Math Terms A to Z: 1,400+ Definitions Explained Simply

What Are Math Terms?

Math terms are the precise words mathematics uses for its numbers, shapes, operations, relationships and methods, such as quotient, hypotenuse, coefficient and derivative. Each one has a single agreed meaning, which is why a problem that says "find the product" leaves no room for guessing.

Many of them look like ordinary English and mean something narrower. "Odd" is not strange, "mean" is not unkind, and a "product" is not something you buy. Most of the confusion students run into with word problems starts right there.

ES = 0.51: the average strength of the link between language skills and math performance across 12 meta-analyses covering more than 3.6 million participants. Geçici, Geçici and Kaya, Journal of Intelligence, July 2026

How Do You Use This Math Glossary?

Jump straight to a letter: A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · X · Y · Z

• One line per term. Each entry gives the meaning first, then a small example where an example makes the idea click.

• Other names are listed too. "Global maximum" sits under G and points you to the absolute maximum, so you find a term under whichever name your textbook uses.

• Linked terms go deeper. A linked term opens a full lesson with worked examples, diagrams and common mistakes.

Which Math Terms Should You Learn First?

Start with the words your current math topic uses every day. The table groups the most-used math terms by branch, with the school stage where they usually appear.

Students who meet "perimeter" and "area" in the same week often swap them on the next quiz. Learning them as a pair, with one drawing that shows the fence around a garden and the grass inside it, keeps them apart.

Which Everyday Words Mean Something Different In Math?

These are the math terms that trip readers up most, because the everyday meaning pulls against the math meaning.

Which Math Definitions Are Often Stated Wrongly?

Some short definitions get copied from list to list with a mistake built in. These are the ones worth correcting before they stick.

• "Whole numbers are the positive integers." Whole numbers include zero: 0, 1, 2, 3 and so on.

• "1 is a prime number." A prime has exactly two factors. The number 1 has only one, so it is neither prime nor composite.

• "Irrational numbers have digits that keep repeating." It is the other way round. An irrational number's decimal never ends and never repeats. A repeating decimal such as $0.333\ldots$ equals $\frac{1}{3}$, which is rational.

• "The 4th root of 3 is 81." Reversed. Since $3^{4} = 81$, the 4th root of 81 is 3.

• "A rhombus has no right angles." A rhombus only needs four equal sides. A square has four equal sides and four right angles, so every square is a rhombus.

• "Circumference is the distance around a circle or a square." Circumference belongs to circles. The distance around a square, or any polygon, is its perimeter.

• "A mixed number is a whole number with a fraction or a decimal, like 3.5." A mixed number pairs a whole number with a fraction, such as $3\frac{1}{2}$. The decimal 3.5 has the same value but is not a mixed number.

• "A tangent touches a curve at only one point." That holds for circles, not every curve. The tangent to $y = x^{3}$ at $(1, 1)$ is $y = 3x - 2$, and it crosses the curve again at $(-2, -8)$.

• "A yard is about 91.5 centimeters." A yard is exactly 0.9144 meters, which is 91.44 centimeters, fixed by international agreement in 1959 (NIST).

Wait, one more deserves a note, because textbooks genuinely disagree: the trapezoid. Some books say "exactly one pair of parallel sides", others say "at least one pair", which makes every parallelogram a trapezoid. Check which definition your course uses before a test.

Math Terms Starting With A

• AAS Congruence: A rule for proving two triangles congruent: if two angles and a side that is not between them match the corresponding parts of another triangle, the triangles are congruent.

• AA Similarity: A rule stating that if two angles of one triangle equal two angles of another triangle, the two triangles are similar. The third angles then match automatically.

• Abacus: A counting frame with beads that slide along rods or wires, used for adding, subtracting and other calculations. Each rod stands for a place value such as ones, tens or hundreds.

• Abscissa: The x-coordinate of a point, which gives its horizontal position. For the point $(4, -2)$, the abscissa is 4.

• Absolute Convergence: A series $\sum a_{n}$ converges absolutely when the series of absolute values $\sum |a_{n}|$ also converges. Every absolutely convergent series is convergent, but not every convergent series is absolutely convergent.

• Absolutely Convergent: Describes a series with absolute convergence: the series still converges after every term is replaced by its absolute value.

• Absolute Maximum: The greatest value a function reaches on its whole domain or on a stated interval. For $f(x) = -x^{2}$, the absolute maximum is 0, reached at $x = 0$.

• Absolute Minimum: The smallest value a function reaches on its whole domain or on a stated interval. For $f(x) = x^{2}$, the absolute minimum is 0, reached at $x = 0$.

• Absolute Value: The distance of a number from zero on the number line, so it is never negative. For example, $|-5| = 5$ and $|5| = 5$.

• Absolute Value of a Complex Number: The distance of the complex number $a + bi$ from the origin in the complex plane, equal to $\sqrt{a^{2} + b^{2}}$. For example, $|3 + 4i| = 5$.

• Absolute Value Rules: Facts used to simplify and solve with absolute values, such as $|ab| = |a| \cdot |b|$ and $|a + b| \leq |a| + |b|$. If $|x| = 3$, then $x = 3$ or $x = -3$.

• Acceleration: The rate at which velocity changes over time. In calculus, acceleration is the derivative of velocity, which makes it the second derivative of position.

• Accuracy: How close a measurement or calculated value is to the true value. It differs from precision, which describes how fine or how repeatable a measurement is.

• Acute Angle: An angle that measures more than $0^{\circ}$ and less than $90^{\circ}$. For example, a $45^{\circ}$ angle is acute.

• Acute Triangle: A triangle in which all three angles are acute, meaning each is less than $90^{\circ}$. Every equilateral triangle is an acute triangle.

• Addend: Any of the numbers being added together. In $3 + 5 = 8$, the addends are 3 and 5.

• Addition: The operation of combining two or more numbers to find their total, called the sum. For example, $7 + 6 = 13$.

• Addition Rule: In probability, the rule $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. When the two events cannot happen together, the last term is 0.

• Additive Inverse of a Matrix: The matrix formed by changing the sign of every entry of a given matrix. Adding a matrix to its additive inverse gives the zero matrix.

• Additive Inverse of a Number: The number that gives 0 when added to the original number. The additive inverse of 7 is $-7$, and the additive inverse of $-2$ is 2.

• Additive Property of Equality: Adding the same number to both sides of an equation keeps the equation true. If $x - 4 = 10$, adding 4 to both sides gives $x = 14$.

