, the minimum if $a > 0$ and the maximum if $a < 0$.

Axis of symmetry

$x = -\frac{b}{2a}$

The vertical line through the vertex that splits the parabola into mirror halves. It lies midway between the two roots.

Vertex form

$y = a(x - h)^{2} + k$

$(h, k)$ is the vertex. The parabola opens up when $a > 0$ and down when $a < 0$.

Completing the square

$ax^{2} + bx + c = a\left(x + \frac{b}{2a}\right)^{2} - \frac{b^{2} - 4ac}{4a}$

Rewrites a quadratic in vertex form. It is the step that derives the quadratic formula and finds the vertex.

General polynomial

$p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \dots + a_{1}x + a_{0}$

A polynomial of degree $n$ with constant coefficients $a_{0}$ to $a_{n}$ and leading coefficient $a_{n} \neq 0$.

Division algorithm

$p(x) = g(x) \cdot q(x) + r(x)$

Dividing $p(x)$ by a nonzero $g(x)$ gives quotient $q(x)$ and remainder $r(x)$, where $r(x) = 0$ or its degree is less than that of $g(x)$.

Remainder theorem

Remainder of $p(x) \div (x - a)$ is $p(a)$

Substitute $x = a$ to find the remainder without long division. For a divisor $ax - b$ with $a \neq 0$, the remainder is $p\left(\frac{b}{a}\right)$.

Factor theorem

$(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$

A value that makes the polynomial zero gives a linear factor. Use it to find the first factor of a cubic.

Rational root theorem

$x = \frac{p}{q}$ with $p \mid a_{0}$ and $q \mid a_{n}$

Any rational root in lowest terms has numerator dividing the constant term and denominator dividing the leading coefficient. Needs integer coefficients.

Sum of coefficients

$p(1) = a_{n} + a_{n-1} + \dots + a_{0}$

Substituting $x = 1$ adds all coefficients. If the sum is zero, then $x - 1$ is a factor.

Sum of roots of a cubic

$\alpha + \beta + \gamma = -\frac{b}{a}$

For $ax^{3} + bx^{2} + cx + d = 0$ with $a \neq 0$, the three roots add to minus $b$ over $a$.

Sum of pairwise products for a cubic

$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$

For the same cubic, the products of the roots taken two at a time add to $c$ over $a$.

Product of roots of a cubic

$\alpha\beta\gamma = -\frac{d}{a}$

For the same cubic, the three roots multiply to minus the constant term over the leading coefficient.

Cubic from its roots

$x^{3} - (\alpha + \beta + \gamma)x^{2} + (\alpha\beta + \beta\gamma + \gamma\alpha)x - \alpha\beta\gamma = 0$

Builds a monic cubic with roots $\alpha$, $\beta$, $\gamma$. The signs alternate minus, plus, minus.

Roots of a degree n polynomial

Sum $= -\frac{a_{n-1}}{a_{n}}$ and product $= (-1)^{n}\frac{a_{0}}{a_{n}}$

Extends the quadratic and cubic relations to any degree $n$, counting complex and repeated roots.

Number of roots

Exactly $n$ roots in the complex numbers

A degree $n$ polynomial has $n$ roots counted with multiplicity, so at most $n$ real roots. Non-real roots of real polynomials come in conjugate pairs.

Direct variation

$y = kx$

$y$ varies directly as $x$ with a nonzero constant $k$. The ratio $\frac{y}{x}$ stays fixed and the graph is a line through the origin.

Direct variation between two pairs

$\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}$

Solves a direct variation problem without finding $k$. Both $x$ values must be nonzero.

Direct variation with a power

$y = kx^{n}$

$y$ varies as the $n$th power of $x$, as when the area of a circle varies as the square of its radius.

Inverse variation

$y = \frac{k}{x}$

$y$ varies inversely as $x$ with $k \neq 0$ and $x \neq 0$. The product $xy = k$ stays fixed and one grows as the other shrinks.

Inverse variation between two pairs

$x_{1}y_{1} = x_{2}y_{2}$

Solves an inverse variation problem, such as workers and days, without finding $k$ first.

Inverse square variation

$y = \frac{k}{x^{2}}$

$y$ varies inversely as the square of $x$. Doubling $x$ makes $y$ one quarter. Used for light intensity and gravity.

Joint variation

$z = kxy$

$z$ varies directly as the product of $x$ and $y$, with a nonzero constant $k$.

Combined variation

$z = \frac{kx}{y}$

$z$ varies directly as $x$ and inversely as $y$, with $k \neq 0$ and $y \neq 0$.

Exponential function

$f(x) = a \cdot b^{x}$

$a \neq 0$ is the starting value and $b$ is the growth factor, with $b > 0$ and $b \neq 1$. Growth if $b > 1$, decay if $0 < b < 1$.

Growth at a fixed percent

$A = P(1 + r)^{t}$

$P$ grows by rate $r$, written as a decimal, for each of $t$ periods. Used for population and investment growth.

Decay at a fixed percent

$A = P(1 - r)^{t}$

$P$ shrinks by rate $r$ per period, with $0 < r < 1$. Used for falling values such as a car's price.

Continuous growth or decay

$A = Pe^{kt}$

$e \approx 2.71828$ and $k$ is the continuous rate per unit time. $k > 0$ gives growth and $k < 0$ gives decay.

Half-life form

$N = N_{0}\left(\frac{1}{2}\right)^{\frac{t}{T}}$

$N_{0}$ is the starting amount and $T$ is the half-life. After each interval $T$ the amount halves.

Half-life from a decay constant

$T = \frac{\ln 2}{k}$

For $N = N_{0}e^{-kt}$ with $k > 0$, this is the time to halve. Note $\ln 2 \approx 0.6931$.

Doubling time form

$N = N_{0} \cdot 2^{\frac{t}{T_{d}}}$

$T_{d}$ is the doubling time. The quantity doubles after every interval of length $T_{d}$.

Doubling time for continuous growth

$T_{d} = \frac{\ln 2}{k}$

For $N = N_{0}e^{kt}$ with $k > 0$, this is the time for the quantity to double.

Doubling time for percent growth

$T_{d} = \frac{\ln 2}{\ln(1 + r)}$

The exact number of periods to double at rate $r$ per period, with $r > 0$ written as a decimal.

Rule of 70

$T_{d} \approx \frac{70}{R}$

A quick estimate of doubling time when $R$ is the percent growth rate per period. Accurate for small rates.

Time to reach a value

$t = \frac{1}{k}\ln\left(\frac{A}{P}\right)$

Solves $A = Pe^{kt}$ for time. Needs $A > 0$, $P > 0$ and $k \neq 0$.

Composition of functions

$(f \circ g)(x) = f(g(x))$

Apply $g$ first, then $f$. The input $x$ must be in the domain of $g$ and $g(x)$ in the domain of $f$.

Order matters in composition

$f \circ g \neq g \circ f$ in general

Changing the order of two functions usually changes the result, so always work from the inside out.

Composition is associative

$(f \circ g) \circ h = f \circ (g \circ h)$

Grouping does not matter for three functions applied in a fixed order, which is $h$, then $g$, then $f$.

Inverse functions

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

$f^{-1}$ undoes $f$. It exists only when $f$ is one-to-one, and $f^{-1}(x)$ is not the same as $\frac{1}{f(x)}$.

Finding an inverse

$y = f(x)$, solve for $x$, then swap $x$ and $y$

The standard method to get $f^{-1}(x)$. The domain of $f^{-1}$ is the range of $f$.

Inverse of a linear function

$f(x) = ax + b$ gives $f^{-1}(x) = \frac{x - b}{a}$

Undo the operations in reverse order, subtract $b$ and then divide by $a$. Needs $a \neq 0$.

Inverse of a composition

$(f \circ g)^{-1} = g^{-1} \circ f^{-1}$

To undo $g$ then $f$, undo $f$ first and then $g$. Both functions must be invertible.

Graph of an inverse

Point $(a, b)$ on $f$ gives $(b, a)$ on $f^{-1}$

The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$.

Even function test

$f(-x) = f(x)$

A function is even if this holds for every $x$ in its domain. Its graph is symmetric about the $y$-axis, like $x^{2}$ or $\cos x$.

Odd function test

$f(-x) = -f(x)$

A function is odd if this holds for every $x$ in its domain. Its graph has symmetry about the origin, like $x^{3}$ or $\sin x$.

Products of even and odd functions

even times even is even, odd times odd is even, even times odd is odd

Combine these rules to test products quickly. Sums of two even functions are even and sums of two odd functions are odd.

Even and odd parts

$f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$

Any function on a domain symmetric about zero splits into an even part plus an odd part.

Graph transformations

$y = a \cdot f(x - h) + k$

Shifts the graph of $f$ right by $h$ and up by $k$, and stretches it vertically by a factor of $\lvert a \rvert$. A negative $a$ reflects it in the $x$-axis.

Average rate of change

$\frac{f(b) - f(a)}{b - a}$

The slope of the line joining $(a, f(a))$ and $(b, f(b))$ on the graph. Needs $a \neq b$.

Common difference

$d = a_{n} - a_{n-1}$

The fixed amount added to each term to get the next one. A positive $d$ gives an increasing AP and a negative $d$ a decreasing one.

nth term of an arithmetic sequence

$a_{n} = a + (n-1)d$

Gives the term in position $n$, where $a$ is the first term, $d$ the common difference and $n$ a positive integer.

nth term from the end

$l - (n-1)d$

Counts backwards from the last term $l$ of a finite AP to find the term in position $n$ from the end.

Number of terms

$n = \frac{l - a}{d} + 1$

Finds how many terms a finite AP has when you know the first term $a$, last term $l$ and common difference $d \neq 0$.

Sum of first n terms

$S_{n} = \frac{n}{2}[2a + (n-1)d]$

Adds the first $n$ terms of an AP using the first term $a$ and common difference $d$.

Sum using the last term

$S_{n} = \frac{n}{2}(a + l)$

The same sum written as the number of terms times the average of the first term $a$ and last term $l$.

nth term from the sums

$a_{n} = S_{n} - S_{n-1}$

Recovers any term from the sum formula for $n \geq 2$, with $a_{1} = S_{1}$. Works for any sequence, not only an AP.

Arithmetic mean of two numbers

$A = \frac{a + b}{2}$

The single number that makes $a$, $A$, $b$ an AP. It is the midpoint of $a$ and $b$.

Condition for three terms in AP

$2b = a + c$

Three numbers $a$, $b$, $c$ are in AP exactly when twice the middle one equals the sum of the outer two.

Inserting n arithmetic means

$d = \frac{b - a}{n + 1}$

Common difference needed to place $n$ equally spaced numbers between $a$ and $b$, giving $n + 2$ terms in total.

Convenient choice of terms

$a - d,\ a,\ a + d$ or $a - 3d,\ a - d,\ a + d,\ a + 3d$

Symmetric choices for three or four unknown terms in AP, used when their sum is given so that $d$ cancels.

Common ratio

$r = \frac{a_{n}}{a_{n-1}}$

The fixed factor each term is multiplied by to get the next one. All terms of a GP are nonzero.

Nth term of a GP

$a_{n} = ar^{n-1}$

Gives the term in position $n$, where $a$ is the first term and $r$ the common ratio.

nth term from the end

$\frac{l}{r^{n-1}}$

Term in position $n$ counted backwards from the last term $l$ of a finite GP with common ratio $r$.

Sum of first n terms of a GP

$S_{n} = \frac{a(r^{n} - 1)}{r - 1}$

Adds the first $n$ terms when $r \neq 1$. The equal form $\frac{a(1 - r^{n})}{1 - r}$ is easier to use when $r < 1$.

Sum when the ratio is 1

$S_{n} = na$

If $r = 1$ every term equals $a$, so the sum of $n$ terms is simply $n$ times $a$.

Sum using the last term

$S_{n} = \frac{lr - a}{r - 1}$

Sum of a finite GP from its first term $a$, last term $l$ and ratio $r \neq 1$.

Sum of an infinite GP

$S_{\infty} = \frac{a}{1 - r}$

The value the partial sums approach. It exists only when $\lvert r \rvert < 1$, otherwise the series diverges.

Geometric mean of two numbers

$G = \sqrt{ab}$

For positive $a$ and $b$, the number that makes $a$, $G$, $b$ a GP.

Condition for three terms in GP

$b^{2} = ac$

Three nonzero numbers $a$, $b$, $c$ are in GP exactly when the square of the middle one equals the product of the outer two.