• Adjacent: Next to each other, sharing a side or a vertex. In a right triangle, the adjacent side of an angle is the leg that forms that angle, not the hypotenuse.

• Adjacent Angles: Two angles that share a vertex and a common arm but do not overlap. Two angles sitting side by side on a straight line are an example.

• Adjoint, Classical: For a square matrix, the transpose of its matrix of cofactors, also called the adjugate. It appears in the inverse formula $A^{-1} = \frac{1}{\det A} \operatorname{adj}(A)$.

• Adjugate: The transpose of the cofactor matrix of a square matrix. Dividing the adjugate by the determinant gives the inverse, when the determinant is not zero. See also Adjoint, Classical.

• Affine Transformation: A transformation made of a linear transformation followed by a translation. It keeps straight lines straight and parallel lines parallel, and includes rotations, reflections, stretches and shifts.

• Aleph Null: The size of the set of natural numbers, written $\aleph_{0}$. It is the smallest infinite cardinal number, and any set that can be listed in an endless sequence has this size.

• Algebra: The branch of mathematics that uses letters and symbols to stand for numbers and to describe relationships between them. Solving $2x + 3 = 11$ for $x$ is an algebra problem.

• Algebraic Numbers: Numbers that are roots of a polynomial equation with integer coefficients (not all zero). For example, $\sqrt{2}$ is algebraic because it solves $x^{2} - 2 = 0$, while $\pi$ is not.

• Algorism: An older word for the decimal number system that uses the digits 0 to 9, and for doing arithmetic with it. The word algorithm grew out of it.

• Algorithm: A step-by-step set of instructions that solves a problem in a finite number of steps. Long division is an algorithm for dividing numbers.

• Alpha (α): The first letter of the Greek alphabet. In math it often names an angle, and in statistics it stands for the significance level of a test.

• Alternate Angles: A pair of angles on opposite sides of a transversal that cuts two lines. When the two lines are parallel, alternate angles are equal.

• Alternate Exterior Angles: Two angles outside a pair of lines and on opposite sides of the transversal that crosses them. If the lines are parallel, these angles are equal.

• Alternate Interior Angles: Two angles between a pair of lines and on opposite sides of the transversal that crosses them. If the lines are parallel, these angles are equal.

• Alternating Series: A series whose terms switch between positive and negative. For example, $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ is an alternating series.

• Alternating Series Remainder: The error made when an alternating series is stopped after a certain term. If the series passes the alternating series test, this error is no larger than the absolute value of the first term left out.

• Alternating Series Test: A test for convergence: an alternating series converges if the absolute values of its terms keep decreasing and approach 0. It does not say whether the convergence is absolute.

• Altitude: The perpendicular height of a shape, or the segment that shows that height. It is measured at a right angle to the base.

• Altitude of a Cone: The perpendicular distance from the tip (apex) of a cone to the plane of its circular base.

• Altitude of a Cylinder: The perpendicular distance between the two circular bases of a cylinder.

• Altitude of a Parallelogram: The perpendicular distance between a chosen base of a parallelogram and the side opposite it. Area equals base times this altitude.

• Altitude of a Prism: The perpendicular distance between the two bases of a prism. Volume equals base area times this altitude.

• Altitude of a Pyramid: The perpendicular distance from the apex of a pyramid to the plane of its base.

• Altitude of a Trapezoid: The perpendicular distance between the two parallel sides of a trapezoid.

• Altitude of a Triangle: A segment from a vertex that meets the opposite side, or the line through it, at a right angle. Every triangle has three altitudes, and their lines meet at one point.

• Amplitude: For a periodic graph with a highest and lowest value, such as a sine or cosine wave, half the distance between those two values. The graph of $y = 3\sin x$ has amplitude 3.

• Analytic Geometry: The study of geometry using a coordinate system, so shapes can be described by equations. It is also called coordinate geometry.

• Analytic Methods: Solving a problem with algebra and exact calculation instead of reading the answer off a graph. "Solve analytically" usually means working it out by hand, without a graphing calculator.

• Angle: The figure formed by two rays that share a common endpoint, called the vertex. Angles are measured in degrees or radians.

• Angle Bisector: A ray or line that divides an angle into two equal angles. The bisector of a $60^{\circ}$ angle makes two $30^{\circ}$ angles.

• Angle of Depression: The angle between the horizontal line of sight and the line down to an object below the observer, such as a boat seen from a cliff.

• Angle of Elevation: The angle between the horizontal line of sight and the line up to an object above the observer, such as the top of a tree.

• Angle of Inclination of a Line: The angle a line makes with the positive x-axis, measured counterclockwise, from $0^{\circ}$ up to but not including $180^{\circ}$. The slope of the line equals the tangent of this angle.

• Annulus: The ring-shaped region between two circles that share a center. Its area is $\pi (R^{2} - r^{2})$, where $R$ and $r$ are the outer and inner radii.

• Anticlockwise: Turning in the direction opposite to the hands of a clock, the same as counterclockwise. Positive angles in the coordinate plane are measured anticlockwise.

• Antiderivative of a Function: A function whose derivative is the given function. Both $x^{2}$ and $x^{2} + 5$ are antiderivatives of $2x$, since any constant can be added.

• Antipodal Points: Two points on a sphere (or circle) that are exactly opposite each other, so the line joining them passes through the center. The North and South Poles are antipodal.

• Apex: The highest point or tip of a shape, such as the point of a cone or the vertex of a pyramid opposite its base.

• Apothem: The segment from the center of a regular polygon to the midpoint of one side, meeting that side at a right angle. Its length is used in the polygon's area formula.

• Approximation by Differentials: Estimating a function's value near a known point using its derivative: $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$. It works well when $\Delta x$ is small.

• ARC: Short for average rate of change, the change in a function's output divided by the change in its input over an interval, $\frac{f(b) - f(a)}{b - a}$.

• arccos: The inverse cosine function, which gives the angle between 0 and $\pi$ whose cosine is a given number. For example, $\arccos(0) = \frac{\pi}{2}$.

• Arccos: Another way of writing arccos, the inverse cosine function that returns the angle whose cosine is a given number.

• arccosec: The inverse cosecant function, which gives an angle whose cosecant is a given number. It is defined for inputs $x$ with $|x| \geq 1$.

• Arccosec: Another way of writing arccosec, the inverse cosecant function that returns an angle whose cosecant is a given number.

• arccot: The inverse cotangent function, which gives an angle whose cotangent is a given number. Books differ slightly on which range of angles it returns.

• Arccot: Another way of writing arccot, the inverse cotangent function that returns an angle whose cotangent is a given number.

• arccsc: Another name for arccosec, the inverse cosecant function that returns an angle whose cosecant is a given number.

• Arccsc: Another name for arccosec, the inverse cosecant function, written with a capital letter.

• Arc Length of a Curve: The distance measured along a curve between two points. For $y = f(x)$ from $a$ to $b$, it equals $\int_{a}^{b} \sqrt{1 + (f'(x))^{2}} , dx$.

• Arc of a Circle: A part of a circle's circumference between two points on it. An arc with central angle $\theta$ in radians has length $r\theta$.

• arcsec: The inverse secant function, which gives an angle whose secant is a given number. It is defined for inputs $x$ with $|x| \geq 1$.

• Arcsec: Another way of writing arcsec, the inverse secant function that returns an angle whose secant is a given number.

• arcsin: The inverse sine function, which gives the angle between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ whose sine is a given number. For example, $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$.

• Arcsin: Another way of writing arcsin, the inverse sine function that returns the angle whose sine is a given number.

• arctan: The inverse tangent function, which gives the angle strictly between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ whose tangent is a given number. For example, $\arctan(1) = \frac{\pi}{4}$.

• Arctan: Another way of writing arctan, the inverse tangent function that returns the angle whose tangent is a given number.

• Area: The amount of surface a flat shape covers, measured in square units. A rectangle 3 cm by 4 cm has an area of $12 \text{ cm}^{2}$.

• Area below a Curve: The area of the region between the graph of a function and the x-axis over an interval. When $f(x) \geq 0$, it equals the definite integral $\int_{a}^{b} f(x) , dx$.

• Area between Curves: The area of the region trapped between two graphs over an interval, found by integrating the top function minus the bottom function.

• Area of a Circle: The space inside a circle, given by $A = \pi r^{2}$. A circle of radius 3 has area $9\pi$, about 28.27 square units.

• Area of a Convex Polygon: The space inside a convex polygon. It can be found by splitting the polygon into triangles, or from the vertex coordinates using the shoelace formula.

• Area of a Kite: Half the product of the kite's diagonals, $A = \frac{1}{2} d_{1} d_{2}$. Diagonals of 6 and 8 give an area of 24.

• Area of an Ellipse: Given by $A = \pi a b$, where $a$ and $b$ are the semi-major and semi-minor axes. When $a = b$, this becomes the circle formula.

• Area of an Equilateral Triangle: For side length $s$, the area is $\frac{\sqrt{3}}{4} s^{2}$. A side of 2 gives an area of $\sqrt{3}$.

• Area of a Parabolic Segment: The region between a parabola and a chord across it. When the chord is perpendicular to the axis, the area is $\frac{2}{3}$ of the base times the height of the segment.

• Area of a Parallelogram: Base times the perpendicular height, $A = bh$. A base of 5 and height of 3 give an area of 15.

• Area of a Rectangle: Length times width, $A = lw$. A rectangle 7 by 2 has an area of 14 square units.

• Area of a Regular Polygon: Half the apothem times the perimeter, $A = \frac{1}{2} a P$. This works for any regular polygon, such as a regular hexagon.

• Area of a Rhombus: Half the product of the diagonals, $A = \frac{1}{2} d_{1} d_{2}$, or base times height. Diagonals of 4 and 10 give an area of 20.

• Area of a Sector of a Circle: The area of a pie-shaped slice of a circle. With central angle $\theta$ in radians, it is $\frac{1}{2} r^{2} \theta$, or $\frac{\theta}{360} \pi r^{2}$ with $\theta$ in degrees.

• Area of a Segment of a Circle: The area between a chord and its arc, found as the sector area minus the triangle area. With $\theta$ in radians, it equals $\frac{1}{2} r^{2} (\theta - \sin\theta)$.

• Area of a Trapezoid: Half the sum of the parallel sides times the height, $A = \frac{1}{2}(b_{1} + b_{2}) h$. Parallel sides of 3 and 5 with height 4 give 16.

• Area of a Triangle: Half the base times the height, $A = \frac{1}{2} bh$. A base of 6 and height of 4 give an area of 12.

• Area under a Curve: Another name for the area below a curve: the region between a graph and the x-axis, found with a definite integral.

• Area Using Parametric Equations: For a curve given by $x(t)$ and $y(t)$, the area is found by integrating $y(t) , x'(t)$ with respect to $t$ over the matching range of $t$.

• Area Using Polar Coordinates: For a curve $r = f(\theta)$, the area swept out between two angles is $\frac{1}{2} \int_{\alpha}^{\beta} r^{2} , d\theta$.

• Argand Plane: A coordinate plane for complex numbers, with the real part on the horizontal axis and the imaginary part on the vertical axis. The number $2 + 3i$ is plotted at $(2, 3)$.

• Argument of a Complex Number: The angle from the positive real axis to the line joining the origin and the complex number. The argument of $1 + i$ is $\frac{\pi}{4}$.

• Argument of a Function: The input value a function acts on. In $\sin(2x)$, the argument is $2x$.

• Argument of a Vector: The angle a vector makes with the positive x-axis, measured counterclockwise. It describes the vector's direction.

• Arithmetic: The branch of mathematics that deals with numbers and the operations of addition, subtraction, multiplication and division.

• Arithmetic Mean: The sum of a set of values divided by how many values there are. The arithmetic mean of 2, 4 and 9 is $\frac{15}{3} = 5$.

• Arithmetic Progression: A sequence in which each term is found by adding the same fixed number, called the common difference, to the previous term. For example, 3, 7, 11, 15 has common difference 4.

• Arithmetic Sequence: Another name for an arithmetic progression: a sequence that grows or shrinks by the same amount each step, such as 5, 8, 11, 14.

• Arithmetic Series: The sum of the terms of an arithmetic sequence. For $n$ terms, it equals $\frac{n}{2}(a_{1} + a_{n})$, so $1 + 2 + \dots + 100 = 5050$.

• Arm of an Angle: Either of the two rays that form an angle, also called a side of the angle.

• Arm of a Right Triangle: Either of the two sides that form the right angle, also called a leg. The third side is the hypotenuse.

• Array: An arrangement of objects or numbers in equal rows and columns. Three rows of 4 dots show that $3 \times 4 = 12$.

• ASA Congruence: A rule for proving two triangles congruent: if two angles and the side between them match the corresponding parts of another triangle, the triangles are congruent.

• Associative: Describes an operation where the grouping does not change the result, such as $(a + b) + c = a + (b + c)$. Addition and multiplication are associative, but subtraction and division are not.