Inserting n geometric means

$r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

Common ratio needed to place $n$ numbers between positive $a$ and $b$ so that all $n + 2$ numbers form a GP.

Product of first n terms

$P_{n} = a^{n}r^{\frac{n(n-1)}{2}}$

Multiplies the first $n$ terms of a GP, since the powers of $r$ add up to $0 + 1 + \cdots + (n-1)$.

Convenient choice of terms

$\frac{a}{r},\ a,\ ar$

Symmetric choice for three unknown terms in GP, used when their product is given so that $r$ cancels.

Harmonic progression

$\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \ldots$ form an AP

A sequence of nonzero numbers is an HP when the reciprocals of its terms are in arithmetic progression.

nth term of an HP

$a_{n} = \frac{1}{\frac{1}{a} + (n-1)d}$

Here $a$ is the first term and $d = \frac{1}{a_{2}} - \frac{1}{a_{1}}$ is the common difference of the reciprocals.

Harmonic mean of two numbers

$H = \frac{2ab}{a + b}$

The number that makes $a$, $H$, $b$ an HP. Used for average speed over two equal distances.

Harmonic mean of n numbers

$H = \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + \cdots + \frac{1}{x_{n}}}$

The count of values divided by the sum of their reciprocals. Suits averages of rates.

Condition for three terms in HP

$b = \frac{2ac}{a + c}$

Three nonzero numbers $a$, $b$, $c$ are in HP when the middle one is the harmonic mean of the outer two.

AM, GM, HM inequality

$A \geq G \geq H$

For any set of positive numbers, the arithmetic mean is at least the geometric mean, which is at least the harmonic mean. Equality holds only when all numbers are equal.

AM-GM for n numbers

$\frac{x_{1} + x_{2} + \cdots + x_{n}}{n} \geq \sqrt[n]{x_{1}x_{2}\cdots x_{n}}$

Standard tool for finding the minimum of a sum or the maximum of a product of positive quantities.

Link between the three means

$G^{2} = AH$

For two positive numbers, the square of the geometric mean equals the product of the arithmetic and harmonic means.

Numbers from their means

$x^{2} - 2Ax + G^{2} = 0$

The two numbers whose AM is $A$ and GM is $G$ are the roots of this quadratic.

Sum of first n natural numbers

$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

Adds $1 + 2 + \cdots + n$ for a positive integer $n$. It is the AP sum with $a = 1$ and $d = 1$.

Sum of squares

$\sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6}$

Adds $1^{2} + 2^{2} + \cdots + n^{2}$ for a positive integer $n$. Needed when summing any quadratic expression in $k$.

Sum of cubes

$\sum_{k=1}^{n} k^{3} = \left[\frac{n(n+1)}{2}\right]^{2}$

Adds $1^{3} + 2^{3} + \cdots + n^{3}$. The result is the square of the sum of the first $n$ natural numbers.

Sum of fourth powers

$\sum_{k=1}^{n} k^{4} = \frac{n(n+1)(2n+1)(3n^{2} + 3n - 1)}{30}$

Adds $1^{4} + 2^{4} + \cdots + n^{4}$. Used less often, mainly in JEE series questions.

Sum of first n odd numbers

$\sum_{k=1}^{n} (2k - 1) = n^{2}$

Adds $1 + 3 + 5 + \cdots + (2n - 1)$, the first $n$ odd numbers. The total is always a perfect square.

Sum of first n even numbers

$\sum_{k=1}^{n} 2k = n(n+1)$

Adds $2 + 4 + 6 + \cdots + 2n$, the first $n$ even numbers. It is twice the sum of the first $n$ natural numbers.

Sum of a constant

$\sum_{k=1}^{n} c = nc$

Adding the same number $c$ a total of $n$ times gives $n$ times $c$.

Linearity of sigma notation

$\sum_{k=1}^{n} (p,u_{k} + q,v_{k}) = p\sum_{k=1}^{n} u_{k} + q\sum_{k=1}^{n} v_{k}$

Constants $p$ and $q$ come out of a sum, and a sum of terms splits into separate sums. Used to sum any polynomial in $k$.

Telescoping sum

$\sum_{k=1}^{n} \frac{1}{k(k+1)} = \frac{n}{n+1}$

Write $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$ so that middle terms cancel in pairs.

Infinite arithmetico-geometric series

$S_{\infty} = \frac{a}{1 - r} + \frac{dr}{(1 - r)^{2}}$

Sum of $a + (a + d)r + (a + 2d)r^{2} + \cdots$, where each AP term is multiplied by a GP term. Needs $\lvert r \rvert < 1$.

Binomial theorem

$(a + b)^{n} = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^{r}$

Expands a power of a two-term sum. The powers of $a$ fall from $n$ to 0 while the powers of $b$ rise from 0 to $n$.

General term

$T_{r+1} = \binom{n}{r} a^{n-r}b^{r}$

Gives the term in position $r + 1$ directly, for $0 \leq r \leq n$, without writing the full expansion.

Number of terms

$n + 1$

The expansion of $(a + b)^{n}$ always has one more term than the power $n$.

Middle term, n even

$T_{\frac{n}{2} + 1}$

When $n$ is even there is a single middle term, in position $\frac{n}{2} + 1$.

Middle terms, n odd

$T_{\frac{n+1}{2}}$ and $T_{\frac{n+3}{2}}$

When $n$ is odd there are two middle terms, in positions $\frac{n+1}{2}$ and $\frac{n+3}{2}$.

rth term from the end

$T_{n - r + 2}$

The term in position $r$ counted from the end is the term in position $n - r + 2$ counted from the start.

Sum of binomial coefficients

$\binom{n}{0} + \binom{n}{1} + \cdots + \binom{n}{n} = 2^{n}$

Found by putting $a = b = 1$. For any polynomial $p(x)$, the sum of its coefficients is $p(1)$.

Alternating sum of coefficients

$\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \cdots + (-1)^{n}\binom{n}{n} = 0$

Found by putting $a = 1$ and $b = -1$, valid for $n \geq 1$.

Even and odd position coefficients

$\binom{n}{0} + \binom{n}{2} + \cdots = \binom{n}{1} + \binom{n}{3} + \cdots = 2^{n-1}$

The coefficients in odd positions and in even positions each add up to half of $2^{n}$, for $n \geq 1$.

Greatest binomial coefficient

$\binom{n}{\frac{n}{2}}$ for even $n$, $\binom{n}{\frac{n-1}{2}} = \binom{n}{\frac{n+1}{2}}$ for odd $n$

The largest coefficient sits in the middle of the expansion.

Term independent of x

Set the power of $x$ in $T_{r+1}$ equal to 0

Solve for $r$, then substitute it into the general term to get the constant term.

Binomial expansion of $(1 + x)^{n}$

$1 + nx + \frac{n(n-1)}{2!}x^{2} + \frac{n(n-1)(n-2)}{3!}x^{3} + \cdots$

Ends after $n + 1$ terms for a positive integer $n$. For any real $n$ it is an infinite series valid when $\lvert x \rvert < 1$.

Factorial

$n! = n(n-1)(n-2)\cdots 2 \cdot 1$

The product of all positive integers up to $n$. By definition $0! = 1$.

Recursive factorial

$n! = n \times (n-1)!$

Each factorial is $n$ times the one before it, for $n \geq 1$. Useful for cancelling factorials in fractions.

nPr formula

$^{n}P_{r} = \frac{n!}{(n-r)!}$

Number of ordered arrangements of $r$ objects chosen from $n$ distinct objects, for $0 \leq r \leq n$.

Arranging all n objects

$^{n}P_{n} = n!$

Number of ways to line up $n$ distinct objects in a row.

Permutations with repetition allowed

$n^{r}$

Number of ordered arrangements of length $r$ when each place can be filled by any of $n$ objects again and again.

Permutations with identical objects

$\frac{n!}{p!,q!,s!}$

Arrangements of $n$ objects in which $p$ are alike of one kind, $q$ of another and $s$ of a third.

Circular permutations

$(n-1)!$

Ways to seat $n$ distinct people around a round table, where rotations of the same order count once.

Circular arrangements that can be flipped

$\frac{(n-1)!}{2}$

For beads on a necklace or flowers in a garland, clockwise and anticlockwise orders count once. Valid for $n \geq 3$.

Particular object always included

$r \cdot {}^{n-1}P_{r-1}$

Arrangements of $r$ out of $n$ distinct objects in which one chosen object must appear.

Particular object never included

$^{n-1}P_{r}$

Arrangements of $r$ out of $n$ distinct objects in which one chosen object is left out.

Derangements

$D_{n} = n!\left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^{n}}{n!}\right)$

Number of ways to arrange $n$ objects so that none is in its original place. For example $D_{3} = 2$ and $D_{4} = 9$.

Power of a prime in n factorial

$\left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^{2}} \right\rfloor + \left\lfloor \frac{n}{p^{3}} \right\rfloor + \cdots$

Exponent of the prime $p$ in $n!$, where the brackets mean the greatest integer. Used to count trailing zeros with $p = 5$.

nCr formula

$^{n}C_{r} = \binom{n}{r} = \frac{n!}{r!,(n-r)!}$

Number of ways to choose $r$ objects from $n$ distinct objects when order does not matter, for $0 \leq r \leq n$.

Link between nPr and nCr

$^{n}P_{r} = r! \times {}^{n}C_{r}$

Each selection of $r$ objects can be ordered in $r!$ ways, which turns a combination count into a permutation count.

Complementary selections

$^{n}C_{r} = {}^{n}C_{n-r}$

Choosing $r$ objects to take is the same as choosing $n - r$ objects to leave behind.

Special values

$^{n}C_{0} = {}^{n}C_{n} = 1$ and $^{n}C_{1} = n$

There is one way to choose nothing or everything, and $n$ ways to choose a single object.

Equal combinations

If $^{n}C_{x} = {}^{n}C_{y}$ then $x = y$ or $x + y = n$

Used to solve equations in which two combinations with the same $n$ are equal.

Pascal's rule

$^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r}$

Two neighbouring entries in one row of Pascal's triangle add to the entry below them, for $1 \leq r \leq n$.

Ratio of consecutive terms

$\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}} = \frac{n - r + 1}{r}$

Lets you get each coefficient from the previous one and find where the coefficients stop increasing.

Absorption identity

$r \cdot {}^{n}C_{r} = n \cdot {}^{n-1}C_{r-1}$

Moves a factor of $r$ inside a combination. Often used when summing $r\binom{n}{r}$ over $r$.

Selecting at least one

$^{n}C_{1} + {}^{n}C_{2} + \cdots + {}^{n}C_{n} = 2^{n} - 1$

Number of nonempty selections from $n$ distinct objects, since each object is either taken or not.

Identical objects into distinct groups

$^{n+r-1}C_{r-1}$

Ways to share $n$ identical objects among $r$ people when a person may get none, also called stars and bars.

Each group gets at least one

$^{n-1}C_{r-1}$

Ways to share $n$ identical objects among $r$ people when each person gets at least one, for $n \geq r$.

Lines and triangles from points

$^{n}C_{2}$ lines and $^{n}C_{3}$ triangles

Counts from $n$ points in a plane with no three collinear, because two points fix a line and three fix a triangle.

Imaginary unit

$i = \sqrt{-1}$, so $i^{2} = -1$

The number whose square is $-1$. It is also called iota in Indian textbooks.

Powers of i

$i^{4k} = 1$, $i^{4k+1} = i$, $i^{4k+2} = -1$, $i^{4k+3} = -i$

Powers of $i$ repeat every 4 steps, so divide the power by 4 and use the remainder. Here $k$ is an integer.

Standard form

$z = a + bi$

A complex number with real part $\text{Re}(z) = a$ and imaginary part $\text{Im}(z) = b$, where $a$ and $b$ are real.

Equality of complex numbers

$a + bi = c + di$ if and only if $a = c$ and $b = d$

Two complex numbers are equal only when both their real parts and their imaginary parts match.

Addition and subtraction

$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$

Add or subtract the real parts and the imaginary parts separately.

Multiplication

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Expand like two binomials and replace $i^{2}$ with $-1$.

Conjugate

$\bar{z} = a - bi$

Flips the sign of the imaginary part. Also $z + \bar{z} = 2a$ and $z\bar{z} = a^{2} + b^{2}$.

Modulus of a complex number

$\lvert z \rvert = \sqrt{a^{2} + b^{2}}$

The distance of the point $(a, b)$ from the origin in the Argand plane. It is never negative.