• Asymptote: A line that a curve gets closer and closer to as it heads off toward infinity. The graph of $y = \frac{1}{x}$ has asymptotes $x = 0$ and $y = 0$.

• Attribute: A feature of an object, such as its color, size, shape or number of sides, often used to sort or classify objects.

• Augmented Matrix: A matrix that holds the coefficients of a system of linear equations with an extra column for the constants. It is used to solve the system by row operations.

• Average: A single number that represents the center of a set of data. It usually means the arithmetic mean, though the median and mode are also kinds of average.

• Average Rate of Change: The full name for ARC: how much a function's output changes per unit of input over an interval, equal to the slope of the secant line.

• Average Value of a Function: The height of the rectangle with the same area as the region under a curve over $[a, b]$, given by $\frac{1}{b - a} \int_{a}^{b} f(x) , dx$.

• Axes: The plural of axis. In a coordinate plane, the axes are the horizontal x-axis and the vertical y-axis, which meet at the origin.

• Axiom: A statement accepted as true without proof and used as a starting point for reasoning. For example, one axiom states that exactly one straight line passes through any two points.

• Axis of a Cylinder: The line joining the centers of the two circular bases of a cylinder.

• Axis of Reflection: The line a shape is flipped across in a reflection, also called the mirror line. Each point and its image are the same distance from it.

• Axis of Rotation: The fixed line that a flat shape is turned around in space to create a solid. Spinning a rectangle around one of its sides traces out a cylinder.

• Axis of Symmetry: A line that divides a figure into two halves that are mirror images of each other. A square has four axes of symmetry.

• Axis of Symmetry of a Parabola: The line through the vertex of a parabola that splits it into two mirror halves. For $y = ax^{2} + bx + c$, it is the vertical line $x = -\frac{b}{2a}$.

Math Terms Starting With B

• Back Substitution: A method for finishing a system of equations in triangular form: solve the last equation first, then substitute that value into the equations above, one at a time.

• Bar Graph: A chart that uses rectangular bars of different lengths to compare amounts across categories. Taller or longer bars show larger values.

• Base: A word with several meanings. In geometry, it is a chosen side or face of a shape. In powers, it is the number being multiplied, and in number systems it is how many digits are used.

• Base 10: The place value system that uses the ten digits 0 to 9, where each place is worth ten times the place to its right. It is also called the decimal system.

• Base (Geometry): A side or face of a shape chosen as the bottom, from which height is measured. Prisms and cylinders have two bases, while pyramids and cones have one.

• Base of an Exponential Expression: The number that is multiplied by itself in a power. In $2^{5}$, the base is 2 and the exponent is 5.

• Base of an Isosceles Triangle: The side of an isosceles triangle that is not one of the two equal sides. The two angles touching the base are equal.

• Base of a Trapezoid: Either of the two parallel sides of a trapezoid.

• Base of a Triangle: Any side of a triangle chosen as the bottom, with the height measured perpendicular to it. Any of the three sides can serve as the base.

• Bearing: A direction given as an angle. A three-figure bearing is measured clockwise from north, so east is $090^{\circ}$. A compass bearing such as S $34^{\circ}$ E means $34^{\circ}$ from due south toward east.

• BEDMAS: A memory aid for the order of operations: Brackets, Exponents, Division and Multiplication, Addition and Subtraction. Division and multiplication share one rank and are done left to right, as are addition and subtraction.

• Bell Curve: The symmetric, bell-shaped graph of a normal distribution. Most values sit near the mean in the middle, with fewer values toward either end.

• Bernoulli Trials: Repeated independent experiments, each with only two outcomes (success or failure) and the same probability of success every time. Flipping a coin 10 times is an example.

• Beta (β): The second letter of the Greek alphabet. In math it often names an angle, and in statistics it stands for the probability of a Type II error.

• Between: Point $B$ lies between points $A$ and $C$ when all three are on one line and $AB + BC = AC$.

• Biconditional: A statement of the form "P if and only if Q", meaning P and Q are either both true or both false. It combines a conditional with its converse.

• Binomial: A polynomial with exactly two terms. For example, $x + 3$ and $4a^{2} - b$ are binomials.

• Binomial Coefficients: The numbers that appear in the expansion of $(a + b)^{n}$, written $\binom{n}{k}$. Each one also counts the ways to choose $k$ items from $n$, so $\binom{4}{2} = 6$.

• Binomial Coefficients in Pascal's Triangle: Row $n$ of Pascal's triangle lists the binomial coefficients $\binom{n}{0}$ to $\binom{n}{n}$. Row 3 is 1, 3, 3, 1, matching $(a + b)^{3}$.

• Binomial Probability Formula: The probability of exactly $k$ successes in $n$ Bernoulli trials, $P = \binom{n}{k} p^{k} (1 - p)^{n - k}$, where $p$ is the chance of success on each trial.

• Binomial Theorem: A formula for expanding any power of a binomial, $(a + b)^{n}$, using binomial coefficients. For example, $(a + b)^{2} = a^{2} + 2ab + b^{2}$.

• Bisect: To divide something into two equal parts. The midpoint of a segment bisects it.

• Bisector: A line, ray or segment that divides a segment or an angle into two equal parts. A perpendicular bisector also meets the segment at a right angle.

• Boundary Value Problem: A differential equation together with conditions the solution must meet at two or more points, often the ends of an interval. It differs from an initial value problem, where conditions are given at one point.

• Bounded Function: A function whose values never go above some fixed number or below some other fixed number. The function $\sin x$ is bounded because it always stays between $-1$ and 1.

• Bounded Sequence: A sequence whose terms all lie between two fixed numbers. The sequence $1, \frac{1}{2}, \frac{1}{3}, \dots$ is bounded between 0 and 1.

• Bounded Set of Geometric Points: A set of points that fits entirely inside some circle, or sphere in space, of finite radius. A line segment is bounded, but a whole line is not.

• Bounded Set of Numbers: A set of numbers that has both an upper bound and a lower bound. The set of all numbers between 2 and 5 is bounded.

• Bounds of Integration: The numbers $a$ and $b$ in a definite integral $\int_{a}^{b} f(x) , dx$, which mark where the integration starts and stops. They are also called limits of integration.

• Box: A solid with six rectangular faces, also called a rectangular prism or cuboid. A shoebox is a common example.

• Box and Whisker Plot: A graph that shows data using five values: minimum, lower quartile, median, upper quartile and maximum. The box covers the middle half of the data, and the whiskers reach the extremes.

• Boxplot: A graph with a box from the lower to upper quartile and a line at the median. In a modified boxplot, whiskers stop at the last values that are not outliers, and outliers are marked as separate dots. See also Box and Whisker Plot.

• Braces: The curly symbols ${ }$, used to list the members of a set, as in ${1, 2, 3}$, and sometimes to group terms.

• Brachistochrone: The curve along which an object slides without friction, under gravity alone, from one point to a lower point in the shortest time. The answer is part of a cycloid, not a straight line.

• Brackets: The square symbols $[ ]$, used to group parts of an expression, often around parentheses, as in $2[3(4 + 1) - 5]$. In interval notation, a square bracket means the endpoint is included.

Math Terms Starting With C

• Calculus: The branch of mathematics that studies change and accumulation. Derivatives measure rates of change, and integrals add up quantities such as the area under a curve.

• Capacity: The amount a container can hold, usually measured in liters, milliliters, gallons or cups. For example, a bottle with a capacity of 2 liters holds 2 liters of water when full.

• Cardinality: The number of elements in a set, written $n(A)$ or $|A|$. For example, the set ${2, 4, 6}$ has cardinality 3.

• Cardinal Numbers: Numbers that tell how many things are in a group, such as 1, 2, 3 and so on, with 0 used for an empty group. In set theory, cardinal numbers also measure the size of infinite sets.

• Cardioid: A heart-shaped curve traced by a point on a circle as that circle rolls around another circle of the same size. One polar equation for it is $r = 1 + \cos\theta$.

• Cartesian Coordinates: An ordered pair $(x, y)$ that locates a point by its signed distances from two perpendicular axes. In three dimensions, a third number is added, giving $(x, y, z)$.

• Cartesian Form: A way of writing an equation using $x$, $y$ (and $z$) coordinates, such as $2x + 3y = 6$ for a line. For complex numbers, Cartesian form means writing the number as $a + bi$.

• Cartesian Plane: A flat surface formed by a horizontal x-axis and a vertical y-axis that cross at right angles at the origin $(0, 0)$. It is used to plot points and graphs.

• Catenary: The U-shaped curve formed by a chain or cable hanging freely between two supports under its own weight. Its equation is $y = a\cosh\left(\frac{x}{a}\right)$, and although it looks like a parabola, it is a different curve.

• Cavalieri’s Principle: If two solids have the same height and their cross-sections at every level have equal areas, then the solids have equal volumes. It explains why a slanted stack of coins has the same volume as a straight one.

• Ceiling Function: The function that rounds a number up to the nearest integer, written $\lceil x \rceil$. For example, $\lceil 2.3 \rceil = 3$ and $\lceil -2.3 \rceil = -2$.

• Center of Mass Formula: Formulas that locate the balance point of a flat shape of uniform density. For a region of area $A$, $\bar{x} = \frac{1}{A} \int x , dA$, the average x-value over the region, and $\bar{y}$ is found the same way.

• Center of Rotation: The fixed point that a figure turns around during a rotation. Every other point stays the same distance from this center as it moves.

• Centers of a Triangle: Special points built from a triangle's lines: the centroid (where medians meet), circumcenter (perpendicular bisectors), incenter (angle bisectors) and orthocenter (altitudes).

• Centimeter: A metric unit of length equal to one hundredth of a meter, written cm. There are 100 centimeters in a meter, and 1 inch is exactly 2.54 centimeters.

• Central Angle: An angle whose vertex is at the center of a circle and whose sides are two radii. Its measure equals the measure of the arc it cuts off.

• Centroid: The point where the three medians of a triangle meet, dividing each median in the ratio 2 to 1 from the vertex. More generally, it is the balance point (center of mass) of any shape of uniform density.

• Centroid Formula: The rule that finds a triangle's centroid by averaging the coordinates of its vertices: for vertices $(x_{1}, y_{1})$, $(x_{2}, y_{2})$ and $(x_{3}, y_{3})$, it is $\left(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}\right)$.

• Ceva’s Theorem: For triangle $ABC$ with points $D$, $E$ and $F$ on sides $BC$, $CA$ and $AB$, the lines $AD$, $BE$ and $CF$ meet at one point exactly when $\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1$.

• Cevian: A line segment from a vertex of a triangle to a point on the opposite side, or on that side extended. Medians, altitudes and angle bisectors are all cevians.

• Chain Rule: A derivative rule for composite functions: if $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$. For example, the derivative of $(3x + 1)^{2}$ is $2(3x + 1) \cdot 3$.

• Change of Base Formula: A rule that rewrites a logarithm in a different base: $\log_{b} x = \frac{\log_{k} x}{\log_{k} b}$. For example, $\log_{2} 8 = \frac{\log 8}{\log 2} = 3$.

• Check a Solution: To substitute an answer back into the original equation or problem to confirm it makes the statement true. For example, $x = 4$ checks in $2x + 1 = 9$ because $2(4) + 1 = 9$.

• Chi (χ): The 22nd letter of the Greek alphabet. In statistics, $\chi^{2}$ (chi-square) names a test and a distribution used to compare observed counts with expected counts.

• Chord: A line segment whose endpoints both lie on a circle or another curve. The longest chord of a circle is a diameter.

• Circle: The set of all points in a plane that are the same distance, called the radius, from a fixed point called the center. A circle with radius $r$ has area $\pi r^{2}$.

• Circle Identities: The trigonometric identities that come from the unit circle equation $x^{2} + y^{2} = 1$, mainly $\sin^{2}\theta + \cos^{2}\theta = 1$. They are also called the Pythagorean identities.

• Circle Trig Definitions: Definitions of the trig functions using a point $(x, y)$ on a circle of radius $r$ centered at the origin: $\sin\theta = \frac{y}{r}$, $\cos\theta = \frac{x}{r}$ and $\tan\theta = \frac{y}{x}$. They work for any angle, not only acute ones.

• Circular Cone: A cone whose base is a circle. When the apex sits directly above the center of the base, it is a right circular cone, the usual cone of school geometry.

• Circular Cylinder: A cylinder whose two bases are congruent circles. Its volume is $\pi r^{2} h$, where $h$ is the perpendicular height, and in a right circular cylinder the curved side meets the bases at right angles.

• Circular Functions: Another name for the trigonometric functions (sine, cosine, tangent and the rest), so called because they can be defined using points on the unit circle.

• Circumcenter: The point where the perpendicular bisectors of a triangle's sides meet. It is the same distance from all three vertices and is the center of the circumcircle.

• Circumcircle: The circle that passes through every vertex of a polygon, when such a circle exists. Every triangle has exactly one, centered at its circumcenter.