Division

$\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^{2} + d^{2}}$

Found by multiplying top and bottom by the conjugate $c - di$ of the denominator. Needs $c + di \neq 0$.

Reciprocal

$\frac{1}{z} = \frac{\bar{z}}{\lvert z \rvert^{2}}$

The multiplicative inverse of any nonzero complex number $z$.

Modulus rules

$\lvert z_{1}z_{2} \rvert = \lvert z_{1} \rvert \lvert z_{2} \rvert$ and $\left\lvert \frac{z_{1}}{z_{2}} \right\rvert = \frac{\lvert z_{1} \rvert}{\lvert z_{2} \rvert}$

The modulus of a product or quotient is the product or quotient of the moduli, with $z_{2} \neq 0$.

Conjugate rules

$\overline{z_{1} + z_{2}} = \overline{z_{1}} + \overline{z_{2}}$ and $\overline{z_{1}z_{2}} = \overline{z_{1}},\overline{z_{2}}$

Taking the conjugate can be done before or after adding or multiplying.

Triangle inequality

$\lvert z_{1} + z_{2} \rvert \leq \lvert z_{1} \rvert + \lvert z_{2} \rvert$

The length of one side of a triangle is at most the sum of the other two. Equality holds when $z_{1}$ and $z_{2}$ point the same way.

Distance between two points

$\lvert z_{1} - z_{2} \rvert$

The distance between the points $z_{1}$ and $z_{2}$ in the Argand plane.

Square root of a complex number

$\sqrt{a + bi} = \pm\left(\sqrt{\frac{\lvert z \rvert + a}{2}} + i,\frac{b}{\lvert b \rvert}\sqrt{\frac{\lvert z \rvert - a}{2}}\right)$

Both square roots of $z = a + bi$ with $b \neq 0$. The factor $\frac{b}{\lvert b \rvert}$ is just the sign of $b$.

Argument

$\tan\theta = \frac{b}{a}$, with $-\pi < \theta \leq \pi$

The angle $z = a + bi$ makes with the positive real axis. Pick the quadrant from the signs of $a$ and $b$ to get the principal argument.

Polar form

$z = r(\cos\theta + i\sin\theta)$

Writes $z$ using its modulus $r = \lvert z \rvert$ and argument $\theta$, so $a = r\cos\theta$ and $b = r\sin\theta$.

Euler's formula

$e^{i\theta} = \cos\theta + i\sin\theta$

Links the exponential function to sine and cosine for any real $\theta$.

Euler (exponential) form

$z = re^{i\theta}$

A short way to write the polar form. Makes multiplication, division and powers easy.

Euler's identity

$e^{i\pi} + 1 = 0$

Euler's formula with $\theta = \pi$. It connects $e$, $i$, $\pi$, 1 and 0.

Product in polar form

$z_{1}z_{2} = r_{1}r_{2}[\cos(\theta_{1} + \theta_{2}) + i\sin(\theta_{1} + \theta_{2})]$

Multiply the moduli and add the arguments. So $\arg(z_{1}z_{2}) = \arg z_{1} + \arg z_{2}$, up to a multiple of $2\pi$.

Quotient in polar form

$\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}[\cos(\theta_{1} - \theta_{2}) + i\sin(\theta_{1} - \theta_{2})]$

Divide the moduli and subtract the arguments, with $z_{2} \neq 0$.

De Moivre's theorem

$(\cos\theta + i\sin\theta)^{n} = \cos n\theta + i\sin n\theta$

Raises a complex number in polar form to an integer power $n$ by multiplying the angle by $n$.

nth roots of a complex number

$r^{\frac{1}{n}}\left[\cos\frac{\theta + 2k\pi}{n} + i\sin\frac{\theta + 2k\pi}{n}\right]$

Gives all $n$ distinct roots of $z = re^{i\theta}$ as $k$ runs through $0, 1, \ldots, n-1$.

Cube roots of unity

$1$, $\omega = \frac{-1 + i\sqrt{3}}{2}$, $\omega^{2} = \frac{-1 - i\sqrt{3}}{2}$

The three solutions of $z^{3} = 1$. The two complex ones are roots of $z^{2} + z + 1 = 0$ and are conjugates of each other.

Properties of omega

$1 + \omega + \omega^{2} = 0$ and $\omega^{3} = 1$

The key facts used to simplify any expression in $\omega$. Also $\lvert \omega \rvert = 1$ and $\bar{\omega} = \omega^{2}$.

nth roots of unity

$e^{\frac{2k\pi i}{n}}$ for $k = 0, 1, \ldots, n-1$

The $n$ solutions of $z^{n} = 1$, equally spaced on the unit circle. Their sum is 0 for $n \geq 2$ and their product is $(-1)^{n-1}$.

Order of a matrix

$m \times n$ matrix has $mn$ entries

A matrix with $m$ rows and $n$ columns. The entry in row $i$ and column $j$ is written $a_{ij}$.

Matrix addition

$(A + B){ij} = a{ij} + b_{ij}$

Add entries in the same position. Defined only when $A$ and $B$ have the same order.

Scalar multiplication

$(kA){ij} = k,a{ij}$

Multiply every entry of $A$ by the number $k$.

Multiplication of matrices

$(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$

Entry $(i, j)$ of the product is row $i$ of $A$ times column $j$ of $B$, added term by term.

Order of a product

$A_{m \times n}B_{n \times p} = C_{m \times p}$

$AB$ exists only when the number of columns of $A$ equals the number of rows of $B$.

Laws of matrix multiplication

$(AB)C = A(BC)$, $A(B + C) = AB + AC$, but $AB \neq BA$ in general

Matrix multiplication is associative and distributive but usually not commutative, so the order of factors matters.

Identity matrix

$AI = IA = A$

$I$ has 1 on the main diagonal and 0 elsewhere. It acts like the number 1 for square matrices of the same order.

Transpose of a matrix

$(A^{T}){ij} = a{ji}$

Rows become columns. An $m \times n$ matrix has an $n \times m$ transpose, also written $A'$.

Transpose rules

$(A^{T})^{T} = A$, $(A + B)^{T} = A^{T} + B^{T}$, $(kA)^{T} = kA^{T}$

Transposing twice returns the original, and transpose passes through sums and scalar multiples.

Reversal law for transpose

$(AB)^{T} = B^{T}A^{T}$

The transpose of a product is the product of the transposes in reverse order.

Symmetric matrix

$A^{T} = A$

A square matrix equal to its own transpose, so $a_{ij} = a_{ji}$ for all $i$ and $j$.

Skew symmetric matrix

$A^{T} = -A$

A square matrix with $a_{ij} = -a_{ji}$. All its diagonal entries are 0.

Symmetric plus skew symmetric split

$A = \frac{1}{2}(A + A^{T}) + \frac{1}{2}(A - A^{T})$

Every square matrix is the sum of a symmetric part and a skew symmetric part, and this split is unique.

Trace

$\text{tr}(A) = \sum_{i=1}^{n} a_{ii}$

Sum of the main diagonal entries of a square matrix. Also $\text{tr}(AB) = \text{tr}(BA)$.

Orthogonal matrix

$AA^{T} = A^{T}A = I$

A square matrix whose transpose is its inverse. Its determinant is $1$ or $-1$.

2 by 2 determinant

$\det A = ad - bc$

For the matrix with rows $(a, b)$ and $(c, d)$: the product of the main diagonal minus the product of the other diagonal.

3 by 3 determinant

$a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$

Expansion along the first row. The signs alternate plus, minus, plus.

Minor and cofactor

$C_{ij} = (-1)^{i+j}M_{ij}$

$M_{ij}$ is the determinant left after deleting row $i$ and column $j$. The cofactor attaches the checkerboard sign.

Expansion along any row

$\det A = \sum_{j=1}^{n} a_{ij}C_{ij}$

Any row or column gives the same value, so expand along the one with the most zeros.

Determinant of a transpose

$\det(A^{T}) = \det A$

Swapping rows and columns does not change the determinant, so every row rule also holds for columns.

Product rule

$\det(AB) = \det A \cdot \det B$

The determinant of a product of two square matrices of the same order is the product of their determinants.

Scalar multiple

$\det(kA) = k^{n}\det A$

For an $n \times n$ matrix, multiplying the whole matrix by $k$ multiplies the determinant by $k^{n}$.

Row operation rules

Swap two rows gives $-\det A$, scale one row by $k$ gives $k\det A$

Adding a multiple of one row to another leaves the determinant unchanged. These rules speed up evaluation.

Zero determinant cases

$\det A = 0$

Always happens when two rows are equal or proportional, or a whole row is zero, though these are not the only cases. Such a matrix is called singular.

Triangular matrix

$\det A = a_{11}a_{22}\cdots a_{nn}$

For an upper or lower triangular or diagonal matrix, the determinant is the product of the diagonal entries.

Determinant of an inverse

$\det(A^{-1}) = \frac{1}{\det A}$

Holds for any invertible matrix, since $AA^{-1} = I$ and $\det I = 1$.

Area of triangle in determinant form

$\text{Area} = \frac{1}{2}\lvert x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}) \rvert$

Half the absolute value of the determinant with rows $(x_{1}, y_{1}, 1)$, $(x_{2}, y_{2}, 1)$, $(x_{3}, y_{3}, 1)$. It equals 0 when the points are collinear.

Adjoint of a matrix

$\text{adj},A = C^{T}$

The transpose of the cofactor matrix $C$. Also called the adjugate.

Adjoint identity

$A(\text{adj},A) = (\text{adj},A)A = (\det A)I$

Holds for every square matrix and leads straight to the inverse formula.

Inverse of a matrix

$A^{-1} = \frac{1}{\det A},\text{adj},A$

The matrix with $AA^{-1} = A^{-1}A = I$. It exists only when $\det A \neq 0$.

Inverse of a 2x2 matrix

$A^{-1} = \frac{1}{ad - bc}$ times the matrix with rows $(d, -b)$ and $(-c, a)$

For $A$ with rows $(a, b)$ and $(c, d)$: swap the diagonal entries, negate the other two and divide by $ad - bc \neq 0$.

Inverse of a product

$(AB)^{-1} = B^{-1}A^{-1}$

The inverse of a product is the product of the inverses in reverse order, when both are invertible.

Inverse of a transpose

$(A^{T})^{-1} = (A^{-1})^{T}$

Transposing and inverting can be done in either order.

Determinant of the adjoint

$\det(\text{adj},A) = (\det A)^{n-1}$

For an $n \times n$ matrix. For a 3 by 3 matrix this gives $(\det A)^{2}$.

Adjoint of the adjoint

$\text{adj}(\text{adj},A) = (\det A)^{n-2}A$

For an $n \times n$ matrix with $n \geq 2$. A frequent JEE result.

Matrix method for linear equations

$AX = B \Rightarrow X = A^{-1}B$

Solves a square system where $A$ holds the coefficients, $X$ the unknowns and $B$ the constants, when $\det A \neq 0$.

Cramer's rule

$x = \frac{D_{x}}{D}$, $y = \frac{D_{y}}{D}$, $z = \frac{D_{z}}{D}$

$D$ is the coefficient determinant and $D_{x}$ replaces the $x$ column with the constants, and so on. Needs $D \neq 0$.

Consistency test

$D \neq 0$ gives a unique solution

If $D = 0$ and some $D_{x}$, $D_{y}$ or $D_{z}$ is nonzero, there is no solution. If all are zero, there are infinitely many solutions or none.

A union B formula

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Counts elements in at least one of two finite sets. The overlap is subtracted once because it was counted twice.

Disjoint sets

$n(A \cup B) = n(A) + n(B)$

When $A \cap B = \emptyset$ there is no overlap, so the counts simply add.

Union of three sets

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

Inclusion-exclusion for three sets. Add singles, subtract pairs, add back the triple overlap.

Only A

$n(A - B) = n(A) - n(A \cap B)$

Elements in $A$ but not in $B$, the difference of sets $A - B$.

Symmetric difference

$n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$

Elements in exactly one of $A$ and $B$, written $A \Delta B = (A - B) \cup (B - A)$.

Complement of a set

$n(A') = n(U) - n(A)$

Elements of the universal set $U$ that are not in $A$. Also written $A^{c}$.

Neither A nor B

$n(A' \cap B') = n(U) - n(A \cup B)$

Elements of the universal set that lie outside both sets.

De Morgan's laws for sets

$(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$

The complement of a union is the intersection of complements, and the complement of an intersection is the union of complements.

Distributive laws

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

Union spreads over intersection and intersection spreads over union, like multiplication over addition.

Exactly one of three sets

$S_{1} - 2S_{2} + 3S_{3}$

$S_{1}$ is the sum of the three single counts, $S_{2}$ the sum of the three pair overlaps and $S_{3} = n(A \cap B \cap C)$.

Exactly two of three sets

$S_{2} - 3S_{3}$

Elements in exactly two of $A$, $B$, $C$, using the same $S_{2}$ and $S_{3}$ as the row above.

Number of subsets

$2^{m}$

A set with $m$ elements has $2^{m}$ subsets, because each element is either in or out.

Proper subsets

$2^{m} - 1$

All subsets except the set itself. Nonempty proper subsets number $2^{m} - 2$ for $m \geq 1$.

Subsets of a fixed size

$^{m}C_{r}$

Number of subsets with exactly $r$ elements, for $0 \leq r \leq m$.

Power set

$n(P(A)) = 2^{m}$

$P(A)$ is the set of all subsets of $A$, including $\emptyset$ and $A$ itself.

Cartesian product

$n(A \times B) = mn$

Number of ordered pairs $(a, b)$ with $a$ in $A$ and $b$ in $B$.

Number of relations from A to B

$2^{mn}$

A relation is any subset of $A \times B$, so the count is the number of subsets of a set of size $mn$.

Relations on a set

$2^{m^{2}}$

Number of relations from a set with $m$ elements to itself.

Reflexive relations, symmetric relations

$2^{m^{2} - m}$ reflexive, $2^{\frac{m(m+1)}{2}}$ symmetric

Counts of relations on a set of $m$ elements that are reflexive, or separately that are symmetric.

Number of functions from A to B

$n^{m}$

Each of the $m$ elements of $A$ can be sent to any of the $n$ elements of $B$.

One-to-one functions

$^{n}P_{m} = \frac{n!}{(n-m)!}$

Number of injective functions from $A$ to $B$ when $m \leq n$. There are none if $m > n$.

Bijections

$n!$

Number of one-to-one and onto functions between two sets that both have $n$ elements.

Onto functions

$\sum_{k=0}^{n} (-1)^{k}\binom{n}{k}(n - k)^{m}$

Number of surjective functions from $A$ onto $B$, for $m \geq n$, found by inclusion-exclusion.

Binary operations on a set

$m^{m^{2}}$

Number of binary operations on a set with $m$ elements, since each of the $m^{2}$ ordered pairs gets one of $m$ outputs.

Negation

$\sim p$

True when $p$ is false and false when $p$ is true. Also written $\neg p$.

Conjunction (AND)

$p \wedge q$

True only when both $p$ and $q$ are true.

Disjunction (OR)

$p \vee q$

False only when both $p$ and $q$ are false. This is the inclusive or.

Implication (if p then q)

$p \Rightarrow q \equiv \sim p \vee q$

False only when $p$ is true and $q$ is false. A false $p$ makes the implication true.

Biconditional (if and only if)

$p \Leftrightarrow q \equiv (p \Rightarrow q) \wedge (q \Rightarrow p)$

True when $p$ and $q$ have the same truth value.

Converse

$q \Rightarrow p$

Swaps the two parts of $p \Rightarrow q$. It is not equivalent to the original.

Inverse

$\sim p \Rightarrow \sim q$

Negates both parts of $p \Rightarrow q$. It is equivalent to the converse, not to the original.

Contrapositive

$\sim q \Rightarrow \sim p$

Swaps and negates both parts. It always has the same truth value as $p \Rightarrow q$, so it is used in indirect proofs.

De Morgan's laws for statements

$\sim(p \wedge q) \equiv \sim p \vee \sim q$ and $\sim(p \vee q) \equiv \sim p \wedge \sim q$

To negate an AND statement, negate each part and use OR, and the reverse for an OR statement.

Negation of an implication

$\sim(p \Rightarrow q) \equiv p \wedge \sim q$

The only way an implication fails is when $p$ holds and $q$ does not.

Double negation

$\sim(\sim p) \equiv p$

Negating a statement twice gives back the original statement.

Tautology and contradiction

$p \vee \sim p$ is always T, $p \wedge \sim p$ is always F

A tautology is true in every row of its truth table and a contradiction is false in every row.

Negation of quantifiers

$\sim(\forall x,, P(x)) \equiv \exists x,, \sim P(x)$

The negation of "for all" is "there exists one that fails", and the negation of "there exists" is "for all, it fails".

Rows in a truth table

$2^{n}$

A compound statement built from $n$ simple statements needs $2^{n}$ rows to list every combination.

Complementary angles

$A + B = 90^{\circ}$

Two angles are complementary when they add to $90^{\circ}$, so the complement of an angle $x$ is $90^{\circ} - x$.

Supplementary angles

$A + B = 180^{\circ}$

Two angles are supplementary when they add to $180^{\circ}$, so the supplement of an angle $x$ is $180^{\circ} - x$.

Linear pair of angles

$\angle 1 + \angle 2 = 180^{\circ}$

Two adjacent angles that sit on a straight line always add to $180^{\circ}$.

Vertical angles

$\angle 1 = \angle 3$

When two straight lines cross, each pair of vertically opposite angles is equal.

Angles at a point (complete angle)

$\angle 1 + \angle 2 + \dots + \angle k = 360^{\circ}$

All the angles around one point add to a full turn of $360^{\circ}$, whatever their number $k$.

Corresponding angles

$\angle a = \angle b$

When a transversal cuts two parallel lines, the angles in matching positions at each crossing are equal.

Alternate interior angles

$\angle a = \angle b$

With parallel lines and a transversal, angles on opposite sides of the transversal and between the lines are equal.

Co-interior angles (same-side interior angles)

$\angle a + \angle b = 180^{\circ}$

With parallel lines and a transversal, the two interior angles on the same side of the transversal are supplementary.

Sum of interior angles of a polygon

$S = (n - 2) \times 180^{\circ}$

For a polygon with $n$ sides, $n \geq 3$, because it splits into $n - 2$ triangles from one vertex.

Each interior angle of a regular polygon

$\frac{(n - 2) \times 180^{\circ}}{n}$

A regular polygon has $n$ equal angles, so divide the interior angle sum by $n$.

Sum of exterior angles of a polygon

$360^{\circ}$

Taking one exterior angle at each vertex of a convex polygon, the total is always $360^{\circ}$, for any $n$.

Each exterior angle of a regular polygon

$\frac{360^{\circ}}{n}$

$n$ is the number of sides. Also used in reverse to find it: $n = \frac{360^{\circ}}{\text{exterior angle}}$.

Interior and exterior angle at a vertex

$\text{interior} + \text{exterior} = 180^{\circ}$

At each vertex the interior angle and its exterior angle form a linear pair on the extended side.

Number of diagonals of a polygon

$\frac{n(n - 3)}{2}$

Each of the $n$ vertices joins $n - 3$ non-adjacent vertices, and every diagonal gets counted twice, hence the division by 2.

Area of a triangle

$A = \frac{1}{2} \times b \times h$

$b$ is any side taken as the base and $h$ is the perpendicular height to that side from the opposite vertex.

Heron's formula

$A = \sqrt{s(s - a)(s - b)(s - c)}$

Gives the area when all three sides $a$, $b$, $c$ are known, with $s$ the semi-perimeter. No height is needed.

Semi-perimeter

$s = \frac{a + b + c}{2}$

Half the perimeter of the triangle. It appears in Heron's formula and in the inradius formula.

Area from two sides and the included angle

$A = \frac{1}{2}ab\sin C$

$a$ and $b$ are two sides and $C$ is the angle between them. Works for every triangle, acute or obtuse.

Perimeter of a triangle

$P = a + b + c$

The perimeter is the total length of the boundary, found by adding the three side lengths.

Angle sum of a triangle (triangle sum theorem)

$A + B + C = 180^{\circ}$

The three interior angles of any triangle add to $180^{\circ}$, which is $\pi$ radians.

Exterior angle theorem

$\angle \text{exterior} = \angle A + \angle B$

An exterior angle of a triangle equals the sum of the two interior angles that are not next to it.

Triangle inequality theorem

$\lvert a - b \rvert < c < a + b$

Any two sides together must be longer than the third. Use it to test whether three lengths can form a triangle.

Area of an equilateral triangle

$A = \frac{\sqrt{3}}{4}a^{2}$

$a$ is the side length. All three sides are equal and each angle is $60^{\circ}$.

Height of an equilateral triangle

$h = \frac{\sqrt{3}}{2}a$

For side $a$, the altitude bisects the base, so Pythagoras on half the triangle gives $h^{2} = a^{2} - \frac{a^{2}}{4}$.

Perimeter of an equilateral triangle

$P = 3a$

Three equal sides of length $a$, so the perimeter is three times one side.

Area of an isosceles triangle (isosceles triangle formula)

$A = \frac{b}{4}\sqrt{4a^{2} - b^{2}}$

$a$ is each of the two equal sides and $b$ is the base, which must satisfy $b < 2a$.

Pythagorean theorem (Pythagoras theorem)

$a^{2} + b^{2} = c^{2}$

In a right triangle, $a$ and $b$ are the legs and $c$ is the hypotenuse, the side opposite the right angle.

Hypotenuse and missing leg

$c = \sqrt{a^{2} + b^{2}}$, $a = \sqrt{c^{2} - b^{2}}$

Rearranged forms of Pythagoras for finding an unknown side once the other two are known.

Converse of Pythagoras theorem

$a^{2} + b^{2} = c^{2} \Rightarrow \angle C = 90^{\circ}$

If the squares of two sides add to the square of the third, the angle opposite that third side is a right angle.

Acute or obtuse test

$c^{2} < a^{2} + b^{2}$ (acute), $c^{2} > a^{2} + b^{2}$ (obtuse)

Here $c$ is the longest side. Compare $c^{2}$ with $a^{2} + b^{2}$ to classify the largest angle.

Area of a right triangle (right triangle formulas)

$A = \frac{1}{2}ab$

The two legs $a$ and $b$ are perpendicular, so one acts as the base and the other as the height.

45-45-90 triangle (isosceles right triangle)

$x : x : x\sqrt{2}$

Both legs equal $x$ and the hypotenuse is $x\sqrt{2}$. This is half of a square cut along its diagonal.

30-60-90 triangle

$x : x\sqrt{3} : 2x$

The side opposite $30^{\circ}$ is $x$, opposite $60^{\circ}$ is $x\sqrt{3}$, and the hypotenuse is $2x$.

Altitude to the hypotenuse

$h = \frac{ab}{c}$

The height from the right angle to the hypotenuse, found by writing the area two ways: $\frac{1}{2}ab = \frac{1}{2}ch$.

Hypotenuse segment relations

$h^{2} = pq$, $a^{2} = pc$, $b^{2} = qc$

The altitude splits the hypotenuse into $p$ (next to leg $a$) and $q$ (next to leg $b$), with $p + q = c$.

Circumradius of a right triangle

$R = \frac{c}{2}$

The hypotenuse is a diameter of the circumcircle, so the circumcentre is the midpoint of the hypotenuse.

Inradius of a right triangle

$r = \frac{a + b - c}{2}$

Radius of the circle touching all three sides, with legs $a$, $b$ and hypotenuse $c$.

Pythagorean triples

$(m^{2} - n^{2}, , 2mn, , m^{2} + n^{2})$

For whole numbers $m > n > 0$ this gives integer sides of a right triangle, such as $(3, 4, 5)$ from $m = 2$, $n = 1$.

Centroid of a triangle

$AG : GM = 2 : 1$

The three medians meet at the centroid $G$, which divides each median in the ratio $2 : 1$ measured from the vertex.

Length of a median

$m_{a} = \frac{1}{2}\sqrt{2b^{2} + 2c^{2} - a^{2}}$

The median from vertex $A$ to the midpoint of side $a$. This comes from Apollonius' theorem.

Altitude of a triangle

$h_{a} = \frac{2A}{a}$

The perpendicular height onto side $a$, found by rearranging $A = \frac{1}{2}ah_{a}$.

Circumradius (circumcentre)

$R = \frac{abc}{4A}$

Radius of the circle through all three vertices. Its centre is where the perpendicular bisectors of the sides meet.

Circumradius from the sine rule

$R = \frac{a}{2\sin A}$

Here $A$ is the angle opposite side $a$. Any side and its opposite angle give the same $R$.