• Circumference: The distance around a circle, found with $C = 2\pi r$ or $C = \pi d$. For example, a circle with radius 5 cm has circumference $10\pi \approx 31.42$ cm.

• Circumscribable: Describes a polygon that can have a circle drawn through all of its vertices. Every triangle is circumscribable, and a quadrilateral is circumscribable exactly when its opposite angles add up to $180^{\circ}$.

• Circumscribed: Drawn around another figure so that it encloses it and touches it without cutting into it. A circle through all the vertices of a polygon is circumscribed about it, and a polygon whose sides all touch a circle is circumscribed about the circle.

• Circumscribed Circle: Another name for the circumcircle: the circle that passes through every vertex of a polygon.

• Cis: Shorthand used with complex numbers, where $\text{cis},\theta = \cos\theta + i\sin\theta$. It lets a complex number be written in polar form as $r,\text{cis},\theta$.

• Classical Adjoint: Another name for the adjugate of a square matrix: the transpose of its cofactor matrix, used to find the inverse.

• Clockwise: Turning in the same direction as the hands of a clock, from the top toward the right. In math, clockwise rotations are usually measured as negative angles.

• Closed Interval: The set of all real numbers between two endpoints, including both endpoints, written $[a, b]$. For example, $[1, 3]$ contains 1, 3 and every number in between.

• Coefficient: The number multiplying a variable in a term. In $3y$, the coefficient is 3, and in $-x^{2}$, the coefficient is $-1$.

• Coefficient Matrix: The matrix formed from only the coefficients of the variables in a system of linear equations. For $2x + 3y = 5$ and $x - y = 1$, its rows are 2, 3 and 1, $-1$.

• Cofactor: For an entry of a square matrix, the minor of that entry (the determinant left after deleting its row and column) multiplied by $(-1)^{i + j}$, where $i$ and $j$ are the row and column numbers. Cofactors are used to compute determinants.

• Cofactor Matrix: The matrix made by replacing each entry of a square matrix with its cofactor. Its transpose is the adjugate, which is used to find the inverse matrix.

• Cofunction Identities: Identities showing that a trig function of an angle equals its cofunction of the complementary angle, such as $\sin\theta = \cos(90^{\circ} - \theta)$ and $\tan\theta = \cot(90^{\circ} - \theta)$.

• Coincident: Lying exactly on top of each other. Coincident lines are the same line and share every point, as with $y = 2x + 1$ and $2y = 4x + 2$.

• Collinear: Describes points that lie on the same straight line. For example, $(0, 0)$, $(1, 2)$ and $(2, 4)$ are collinear.

• Column of a Matrix: A vertical line of entries in a matrix. A matrix with $m$ rows and $n$ columns has $n$ columns, each holding $m$ entries.

• Combination: A selection of items where order does not matter. Choosing 2 letters from A, B and C gives 3 combinations: AB, AC and BC.

• Combination Formula: The count of ways to choose $r$ items from $n$ when order does not matter, equal to n factorial divided by r factorial times (n minus r) factorial. For example, $\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10$.

• Combinatorics: The branch of mathematics that studies counting, arranging and selecting objects, including permutations, combinations and the structure of networks called graphs.

• Common Factor: A number or expression that divides two or more numbers or expressions exactly. The common factors of 12 and 18 are 1, 2, 3 and 6.

• Common Logarithm: A logarithm with base 10, written $\log x$ or $\log_{10} x$. For example, $\log 1000 = 3$ because $10^{3} = 1000$.

• Common Ratio: The fixed number that each term of a geometric sequence is multiplied by to get the next term. In 3, 6, 12, 24, ... the common ratio is 2.

• Commutative: Describes an operation where changing the order of the two inputs does not change the result. Addition and multiplication are commutative ($3 + 5 = 5 + 3$), but subtraction and division are not.

• Comparison Test: A test for series with nonnegative terms. If each term is at most the matching term of a known convergent series, the series converges, and if each term is at least the matching term of a known divergent series, it diverges.

• Compatible Matrices: Two matrices whose sizes allow an operation between them. For the product $AB$, the number of columns of $A$ must equal the number of rows of $B$, and for addition the sizes must match.

• Complementary Angles: Two angles whose measures add up to $90^{\circ}$. For example, $35^{\circ}$ and $55^{\circ}$ are complementary.

• Complement of an Angle: The angle that must be added to a given angle to make $90^{\circ}$. The complement of $30^{\circ}$ is $60^{\circ}$.

• Complement of an Event: All the outcomes in which an event does not happen, with probability $P(A') = 1 - P(A)$. If the chance of rain is 0.3, the chance of no rain is 0.7.

• Complement of a Set: The set of all elements of the universal set that are not in the given set, written $A'$ or $A^{c}$. If $U = {1, 2, 3, 4}$ and $A = {1, 2}$, then $A' = {3, 4}$.

• Complex Conjugate: The complex number formed by changing the sign of the imaginary part. The conjugate of $3 + 4i$ is $3 - 4i$, and their product, 25, is a real number.

• Complex Fraction: A fraction whose numerator, denominator or both contain a fraction, such as $\frac{1/2}{3/4}$, which simplifies to $\frac{2}{3}$.

• Complex Number Formulas: Key rules for working with complex numbers, such as $i^{2} = -1$, $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$ and the modulus $|a + bi| = \sqrt{a^{2} + b^{2}}$.

• Complex Numbers: Numbers of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit with $i^{2} = -1$. Every real number is a complex number with $b = 0$.

• Complex Plane: Another name for the Argand plane: a coordinate plane whose horizontal axis shows the real part and vertical axis the imaginary part of complex numbers.

• Composite: In number work, describes a whole number greater than 1 with more than two factors, such as 9. In functions, a composite function is one function applied to the output of another.

• Composite Number: A whole number greater than 1 that has more than two factors. For example, 12 is composite (factors 1, 2, 3, 4, 6 and 12), while 1 is neither prime nor composite.

• Composition: Combining two functions by using the output of one as the input of the other, written $(f \circ g)(x) = f(g(x))$. If $f(x) = x + 1$ and $g(x) = 2x$, then $f(g(3)) = 7$.

• Compounded: Used for compound interest: interest added to the balance at set periods, so that later interest is also earned on earlier interest.

• Compounded Continuously: Interest that is compounded at every instant rather than at set periods. The amount after $t$ years is $A = Pe^{rt}$, where $P$ is the starting amount and $r$ is the yearly rate as a decimal.

• Compound Fraction: Another name for a complex fraction: a fraction with a fraction in its numerator, denominator or both.