Inradius (incentre)

$r = \frac{A}{s}$

Radius of the circle touching all three sides. Its centre is where the angle bisectors meet, and $A = rs$.

Circumradius and inradius of an equilateral triangle

$R = \frac{a}{\sqrt{3}}$, $r = \frac{a}{2\sqrt{3}}$

For side $a$ the centroid, circumcentre and incentre coincide, so $R = 2r$.

Euler's distance between centres

$OI^{2} = R^{2} - 2Rr$

$O$ is the circumcentre and $I$ the incentre. It shows $R \geq 2r$, with equality only for an equilateral triangle.

Ratio of lengths

$\frac{a'}{a} = k$

Every pair of matching sides, heights, medians or radii in similar figures has the same ratio $k$.

Ratio of perimeters

$\frac{P'}{P} = k$

Perimeters are lengths, so they scale by the same factor $k$ as the sides.

Ratio of areas (area of similar triangles)

$\frac{A'}{A} = k^{2}$

Areas of similar figures scale by the square of the side ratio. Surface areas of similar solids also scale by $k^{2}$.

Ratio of volumes

$\frac{V'}{V} = k^{3}$

Volumes of similar solids scale by the cube of the length ratio, so doubling every length multiplies volume by 8.

Basic proportionality theorem (Thales)

$\frac{AD}{DB} = \frac{AE}{EC}$

If a line parallel to $BC$ cuts $AB$ at $D$ and $AC$ at $E$, it divides those two sides in the same ratio.

Midsegment of a triangle

$DE = \frac{1}{2}BC$

The segment joining the midpoints of two sides is parallel to the third side and half its length.

Angle bisector theorem

$\frac{BD}{DC} = \frac{AB}{AC}$

The bisector of angle $A$ meets $BC$ at $D$ and splits it in the ratio of the two sides next to angle $A$.

Area of a square

$A = a^{2}$

$a$ is the side length. The side from a known area is $a = \sqrt{A}$.

Perimeter of a square

$P = 4a$

A square has four equal sides of length $a$, so the boundary is four times one side.

Diagonal of a square

$d = a\sqrt{2}$

Both diagonals are equal, bisect each other at right angles and follow from Pythagoras on two sides.

Area of a square from its diagonal

$A = \frac{d^{2}}{2}$

Useful when only the diagonal $d$ is given. It matches the rhombus rule $\frac{1}{2}d_{1}d_{2}$ with equal diagonals.

Area of a rectangle

$A = l \times w$

$l$ is the length and $w$ the width (breadth), measured in the same unit.

Perimeter of a rectangle

$P = 2(l + w)$

Two lengths plus two widths. Use it for fencing, borders and frame problems.

Diagonal of a rectangle

$d = \sqrt{l^{2} + w^{2}}$

The diagonal is the hypotenuse of a right triangle with legs $l$ and $w$. Both diagonals are equal.

Area of a path around a rectangle

$(l + 2x)(w + 2x) - lw$

Area of a uniform border of width $x$ outside an $l$ by $w$ rectangle: outer rectangle minus inner rectangle.

Area of a parallelogram

$A = b \times h$

$b$ is a base and $h$ is the perpendicular distance between that base and the opposite side, not the slanted side.

Area of a parallelogram from sides and angle

$A = ab\sin\theta$

$a$ and $b$ are adjacent sides and $\theta$ is the angle between them.

Perimeter of a parallelogram

$P = 2(a + b)$

Opposite sides are equal, so add the two adjacent sides $a$ and $b$ and double.

Diagonals of a parallelogram

$d_{1}^{2} + d_{2}^{2} = 2(a^{2} + b^{2})$

The parallelogram law: the squares of the diagonals add to twice the sum of the squares of the adjacent sides.

Area of a rhombus (rhombus formula)

$A = \frac{1}{2}d_{1}d_{2}$

$d_{1}$ and $d_{2}$ are the diagonals. The rule works because a rhombus has perpendicular diagonals.

Perimeter of a rhombus

$P = 4a$

All four sides of a rhombus are equal to $a$.

Side of a rhombus from its diagonals (diagonal of rhombus)

$a = \frac{1}{2}\sqrt{d_{1}^{2} + d_{2}^{2}}$

The diagonals bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs $\frac{d_{1}}{2}$ and $\frac{d_{2}}{2}$.

Area of a trapezium (trapezoid)

$A = \frac{1}{2}(a + b)h$

$a$ and $b$ are the two parallel sides and $h$ is the perpendicular distance between them.

Midsegment of a trapezium

$m = \frac{a + b}{2}$

The segment joining the midpoints of the non-parallel sides. It is the mean of the parallel sides, and $A = mh$.

Diagonal of an isosceles trapezoid

$d = \sqrt{c^{2} + ab}$

$a$ and $b$ are the parallel sides and $c$ is each equal leg. Both diagonals have this length.

Area of a kite (properties of a kite)

$A = \frac{1}{2}d_{1}d_{2}$

$d_{1}$ and $d_{2}$ are the diagonals, which meet at right angles in every kite.

Perimeter of a kite

$P = 2(a + b)$

A kite has two pairs of equal adjacent sides, $a$ and $b$.

Area of any quadrilateral from a diagonal

$A = \frac{1}{2}d(h_{1} + h_{2})$

$d$ is one diagonal and $h_{1}$, $h_{2}$ are the perpendiculars to it from the other two vertices.

Angle sum of a quadrilateral (angles of quadrilateral)

$A + B + C + D = 360^{\circ}$

Any quadrilateral splits into two triangles along a diagonal, giving $2 \times 180^{\circ}$.

Area of a regular polygon (area of polygons)

$A = \frac{1}{2} \times P \times a$

$P$ is the perimeter and $a$ is the apothem, the perpendicular distance from the centre to a side.

Area of a regular polygon from its side

$A = \frac{ns^{2}}{4\tan\left(\frac{180^{\circ}}{n}\right)}$

Use when only the side $s$ and the number of sides $n$ are known.

Apothem

$a = \frac{s}{2\tan\left(\frac{180^{\circ}}{n}\right)}$

Distance from the centre to the midpoint of a side. It is also the radius of the inscribed circle.

Circumradius of a regular polygon

$R = \frac{s}{2\sin\left(\frac{180^{\circ}}{n}\right)}$

Distance from the centre to a vertex, which is the radius of the circle through all vertices.

Perimeter of a regular polygon

$P = ns$

Multiply the number of sides by the side length.

Area of a regular hexagon

$A = \frac{3\sqrt{3}}{2}s^{2}$

A regular hexagon splits into six equilateral triangles of side $s$, and its circumradius equals $s$.

Diagonals of a regular hexagon

$d_{\text{long}} = 2s$, $d_{\text{short}} = s\sqrt{3}$

The long diagonals pass through the centre. The short diagonals skip one vertex.

Area of a regular pentagon

$A = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})},s^{2} \approx 1.7205s^{2}$

Exact area of a regular pentagon with side $s$. Each interior angle is $108^{\circ}$.

Perimeter of a pentagon

$P = 5s$

For a regular pentagon. For an irregular one, add the five side lengths.

Area of a regular octagon

$A = 2(1 + \sqrt{2})s^{2} \approx 4.8284s^{2}$

Exact area of a regular octagon with side $s$. Each interior angle is $135^{\circ}$.

Diameter (diameter formula)

$d = 2r$

The diameter is a chord through the centre and is twice the radius $r$.

Circumference of a circle

$C = 2\pi r = \pi d$

The perimeter of a circle. Use $\pi \approx 3.1416$ or $\frac{22}{7}$ when the question says so.

Area of a circle (circle area formulas)

$A = \pi r^{2} = \frac{\pi d^{2}}{4}$

Area enclosed by a circle of radius $r$ or diameter $d$.

Area from the circumference

$A = \frac{C^{2}}{4\pi}$

Finds the area directly from a known circumference $C$, without first finding $r$.

Radius from the area

$r = \sqrt{\frac{A}{\pi}}$

Rearranged area formula for when the area is given and the radius is unknown.

Area of a semicircle (semicircle formulas)

$A = \frac{1}{2}\pi r^{2}$

Half of a full circle of radius $r$, cut along a diameter.

Perimeter of a semicircle

$P = \pi r + 2r$

The curved half of the circumference plus the straight diameter. Using $\pi r$ alone gives only the arc.

Area of a quadrant

$A = \frac{1}{4}\pi r^{2}$

A quarter circle, which is a sector with a central angle of $90^{\circ}$.

Perimeter of a quadrant

$P = \frac{\pi r}{2} + 2r$

One quarter of the circumference plus the two bounding radii.

Area of an annulus (ring)

$A = \pi(R^{2} - r^{2}) = \pi(R + r)(R - r)$

Area between two concentric circles with outer radius $R$ and inner radius $r$, where $R > r$.

Distance covered by a wheel

$D = N \times 2\pi r$

A wheel of radius $r$ travels one circumference per revolution, so $N$ revolutions cover $D$.

Arc length (degrees)

$L = \frac{\theta}{360^{\circ}} \times 2\pi r$

On a circle of radius $r$, the arc is the same fraction of the circumference as $\theta$ is of a full turn.

Arc length (radians)

$L = r\theta$

The simplest form, valid only when $\theta$ is measured in radians.

Area of a sector (degrees)

$A = \frac{\theta}{360^{\circ}} \times \pi r^{2}$

A sector is the slice between two radii, so its area is that fraction of the full circle.

Area of a sector (radians)

$A = \frac{1}{2}r^{2}\theta = \frac{1}{2}Lr$

With $\theta$ in radians, or with arc length $L$ known and no angle needed.

Perimeter of a sector

$P = 2r + L$

Two radii plus the arc length. Students often forget the two straight edges.

Area of a segment (radians)

$A = \frac{1}{2}r^{2}(\theta - \sin\theta)$

The region between a chord and its arc, equal to the sector minus the triangle, with $\theta$ in radians.

Area of a segment (degrees)

$A = \frac{\theta}{360^{\circ}}\pi r^{2} - \frac{1}{2}r^{2}\sin\theta$

Same idea in degrees: sector area minus the area of the triangle formed by the two radii.

Chord length from the central angle

$c = 2r\sin\left(\frac{\theta}{2}\right)$

Length of the chord cut off by a central angle $\theta$ in a circle of radius $r$.

Chord length from its distance to the centre

$c = 2\sqrt{r^{2} - d^{2}}$

$d$ is the perpendicular distance from the centre to the chord, with $0 \leq d \leq r$. That perpendicular bisects the chord.

Central angle from arc length

$\theta = \frac{L}{r}$

Gives $\theta$ in radians. Multiply by $\frac{180^{\circ}}{\pi}$ to convert to degrees.

Inscribed angle theorem

$\angle ACB = \frac{1}{2}\angle AOB$

An angle at the circle is half the angle at the centre $O$ standing on the same arc $AB$.

Angle in a semicircle

$\angle ACB = 90^{\circ}$

If $AB$ is a diameter, the angle it makes at any other point $C$ on the circle is a right angle.

Angles in the same segment (arcs and subtended angles)

$\angle ACB = \angle ADB$

Angles at $C$ and $D$ on the same side of chord $AB$, standing on the same arc, are equal.

Cyclic quadrilateral

$\angle A + \angle C = 180^{\circ}$, $\angle B + \angle D = 180^{\circ}$

When all four vertices lie on a circle, each pair of opposite angles is supplementary.

Brahmagupta's formula

$A = \sqrt{(s - a)(s - b)(s - c)(s - d)}$

Area of a cyclic quadrilateral with sides $a$, $b$, $c$, $d$ and semi-perimeter $s = \frac{a + b + c + d}{2}$.

Ptolemy's theorem

$AC \cdot BD = AB \cdot CD + AD \cdot BC$

In a cyclic quadrilateral $ABCD$, the product of the diagonals equals the sum of products of opposite sides.

Tangent perpendicular to radius

$OT \perp PT$

The tangent line $PT$ touching the circle at $T$ meets the radius $OT$ from centre $O$ at a right angle.

Tangents from an external point

$PA = PB = \sqrt{d^{2} - r^{2}}$

From a point $P$ at distance $d > r$ from the centre, both tangents to the circle have equal length.

Intersecting chords theorem

$AE \times EB = CE \times ED$

When chords $AB$ and $CD$ cross at $E$ inside the circle, the products of their parts are equal.

Tangent-secant theorem

$PT^{2} = PA \times PB$

From external point $P$, tangent $PT$ and a secant meeting the circle at $A$ (near) and $B$ (far).

Secant-secant theorem

$PA \times PB = PC \times PD$

Two secants from external point $P$ cut the circle at $A$, $B$ and $C$, $D$. Use whole secant lengths $PB$ and $PD$.

Alternate segment theorem

$\angle TAB = \angle ACB$

The angle between tangent $TA$ and chord $AB$ equals the inscribed angle at $C$ in the opposite segment.

Area of an ellipse

$A = \pi ab$

Exact area of an ellipse. It stretches the circle formula along two axes.

Perimeter of an ellipse (Ramanujan)

$P \approx \pi\left[3(a + b) - \sqrt{(3a + b)(a + 3b)}\right]$

No simple exact formula exists. This approximation is accurate to a tiny fraction of a percent for most ellipses.

Rough perimeter of an ellipse

$P \approx 2\pi\sqrt{\frac{a^{2} + b^{2}}{2}}$

A quicker estimate that works well when $a$ and $b$ are close in size.

Area of a semi-ellipse

$A = \frac{1}{2}\pi ab$

Half of an ellipse cut along either axis.

Circle as a special case

$a = b = r \Rightarrow A = \pi r^{2}$

When both semi-axes are equal, the ellipse is a circle and the formulas reduce to the circle forms.

Area of an elliptical ring

$A = \pi(ab - cd)$

Region between an outer ellipse with semi-axes $a$, $b$ and an inner ellipse with semi-axes $c$, $d$.

Volume of a cube (cube formula)

$V = a^{3}$

$a$ is the edge length. Volume is measured in cubic units.

Lateral surface area of a cube

$\text{LSA} = 4a^{2}$

Area of the four side faces only, leaving out the top and bottom.

Total surface area of a cube

$\text{TSA} = 6a^{2}$

Area of all six square faces of the cube.

Face diagonal of a cube

$a\sqrt{2}$

Diagonal across one square face, found from Pythagoras on two edges.

Space diagonal of a cube

$d = a\sqrt{3}$

Line through the centre joining opposite corners of the cube.

Edge of a cube from its volume

$a = \sqrt[3]{V}$

Take the cube root of the volume to recover the edge.

Volume of a cuboid (rectangular prism)

$V = l \times w \times h$

$l$, $w$ and $h$ are the length, width (breadth) and height in the same unit.

Lateral surface area of a cuboid

$\text{LSA} = 2h(l + w)$

Area of the four walls, as used for painting the walls of a room.

Total surface area of a cuboid

$\text{TSA} = 2(lw + wh + hl)$

Area of all six rectangular faces, which come in three equal pairs.

Space diagonal of a cuboid

$d = \sqrt{l^{2} + w^{2} + h^{2}}$

Longest straight rod that fits inside the box, joining opposite corners.

Total edge length

$4(l + w + h)$ for a cuboid, $12a$ for a cube

A cuboid has 12 edges, four each of length $l$, $w$ and $h$.

Volume of a cylinder

$V = \pi r^{2}h$

Base area $\pi r^{2}$ times height $h$. It also holds for an oblique cylinder if $h$ is the perpendicular height.

Curved surface area of a right circular cylinder

$\text{CSA} = 2\pi rh$

Area of the side only. Unrolled, it is a rectangle of width $2\pi r$ and height $h$.

Total surface area of a cylinder

$\text{TSA} = 2\pi r(r + h)$

Curved surface plus the two circular ends, $2\pi rh + 2\pi r^{2}$.

Volume of a hollow cylinder

$V = \pi(R^{2} - r^{2})h$

Volume of material in a pipe with outer radius $R$, inner radius $r$ and length $h$.

Total surface area of a hollow cylinder

$2\pi(R + r)h + 2\pi(R^{2} - r^{2})$

Outer and inner curved surfaces plus the two ring-shaped ends.

Height of a cylinder from its volume

$h = \frac{V}{\pi r^{2}}$

Rearranged volume formula, used for tank and container filling questions.

Volume of a cone

$V = \frac{1}{3}\pi r^{2}h$

One third of a cylinder with the same base radius $r$ and height $h$.

Slant height of a right circular cone

$l = \sqrt{r^{2} + h^{2}}$

Distance from the apex to the edge of the base, found from the right triangle formed by $r$, $h$ and $l$.

Curved surface area of a cone

$\text{CSA} = \pi rl$

Area of the sloping surface only. It uses the slant height $l$, not the vertical height.

Total surface area of a cone

$\text{TSA} = \pi r(l + r)$

Curved surface plus the circular base, $\pi rl + \pi r^{2}$.

Cone made from a sector

$l = \text{sector radius}$, $r = \frac{\theta}{360^{\circ}} \times l$

Rolling a sector of angle $\theta$ into a cone turns the sector radius into $l$ and the arc into the base circumference.

Volume of a frustum of a cone

$V = \frac{1}{3}\pi h(R^{2} + Rr + r^{2})$

Volume of a cone with its top cut off parallel to the base, such as a bucket or lampshade.

Slant height of a frustum

$l = \sqrt{h^{2} + (R - r)^{2}}$

Found from a right triangle with legs $h$ and $R - r$.

Curved surface area of a frustum

$\text{CSA} = \pi l(R + r)$

Area of the sloping side of the frustum only.

Total surface area of a frustum

$\text{TSA} = \pi l(R + r) + \pi R^{2} + \pi r^{2}$

Curved surface plus both circular ends. Drop $\pi r^{2}$ for an open-topped bucket.

Volume of a sphere

$V = \frac{4}{3}\pi r^{3}$

$r$ is the radius. The volume grows with the cube of the radius.

Surface area of a sphere

$A = 4\pi r^{2}$

Equal to four times the area of a great circle of the same radius.

Volume of a hemisphere

$V = \frac{2}{3}\pi r^{3}$

Half the volume of a sphere of radius $r$.

Curved surface area of a hemisphere

$\text{CSA} = 2\pi r^{2}$

Area of the dome only, which is half the sphere's surface.

Total surface area of a hemisphere

$\text{TSA} = 3\pi r^{2}$

Curved dome $2\pi r^{2}$ plus the flat circular base $\pi r^{2}$.

Volume of a spherical shell

$V = \frac{4}{3}\pi(R^{3} - r^{3})$

Material in a hollow ball with outer radius $R$ and inner radius $r$.

Volume of a spherical cap

$V = \frac{1}{3}\pi h^{2}(3r - h)$

A cap of height $h$ cut from a sphere of radius $r$, with $0 < h \leq 2r$.

Curved surface area of a spherical cap

$A = 2\pi rh$

Area of the curved part of a cap of height $h$ on a sphere of radius $r$.

Volume of an ellipsoid

$V = \frac{4}{3}\pi abc$

$a$, $b$ and $c$ are the three semi-axes. It becomes the sphere formula when all three equal $r$.

Radius of a sphere from its volume

$r = \sqrt[3]{\frac{3V}{4\pi}}$

Rearranged volume formula for when the volume is known.

Volume of a prism

$V = B \times h$

Base area times the perpendicular height $h$ between the two identical end faces.

Lateral surface area of a prism (lateral area formula)

$\text{LSA} = P \times h$

For a right prism, the side faces are rectangles whose total area is the base perimeter times height.

Total surface area of a prism

$\text{TSA} = Ph + 2B$

Lateral area plus the two congruent end faces.

Volume of a triangular prism

$V = \frac{1}{2}bh_{t} \times l$

$b$ and $h_{t}$ are the base and height of the triangular face, and $l$ is the length of the prism.

Volume of a pyramid

$V = \frac{1}{3}Bh$

One third of the prism with the same base and height. $h$ is the perpendicular height to the apex.

Lateral surface area of a regular pyramid

$\text{LSA} = \frac{1}{2}Pl$

$l$ is the slant height of each triangular face, measured from the apex to the midpoint of a base edge.

Total surface area of a regular pyramid

$\text{TSA} = \frac{1}{2}Pl + B$

Area of all the triangular side faces plus the area $B$ of the base.

Square pyramid

$V = \frac{1}{3}a^{2}h$, $l = \sqrt{h^{2} + \frac{a^{2}}{4}}$

$a$ is the base edge, $h$ the height and $l$ the slant height of each face.

Total surface area of a square pyramid

$\text{TSA} = a^{2} + 2al$

Square base $a^{2}$ plus four triangles of area $\frac{1}{2}al$ each.

Volume of a frustum of a pyramid

$V = \frac{h}{3}\left(B_{1} + B_{2} + \sqrt{B_{1}B_{2}}\right)$

$B_{1}$ and $B_{2}$ are the areas of the two parallel faces and $h$ is the perpendicular height between them.

Regular tetrahedron (triangular pyramid)

$V = \frac{a^{3}}{6\sqrt{2}}$, $\text{TSA} = \sqrt{3}a^{2}$

A pyramid whose four faces are equilateral triangles with edge $a$.

Euler's formula for a polyhedron

$V - E + F = 2$

Vertices minus edges plus faces always equals 2 for a convex solid. Use it to find one count from the other two.

Edges from the faces

$E = \frac{1}{2} \sum (\text{sides of each face})$

Each edge is shared by exactly two faces, so adding the sides of all faces counts every edge twice.

Tetrahedron

$V = 4$, $E = 6$, $F = 4$

Four triangular faces. Check: $4 - 6 + 4 = 2$.

Cube (hexahedron)

$V = 8$, $E = 12$, $F = 6$

Six square faces. Check: $8 - 12 + 6 = 2$.

Octahedron

$V = 6$, $E = 12$, $F = 8$

Eight triangular faces, with four meeting at each vertex. Check: $6 - 12 + 8 = 2$.

Dodecahedron

$V = 20$, $E = 30$, $F = 12$

Twelve regular pentagon faces, with three meeting at each vertex.

Icosahedron

$V = 12$, $E = 30$, $F = 20$

Twenty triangular faces, with five meeting at each vertex.

Prism with an $n$-sided base

$V = 2n$, $E = 3n$, $F = n + 2$

Two $n$-gon ends plus $n$ rectangular sides. A triangular prism has 6 vertices, 9 edges and 5 faces.

Pyramid with an $n$-sided base

$V = n + 1$, $E = 2n$, $F = n + 1$

One $n$-gon base plus $n$ triangular faces meeting at the apex.

Distance formula in 2D

$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

Length of the segment joining $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, built from the Pythagorean theorem on the horizontal and vertical gaps.

Euclidean distance formula in 3D

$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}$

Straight-line distance between two points in space. Add the squared gap along the $z$-axis to the 2D formula.

Midpoint formula

$M = \left(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}\right)$

Point exactly halfway between two endpoints, found by averaging the $x$-coordinates and the $y$-coordinates separately.

Midpoint in 3D

$M = \left(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2}\right)$

The same averaging rule applied to all three coordinates of two endpoints in space.

Section formula (internal division)

$P = \left(\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n}\right)$

Point $P$ between $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$ with $AP : PB = m : n$. Note that $m$ multiplies the coordinates of $B$.

Section formula (external division)

$P = \left(\frac{mx_{2} - nx_{1}}{m - n}, \frac{my_{2} - ny_{1}}{m - n}\right)$

Point on line $AB$ extended, outside the segment, with $AP : PB = m : n$. It needs $m \neq n$.

Section formula in 3D

$P = \left(\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n}, \frac{mz_{2} + nz_{1}}{m + n}\right)$

Internal division of a segment in space in the ratio $m : n$, with the same weights applied to the $z$-coordinates.

Ratio in which a line divides a segment

$\frac{m}{n} = -\frac{ax_{1} + by_{1} + c}{ax_{2} + by_{2} + c}$

Ratio in which the line $ax + by + c = 0$ cuts the segment from $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$. A positive value means internal division.

Centroid of a triangle

$G = \left(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}\right)$

Meeting point of the three medians, found by averaging the vertices. It divides each median in the ratio $2 : 1$ from the vertex.

Incentre (incenter) of a triangle

$I = \left(\frac{ax_{1} + bx_{2} + cx_{3}}{a + b + c}, \frac{ay_{1} + by_{2} + cy_{3}}{a + b + c}\right)$

Centre of the inscribed circle, where the angle bisectors meet. Each vertex is weighted by the length of the side opposite it.

Excentre opposite $A$

$I_{1} = \left(\frac{-ax_{1} + bx_{2} + cx_{3}}{-a + b + c}, \frac{-ay_{1} + by_{2} + cy_{3}}{-a + b + c}\right)$

Centre of the circle touching side $BC$ and the extensions of $AB$ and $AC$. Flip the sign of the weight on $A$.