• Compound Inequality: Two inequalities joined by "and" or "or". For example, $-2 < x \leq 5$ means $x > -2$ and $x \leq 5$ at the same time.

• Compound Interest: Interest calculated on the starting amount plus all interest already earned. The formula is $A = P\left(1 + \frac{r}{n}\right)^{nt}$ for $n$ compounding periods per year over $t$ years.

• Compression: A transformation that squeezes a graph or figure toward an axis or a point. For example, $y = \frac{1}{2}f(x)$ is a vertical compression of $y = f(x)$ by a factor of $\frac{1}{2}$.

• Compression of a Geometric Figure: A transformation that shrinks a figure by multiplying distances by a scale factor between 0 and 1. When all directions shrink equally, the result is a smaller figure similar to the original.

• Compression of a Graph: A transformation that squeezes a graph toward an axis. The graph of $y = af(x)$ with $0 < a < 1$ is compressed vertically, and $y = f(bx)$ with $b > 1$ is compressed horizontally.

• Compute: To find a numerical answer or result by calculating. For example, to compute $4 \times 7$ is to work out that it equals 28.

• Concave: In geometry, a polygon is concave if at least one interior angle is greater than $180^{\circ}$, so part of it caves inward. In calculus, a graph is described as concave up or concave down depending on how it bends.

• Concave Down: Describes part of a graph that bends downward like an upside-down bowl, where the slope is decreasing. The graph is concave down wherever the second derivative is negative, as with $y = -x^{2}$.

• Concave Up: Describes part of a graph that bends upward like a bowl, where the slope is increasing. The graph is concave up wherever the second derivative is positive, as with $y = x^{2}$.

• Concentric: Describes circles or other shapes that share the same center. The rings of a dartboard are concentric circles.

• Conclusion: The "then" part of an if-then statement, or the final statement reached in a proof. In "If a number is even, then it is divisible by 2," the conclusion is "it is divisible by 2."

• Concurrent: Describes three or more lines that pass through a single common point. The three medians of any triangle are concurrent at the centroid.

• Conditional: A statement of the form "if $p$, then $q$," written $p \rightarrow q$. It is false only when $p$ is true and $q$ is false.

• Conditional Convergence: The behavior of a series that converges while the series of the absolute values of its terms diverges. For example, $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ converges, but $1 + \frac{1}{2} + \frac{1}{3} + \cdots$ does not.

• Conditional Equation: An equation that is true for some values of the variable but not all. For example, $x + 2 = 5$ is true only when $x = 3$.

• Conditional Inequality: An inequality that is true for some values of the variable but not all. For example, $x + 1 > 4$ holds only when $x > 3$.

• Conditional Probability: The probability that an event happens given that another event has happened, written $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ when $P(B) > 0$.

• Cone: A solid with a flat base, usually a circle, that narrows smoothly to a single point called the apex. Its volume is one third of base area times height, which is $\frac{1}{3}\pi r^{2} h$ for a circular base.

• Cone Angle: The angle of the paper sector left over after the takeout angle is cut away, so the paper rolls into a right circular cone. It equals $\frac{360^{\circ} r}{s}$, with base radius $r$ and slant height $s$.

• Congruence: The relationship between figures that have exactly the same shape and size, shown by the symbol $\cong$. In number theory, two integers are congruent if they leave the same remainder when divided by a given number. See also Congruence Tests for Triangles.

• Congruence Tests for Triangles: Rules that prove two triangles congruent from only some matching parts: SSS, SAS, ASA, AAS and, for right triangles, HL (hypotenuse-leg). SSA and AAA are not valid tests.

• Congruent: Having exactly the same shape and size. Two figures are congruent if one can be moved onto the other by slides, turns and flips, so matching sides and angles are equal.

• Conic Sections: The curves formed when a plane cuts a double cone: circles, ellipses, parabolas and hyperbolas. Each can be written as a second-degree equation in $x$ and $y$.

• Conjecture: A statement believed to be true, based on patterns or evidence, that has not yet been proved. Once it is proved, it becomes a theorem.

• Conjugate Pair Theorem: A theorem stating that if a polynomial has real coefficients and $a + bi$ is a root, then its conjugate $a - bi$ is also a root. So the non-real roots of such polynomials come in pairs.

• Conjugates: Pairs of expressions of the form $a + b$ and $a - b$, often with a square root or $i$ in the second part, such as $3 + \sqrt{2}$ and $3 - \sqrt{2}$. Multiplying them gives $a^{2} - b^{2}$, which removes the root.

• Conjunction: A compound statement that joins two statements with "and," written $p \wedge q$. It is true only when both parts are true.

• Consecutive Interior Angles: When a transversal crosses two lines, the pair of angles on the same side of the transversal and between the two lines. If the lines are parallel, these angles add up to $180^{\circ}$.

• Consistent System of Equations: A system of equations that has at least one solution. It may have exactly one solution, where lines cross, or infinitely many, where lines coincide.

• Constant: A value that does not change, such as 7, $-2$ or $\pi$. In $3x + 5$, the number 5 is a constant.

• Constant Function: A function that gives the same output for every input, such as $f(x) = 4$. Its graph is a horizontal line.

• Constant Term: The term in an expression or polynomial that has no variable. In $x^{2} + 3x - 7$, the constant term is $-7$.

• Continued Sum: A sum of a list of terms, usually written in compact form with sigma notation. For example, $\sum_{k=1}^{4} k = 1 + 2 + 3 + 4 = 10$.

• Continuous: Without breaks, jumps or holes. A continuous set of numbers, such as an interval, has no gaps, unlike the integers, and a continuous graph can be drawn without lifting the pencil.

• Continuous Compounding: Another name for interest compounded continuously: growth at every instant, given by $A = Pe^{rt}$.

• Continuous Function: A function whose graph has no breaks, jumps or holes on its domain. At each point $a$ of the domain, $\lim_{x \to a} f(x) = f(a)$.

• Continuously Compounded Interest: Another name for interest compounded continuously, where the amount grows at every instant according to $A = Pe^{rt}$.

• Continuously Differentiable Function: A function that has a derivative at every point of its domain, where that derivative is itself continuous. Smooth curves such as $y = x^{3}$ are examples.

• Contraction: Another name for a compression: a transformation that squeezes a graph or figure toward an axis or a point.

• Contraction of a Geometric Figure: Another name for the compression of a geometric figure: shrinking it by a scale factor between 0 and 1.