Circumcentre (circumcenter)

$(x - x_{1})^{2} + (y - y_{1})^{2} = (x - x_{2})^{2} + (y - y_{2})^{2} = (x - x_{3})^{2} + (y - y_{3})^{2}$

The point $(x, y)$ equidistant from all three vertices. Expand to get two linear equations. In a right triangle it is the midpoint of the hypotenuse.

Euler line relation

$G = \frac{H + 2O}{3}$

Orthocentre $H$, centroid $G$ and circumcentre $O$ are collinear, and $G$ divides $HO$ in the ratio $2 : 1$. Apply it coordinate by coordinate.

Area of a triangle from coordinates

$\text{Area} = \frac{1}{2}\lvert x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}) \rvert$

Area straight from the three vertices. The absolute value keeps the answer positive whatever order the vertices are listed in.

Shoelace formula for a polygon

$\text{Area} = \frac{1}{2}\left\lvert \sum_{k=1}^{n} (x_{k}y_{k+1} - x_{k+1}y_{k}) \right\rvert$

Area of a simple polygon with $n$ vertices listed in order around the boundary, taking $(x_{n+1}, y_{n+1}) = (x_{1}, y_{1})$.

Collinearity condition

$x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}) = 0$

Three points lie on one straight line exactly when the triangle they form has zero area. Equal slopes of $AB$ and $BC$ give the same test.

Slope from two points

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Rise over run between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$. It needs $x_{1} \neq x_{2}$, because a vertical line has undefined slope.

Slope from the angle of inclination

$m = \tan\theta$

$\theta$ is the angle the line makes with the positive $x$-axis, measured anticlockwise, with $0^{\circ} \leq \theta < 180^{\circ}$ and $\theta \neq 90^{\circ}$.

Slope from the general equation

$m = -\frac{a}{b}$

Slope of the line $ax + by + c = 0$ when $b \neq 0$. If $b = 0$ the line is vertical.

Angle between two lines

$\tan\theta = \left\lvert \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}} \right\rvert$

Acute angle $\theta$ between lines with slopes $m_{1}$ and $m_{2}$. If $m_{1}m_{2} = -1$ the lines are perpendicular and $\theta = 90^{\circ}$.

Angle between lines in general form

$\cos\theta = \frac{\lvert a_{1}a_{2} + b_{1}b_{2} \rvert}{\sqrt{a_{1}^{2} + b_{1}^{2}}\sqrt{a_{2}^{2} + b_{2}^{2}}}$

Acute angle between $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$. It works even when a line is vertical.

Parallel lines (slope test)

$m_{1} = m_{2}$

Two non-vertical lines are parallel exactly when their slopes are equal.

Perpendicular lines (slope test)

$m_{1}m_{2} = -1$

Two lines, neither of them vertical, meet at a right angle exactly when the product of their slopes is $-1$.

Parallel lines in general form

$a_{1}b_{2} = a_{2}b_{1}$

Condition for $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$ to be parallel. If the constants share that ratio, the lines coincide.

Perpendicular lines in general form

$a_{1}a_{2} + b_{1}b_{2} = 0$

Condition for the two lines above to be perpendicular, read from their normal vectors $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$.

Normal (perpendicular) form

$x\cos\alpha + y\sin\alpha = p$

$p \geq 0$ is the perpendicular distance from the origin to the line, and $\alpha$ is the angle that perpendicular makes with the positive $x$-axis.

Parametric (distance) form

$\frac{x - x_{1}}{\cos\theta} = \frac{y - y_{1}}{\sin\theta} = r$

The point $(x_{1} + r\cos\theta, y_{1} + r\sin\theta)$ lies at signed distance $r$ from $(x_{1}, y_{1})$ along a line inclined at $\theta$.

Line parallel to a given line

$ax + by + k = 0$

Every line parallel to $ax + by + c = 0$ keeps $a$ and $b$ and changes only the constant. Find $k$ from one point on it.

Line perpendicular to a given line

$bx - ay + k = 0$

Every line perpendicular to $ax + by + c = 0$. Swap the coefficients of $x$ and $y$ and change one sign, then find $k$.

Point of intersection of two lines

$\left(\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}, \frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}\right)$

Where $a_{1}x + b_{1}y + c_{1} = 0$ meets $a_{2}x + b_{2}y + c_{2} = 0$, valid when $a_{1}b_{2} \neq a_{2}b_{1}$, that is, the lines are not parallel.

Family of lines through an intersection

$(a_{1}x + b_{1}y + c_{1}) + \lambda(a_{2}x + b_{2}y + c_{2}) = 0$

Any line through the meeting point of the two given lines, except the second line itself. One more condition fixes the value of $\lambda$.

Concurrency of three lines

$a_{1}(b_{2}c_{3} - b_{3}c_{2}) - b_{1}(a_{2}c_{3} - a_{3}c_{2}) + c_{1}(a_{2}b_{3} - a_{3}b_{2}) = 0$

Three lines $a_{i}x + b_{i}y + c_{i} = 0$, no two of them parallel, pass through one point exactly when this coefficient determinant equals zero.

Side of a line on which a point lies

$\text{sign of } ax_{1} + by_{1} + c$

Two points are on the same side of $ax + by + c = 0$ when substituting them gives values of the same sign, and on opposite sides otherwise.

Angle bisectors of two lines

$\frac{a_{1}x + b_{1}y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \pm \frac{a_{2}x + b_{2}y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

The two lines that bisect the angles between the given lines. Every point on them is equidistant from both lines.

Pair of straight lines through the origin

$ax^{2} + 2hxy + by^{2} = 0$

Represents two lines through the origin when $h^{2} \geq ab$. The acute angle between them satisfies $\tan\theta = \frac{2\sqrt{h^{2} - ab}}{\lvert a + b \rvert}$.

Distance of a point from a line

$d = \frac{\lvert ax_{1} + by_{1} + c \rvert}{\sqrt{a^{2} + b^{2}}}$

Shortest (perpendicular) distance from the point $(x_{1}, y_{1})$ to the line $ax + by + c = 0$.

Distance between parallel lines

$d = \frac{\lvert c_{1} - c_{2} \rvert}{\sqrt{a^{2} + b^{2}}}$

Gap between $ax + by + c_{1} = 0$ and $ax + by + c_{2} = 0$. Scale one equation first so the $x$ and $y$ coefficients match.

Distance between parallel lines in slope form

$d = \frac{\lvert c_{1} - c_{2} \rvert}{\sqrt{1 + m^{2}}}$

Gap between the parallel lines $y = mx + c_{1}$ and $y = mx + c_{2}$.

Foot of the perpendicular

$\frac{h - x_{1}}{a} = \frac{k - y_{1}}{b} = -\frac{ax_{1} + by_{1} + c}{a^{2} + b^{2}}$

$(h, k)$ is the point on $ax + by + c = 0$ nearest to $(x_{1}, y_{1})$. If $a$ or $b$ is $0$, that coordinate stays unchanged.

Image (reflection) of a point in a line

$\frac{h - x_{1}}{a} = \frac{k - y_{1}}{b} = -\frac{2(ax_{1} + by_{1} + c)}{a^{2} + b^{2}}$

Mirror image $(h, k)$ of $(x_{1}, y_{1})$ in $ax + by + c = 0$. The foot of the perpendicular is the midpoint of the point and its image.

Reflection in the axes and in $y = x$

$(x, y) \to (x, -y), (-x, y), (y, x)$

Images of the point $(x, y)$ in the $x$-axis, the $y$-axis and the line $y = x$, in that order.

Area cut off by a line and the axes

$\text{Area} = \frac{c^{2}}{2\lvert ab \rvert}$

Area of the triangle formed by $ax + by + c = 0$ and the two coordinate axes, with $a \neq 0$ and $b \neq 0$.

Equation of a circle (centre-radius form)

$(x - h)^{2} + (y - k)^{2} = r^{2}$

Circle with centre $(h, k)$ and radius $r > 0$. Every point on it is at distance $r$ from the centre.

General form of a circle

$x^{2} + y^{2} + 2gx + 2fy + c = 0$

Expanded form. The coefficients of $x^{2}$ and $y^{2}$ are equal and there is no $xy$ term.

Centre of a circle and radius from the general form

$(-g, -f)$ and $r = \sqrt{g^{2} + f^{2} - c}$

Read these off the general form. The circle is real only if $g^{2} + f^{2} - c > 0$, and shrinks to a point if it equals $0$.

Diameter form

$(x - x_{1})(x - x_{2}) + (y - y_{1})(y - y_{2}) = 0$

Circle that has the segment from $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$ as a diameter.

Parametric form

$x = h + r\cos\theta$, $y = k + r\sin\theta$

Gives every point of the circle as $\theta$ runs over $0 \leq \theta < 2\pi$. Useful when a point moves around the circle.

Position of a point

$S_{1} = x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c$

Substitute the point into the general form. $S_{1} < 0$ means inside, $S_{1} = 0$ on the circle and $S_{1} > 0$ outside.

Tangent at a point on the circle

$xx_{1} + yy_{1} = r^{2}$

Tangent to $x^{2} + y^{2} = r^{2}$ at $(x_{1}, y_{1})$. For the general form use $xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0$.

Condition of tangency

$c^{2} = r^{2}(1 + m^{2})$

The line $y = mx + c$ touches $x^{2} + y^{2} = r^{2}$ exactly when this holds, which gives the tangents $y = mx \pm r\sqrt{1 + m^{2}}$.

Length of the tangent from an external point

$L = \sqrt{S_{1}}$

Tangent length from an outside point $(x_{1}, y_{1})$ to the point of contact. Equivalently $\sqrt{d^{2} - r^{2}}$, with $d$ the distance to the centre.

Chord of contact

$xx_{1} + yy_{1} = r^{2}$

For an external point $(x_{1}, y_{1})$, this is the chord joining the two points where its tangents touch $x^{2} + y^{2} = r^{2}$.

Orthogonal circles

$2g_{1}g_{2} + 2f_{1}f_{2} = c_{1} + c_{2}$

Two circles in general form cut each other at right angles exactly when this holds, equivalently $d^{2} = r_{1}^{2} + r_{2}^{2}$ for centre distance $d$.

Parabola in standard form

$y^{2} = 4ax$

Vertex at the origin and axis along the $x$-axis. It is the set of points equally far from the focus and the directrix.

Focus of parabola

$(a, 0)$

Fixed point inside the curve on its axis, at distance $a$ from the vertex.

Directrix of parabola

$x = -a$

Fixed line perpendicular to the axis, at distance $a$ from the vertex on the side away from the focus.

Latus rectum of parabola

$\text{length} = 4a$

Focal chord perpendicular to the axis, with endpoints $(a, 2a)$ and $(a, -2a)$.

Other standard orientations

$y^{2} = -4ax$, $x^{2} = 4ay$, $x^{2} = -4ay$

Parabolas opening left with focus $(-a, 0)$, up with focus $(0, a)$ and down with focus $(0, -a)$. The directrix is always opposite the focus.

Parabola with vertex $(h, k)$

$(y - k)^{2} = 4a(x - h)$

Right-opening parabola shifted so the vertex is $(h, k)$, with focus $(h + a, k)$ and directrix $x = h - a$.

Parametric form

$x = at^{2}$, $y = 2at$

The point with parameter $t$ on $y^{2} = 4ax$. Each real value of $t$ gives exactly one point of the curve.

Focal distance of a point

$PS = x_{1} + a$

Distance from a point $P(x_{1}, y_{1})$ on the parabola to the focus $S$, which equals its distance from the directrix.

Tangent at a point

$yy_{1} = 2a(x + x_{1})$

Tangent to $y^{2} = 4ax$ at the point $(x_{1}, y_{1})$ on the curve.

Tangent in slope form

$y = mx + \frac{a}{m}$

Tangent to $y^{2} = 4ax$ with slope $m \neq 0$. It touches the curve at $\left(\frac{a}{m^{2}}, \frac{2a}{m}\right)$.

Normal in slope form

$y = mx - 2am - am^{3}$

Normal to $y^{2} = 4ax$ with slope $m$, meeting the curve at $(am^{2}, -2am)$.

Eccentricity of parabola

$e = 1$

Every point is exactly as far from the focus as from the directrix, so the ratio of the two distances is $1$.

Ellipse in standard form

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

Centre at the origin with $a > b > 0$. The major axis lies along the $x$-axis, $a$ is the semi-major axis and $b$ the semi-minor axis.

Focus distance relation

$c^{2} = a^{2} - b^{2}$

$c$ is the distance from the centre to each focus. Since $a > b$, the foci lie inside the ellipse with $c < a$.

Foci of ellipse

$(\pm c, 0)$

Two fixed points on the major axis, one on each side of the centre.

Eccentricity of ellipse

$e = \frac{c}{a} = \sqrt{1 - \frac{b^{2}}{a^{2}}}$

Measures how stretched the ellipse is, with $0 < e < 1$. Values near $0$ look almost like a circle.

Vertex of ellipse (vertices)

$(\pm a, 0)$

Ends of the major axis, of length $2a$. The ends of the minor axis, of length $2b$, are $(0, \pm b)$.

Directrices

$x = \pm \frac{a}{e}$

Two lines perpendicular to the major axis, outside the ellipse, one paired with each focus.

Latus rectum of ellipse

$\text{length} = \frac{2b^{2}}{a}$

Length of the focal chord through either focus, perpendicular to the major axis.

Sum of focal distances

$PF_{1} + PF_{2} = 2a$

Defining property of every point $P$ on the ellipse. For $P$ with abscissa $x_{1}$, the distances are $a - ex_{1}$ and $a + ex_{1}$.

Vertical ellipse

$\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1$

With $a > b$, the major axis lies along the $y$-axis and the foci are $(0, \pm c)$, still with $c^{2} = a^{2} - b^{2}$.

Ellipse with centre $(h, k)$

$\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1$

The standard ellipse shifted so its centre is $(h, k)$. The foci move to $(h \pm c, k)$.

Parametric form

$x = a\cos\theta$, $y = b\sin\theta$

The point with eccentric angle $\theta$ on the ellipse, for $0 \leq \theta < 2\pi$.

Tangent at a point

$\frac{xx_{1}}{a^{2}} + \frac{yy_{1}}{b^{2}} = 1$

Tangent at $(x_{1}, y_{1})$ on the ellipse. In slope form the tangents are $y = mx \pm \sqrt{a^{2}m^{2} + b^{2}}$.

Hyperbola in standard form

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

Centre at the origin, $a > 0$, $b > 0$, transverse axis along the $x$-axis, with two branches opening left and right.

Focus distance relation

$c^{2} = a^{2} + b^{2}$

$c$ is the distance from the centre to each focus, so $c > a$ and the foci lie beyond the vertices.

Foci of hyperbola

$(\pm c, 0)$

Two fixed points on the transverse axis, each inside one branch.

Eccentricity of hyperbola

$e = \frac{c}{a} = \sqrt{1 + \frac{b^{2}}{a^{2}}}$

Always $e > 1$. A larger $e$ means the branches open more widely.

Vertex of hyperbola (vertices)

$(\pm a, 0)$

Where the branches cross the transverse axis. The transverse axis has length $2a$ and the conjugate axis has length $2b$.

Asymptotes

$y = \pm \frac{b}{a}x$

Lines that the branches approach but never touch as $x$ grows. Setting the right side of the equation to $0$ gives them.

Directrices

$x = \pm \frac{a}{e}$

Two lines perpendicular to the transverse axis, between the centre and the vertices, one for each focus.

Latus rectum

$\text{length} = \frac{2b^{2}}{a}$

Length of the focal chord through either focus, perpendicular to the transverse axis.

Difference of focal distances

$\lvert PF_{1} - PF_{2} \rvert = 2a$

Defining property of every point $P$ on the hyperbola, in contrast to the constant sum for an ellipse.

Vertical hyperbola

$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$

Opens up and down, with foci $(0, \pm c)$ and asymptotes $y = \pm \frac{a}{b}x$.

Conjugate hyperbola

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = -1$

Shares its asymptotes with the standard hyperbola. The two eccentricities satisfy $\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}} = 1$.

Rectangular hyperbola

$x^{2} - y^{2} = a^{2}$

The case $a = b$, with perpendicular asymptotes and $e = \sqrt{2}$. Turning the axes clockwise through $45^{\circ}$ turns it into $xy = \frac{a^{2}}{2}$.

Tangent at a point

$\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1$

Tangent at $(x_{1}, y_{1})$ on the hyperbola. In slope form, $y = mx \pm \sqrt{a^{2}m^{2} - b^{2}}$ with $a^{2}m^{2} > b^{2}$.

Eccentricity (focus-directrix definition)

$PS = e \cdot PM$

A conic is the path of a point $P$ whose distance $PS$ to a fixed focus is $e$ times its distance $PM$ to a fixed directrix.

Type of conic from eccentricity

$e = 0$, $0 < e < 1$, $e = 1$, $e > 1$

These give a circle, an ellipse, a parabola and a hyperbola, in that order.

General second-degree equation

$Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$

Every conic in the plane, including shifted and rotated ones, has this form. The $xy$ term appears only when the axes of the conic are tilted.

Conic section type test

$B^{2} - 4AC$

For a non-degenerate conic, a negative value means an ellipse (a circle if $A = C$ and $B = 0$), zero means a parabola and positive means a hyperbola.

Rotation angle that removes the $xy$ term

$\cot 2\theta = \frac{A - C}{B}$

Turning the axes through this $\theta$, with $B \neq 0$, removes the $xy$ term so the conic can be written in standard form.

Shift of origin

$x = X + h$, $y = Y + k$

Moving the origin to $(h, k)$ without turning the axes. $(x, y)$ are old coordinates and $(X, Y)$ are new ones.

Rotation of axes

$x = X\cos\theta - Y\sin\theta$, $y = X\sin\theta + Y\cos\theta$

Old coordinates in terms of new ones when the axes are turned anticlockwise through $\theta$ about the origin.

Polar to rectangular

$x = r\cos\theta$, $y = r\sin\theta$

Converts polar $(r, \theta)$ to Cartesian $(x, y)$, where $r$ is the distance from the pole and $\theta$ the angle from the positive $x$-axis.

Polar coordinates from rectangular

$r = \sqrt{x^{2} + y^{2}}$, $\tan\theta = \frac{y}{x}$

Converts $(x, y)$ to $(r, \theta)$. Pick $\theta$ in the quadrant of the point, because $\tan^{-1}$ alone gives only angles in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Distance between two polar points

$d = \sqrt{r_{1}^{2} + r_{2}^{2} - 2r_{1}r_{2}\cos(\theta_{1} - \theta_{2})}$

The law of cosines on the triangle formed by the pole and the points $(r_{1}, \theta_{1})$ and $(r_{2}, \theta_{2})$.

Polar equation of a line

$r\cos(\theta - \alpha) = p$

Line at perpendicular distance $p$ from the pole, with that perpendicular at angle $\alpha$. It is the normal form written in polar terms.

Polar equations of circles

$r = a$, $r = 2a\cos\theta$

The first is a circle of radius $a$ centred at the pole. The second has radius $a$, centre $(a, 0)$ and passes through the pole.

Polar equation of a conic

$r = \frac{l}{1 + e\cos\theta}$

Conic with one focus at the pole, eccentricity $e$ and semi-latus rectum $l$. It is the standard form for orbit problems.

Cylindrical coordinates

$x = r\cos\theta$, $y = r\sin\theta$, $z = z$

A 3D system that uses polar coordinates in the $xy$-plane and keeps the height $z$. Suited to cylinders and shapes with an axis.

Spherical coordinates

$x = \rho\sin\phi\cos\theta$, $y = \rho\sin\phi\sin\theta$, $z = \rho\cos\phi$

$\rho$ is distance from the origin, $\phi$ the angle from the positive $z$-axis with $0 \leq \phi \leq \pi$, and $\theta$ the angle in the $xy$-plane.

Components of a vector

$\vec{a} = a_{1}\hat{i} + a_{2}\hat{j} + a_{3}\hat{k}$

$a_{1}$, $a_{2}$, $a_{3}$ are the parts of the vector along the $x$, $y$ and $z$ axes, and $\hat{i}$, $\hat{j}$, $\hat{k}$ are unit vectors along them.

Magnitude of a vector

$\lvert \vec{a} \rvert = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$

Length of the vector, also written $\lVert \vec{a} \rVert$. In 2D, leave out $a_{3}$.

Unit vector

$\hat{a} = \frac{\vec{a}}{\lvert \vec{a} \rvert}$

Vector of length $1$ pointing the same way as $\vec{a}$, defined only for $\vec{a} \neq \vec{0}$.

Position vector of $B$ relative to $A$

$\overrightarrow{AB} = \vec{b} - \vec{a} = (x_{2} - x_{1})\hat{i} + (y_{2} - y_{1})\hat{j} + (z_{2} - z_{1})\hat{k}$

Vector from $A$ to $B$, where $\vec{a}$ and $\vec{b}$ are the position vectors of the two points. Always head minus tail.

Vector addition

$\vec{a} + \vec{b} = (a_{1} + b_{1})\hat{i} + (a_{2} + b_{2})\hat{j} + (a_{3} + b_{3})\hat{k}$

Add matching components. Geometrically this is the triangle law or the parallelogram law of addition.

Scalar multiple

$\lambda\vec{a} = \lambda a_{1}\hat{i} + \lambda a_{2}\hat{j} + \lambda a_{3}\hat{k}$

Multiplies the length by $\lvert \lambda \rvert$. The direction reverses when $\lambda < 0$.

Collinear vectors

$\vec{a} = \lambda\vec{b}$

Non-zero vectors are parallel exactly when one is a scalar multiple of the other, so $\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{a_{3}}{b_{3}}$ when no $b_{i}$ is zero.

Section formula for vectors (internal)

$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m + n}$

Position vector of the point dividing $AB$ internally in the ratio $m : n$.

Section formula for vectors (external)

$\vec{r} = \frac{m\vec{b} - n\vec{a}}{m - n}$

Position vector of the point dividing $AB$ externally in the ratio $m : n$, with $m \neq n$.

Midpoint and centroid in vector form

$\frac{\vec{a} + \vec{b}}{2}$, $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

Position vectors of the midpoint of $AB$ and of the centroid of triangle $ABC$.

Dot product (scalar product)

$\vec{a} \cdot \vec{b} = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos\theta$

$\theta$ is the angle between the vectors, with $0 \leq \theta \leq \pi$. The result is a number, not a vector.

Dot product in components

$\vec{a} \cdot \vec{b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}$

Multiply matching components and add. This is how the dot product is worked out in practice.

Angle between two vectors

$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{\lvert \vec{a} \rvert \lvert \vec{b} \rvert}$

Angle between two non-zero vectors. A negative dot product means the angle is obtuse.

Perpendicular vectors

$\vec{a} \cdot \vec{b} = 0$

Two non-zero vectors are perpendicular (orthogonal) exactly when their dot product is zero.

Dot products of unit vectors

$\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$, $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

The axis unit vectors have length $1$ and are mutually perpendicular, which is why the component formula works.

Dot product of a vector with itself

$\vec{a} \cdot \vec{a} = \lvert \vec{a} \rvert^{2}$

Gives the squared length. The dot product is also commutative, so $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.

Square of a sum of vectors

$\lvert \vec{a} + \vec{b} \rvert^{2} = \lvert \vec{a} \rvert^{2} + \lvert \vec{b} \rvert^{2} + 2\vec{a} \cdot \vec{b}$

Finds the length of a resultant. For $\lvert \vec{a} - \vec{b} \rvert^{2}$, change the last sign to minus.

Scalar projection of $\vec{a}$ on $\vec{b}$

$\frac{\vec{a} \cdot \vec{b}}{\lvert \vec{b} \rvert}$

Signed length of the shadow of $\vec{a}$ along the direction of $\vec{b}$, for $\vec{b} \neq \vec{0}$.

Projection vector of $\vec{a}$ on $\vec{b}$

$\left(\frac{\vec{a} \cdot \vec{b}}{\lvert \vec{b} \rvert^{2}}\right)\vec{b}$

The part of $\vec{a}$ that points along $\vec{b}$, written as a vector.

Cauchy-Schwarz inequality

$\lvert \vec{a} \cdot \vec{b} \rvert \leq \lvert \vec{a} \rvert \lvert \vec{b} \rvert$

Follows from $\lvert \cos\theta \rvert \leq 1$. Equality holds only when the vectors are parallel or one of them is zero.

Work done by a constant force

$W = \vec{F} \cdot \vec{d}$

Work when force $\vec{F}$ moves an object through displacement $\vec{d}$, equal to $\lvert \vec{F} \rvert \lvert \vec{d} \rvert \cos\theta$.