• Contrapositive: The statement formed by swapping and negating both parts of a conditional: the contrapositive of "if $p$, then $q$" is "if not $q$, then not $p$." It is true exactly when the original statement is true.

• Converge: To approach a single finite value. A sequence or series converges if its terms, or its partial sums, get closer and closer to one limit, as $\frac{1}{n}$ approaches 0.

• Converge Absolutely: Another name for absolute convergence: a series converges absolutely when the sum of the absolute values of its terms converges.

• Converge Conditionally: Another name for conditional convergence: a series converges, but the series of its absolute values diverges.

• Convergence Tests: Methods for deciding whether an infinite series converges or diverges, such as the nth-term, comparison, ratio, root, integral and alternating series tests.

• Convergent Sequence: A sequence whose terms approach a single finite limit. For example, $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots$ converges to 1.

• Convergent Series: An infinite series whose partial sums approach a finite limit, called its sum. For example, $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ converges to 1.

• Converse: The statement formed by swapping the two parts of a conditional: the converse of "if $p$, then $q$" is "if $q$, then $p$." A true statement can have a false converse.

• Convex: In geometry, a polygon is convex if every interior angle is less than $180^{\circ}$, so any segment joining two points inside it stays inside. In calculus, a convex function is one that is concave up.

• Coordinate Geometry: Another name for analytic geometry: the study of shapes using coordinates and equations on a coordinate plane.

• Coordinate Plane: Another name for the Cartesian plane: the flat grid formed by a horizontal x-axis and a vertical y-axis, used to plot points.

• Coordinates: The ordered numbers that give a point's position, such as $(3, -2)$, meaning 3 units right along the x-axis and 2 units down. Other systems, such as polar coordinates, use a distance and an angle.

• Coplanar: Describes points, lines or shapes that lie in the same plane. Any three points are always coplanar, but four points may not be.

• Corollary: A result that follows directly from a theorem that has just been proved, with little or no extra work.

• Correlation: A relationship between two variables in which they tend to change together. It is positive when both rise together and negative when one rises as the other falls, and correlation alone does not prove cause.

• Correlation Coefficient: A number $r$ from $-1$ to 1 that measures the strength and direction of a linear relationship between two variables. Values near 1 or $-1$ show a strong relationship, and values near 0 a weak one.

• Corresponding: Matching in position when two figures are compared. In congruent or similar figures, corresponding sides and angles sit in the same relative places, and corresponding angles also appear when a transversal crosses two lines.

• cos: Short for cosine. In a right triangle, $\cos\theta$ equals the adjacent side divided by the hypotenuse, and on the unit circle it is the x-coordinate of the point at angle $\theta$.

• cos-1: Another name for arccos, the inverse cosine: $\cos^{-1} x$ gives the angle whose cosine is $x$, not $\frac{1}{\cos x}$.

• Cos-1: Another name for arccos, written with a capital letter in some books to mean the principal value of the inverse cosine, from 0 to $\pi$.

• cosec: Short for cosecant, the reciprocal of sine: $\csc\theta = \frac{1}{\sin\theta}$. In a right triangle, it equals the hypotenuse divided by the opposite side.

• cosec-1: Another name for arccosec, the inverse cosecant: it gives the angle whose cosecant is $x$, for $|x| \geq 1$.

• Cosec-1: Another name for arccosec, written with a capital letter in some books to mean the principal value of the inverse cosecant.

• cosecant: Another name for cosec: the reciprocal of sine, equal to the hypotenuse over the opposite side in a right triangle.

• cosine: Another name for cos: the trig function equal to the adjacent side over the hypotenuse in a right triangle.

• cot: Short for cotangent, the reciprocal of tangent: $\cot\theta = \frac{\cos\theta}{\sin\theta}$. In a right triangle, it equals the adjacent side divided by the opposite side.

• cot-1: Another name for arccot, the inverse cotangent: $\cot^{-1} x$ gives the angle whose cotangent is $x$.

• Cot-1: Another name for arccot, written with a capital letter in some books to mean the principal value of the inverse cotangent.

• Cotangent: Another name for cot: the reciprocal of tangent, equal to the adjacent side over the opposite side in a right triangle.

• Coterminal: Describes angles in standard position that share the same terminal side. They differ by a multiple of $360^{\circ}$, so $30^{\circ}$, $390^{\circ}$ and $-330^{\circ}$ are coterminal.

• Countable: Describes a set whose elements can be listed one by one and matched with the counting numbers 1, 2, 3, and so on. Finite sets and the integers are countable, but the real numbers are not.

• Countably Infinite: Describes an infinite set that can be matched one-to-one with the counting numbers. The integers and the rational numbers are countably infinite.

• Counterclockwise: Another name for anticlockwise: turning opposite to the hands of a clock, the direction of positive angles in math.

• Counterexample: A single example that shows a general statement is false. The number 2 is a counterexample to the claim "all prime numbers are odd."

• Counting Numbers: The numbers 1, 2, 3, 4 and so on, used for counting. They are the natural numbers (some books also include 0 in the natural numbers).

• CPCFC: Short for "corresponding parts of congruent figures are congruent." Once two figures are shown to be congruent, every pair of matching sides and angles is equal.

• CPCTC: Short for "corresponding parts of congruent triangles are congruent." After two triangles are proved congruent, it justifies that their matching sides and angles are equal.

• Cramer’s Rule: A method for solving a system of linear equations with determinants: each variable equals the determinant of the coefficient matrix with that variable's column replaced by the constants, divided by the coefficient matrix's determinant. It works only when that determinant is not zero.

• Critical Number: A value $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist. Local maximums and minimums inside the domain can occur only at critical numbers.

• Critical Point: A point on a graph where the derivative is zero or does not exist, located at a critical number. For $y = x^{2}$, the critical point is $(0, 0)$.

• Critical Value: In statistics, a cutoff score such as $z^{}$or$t^{}$that separates the region where a hypothesis is rejected from the region where it is not. In calculus, the term also means a critical number or the output$f(c)$ there.

• Cross Product: An operation on two vectors in three dimensions that gives a vector perpendicular to both, written $\mathbf{a} \times \mathbf{b}$. Its length, $|\mathbf{a}||\mathbf{b}|\sin\theta$, equals the area of the parallelogram the vectors form.

• csc: Another name for cosec, the cosecant: the reciprocal of sine, written $\csc\theta = \frac{1}{\sin\theta}$.

• csc-1: Another name for arccosec, the inverse cosecant, also written $\csc^{-1} x$: the angle whose cosecant is $x$.

• Csc-1: Another name for arccosec, with a capital letter used in some books for the principal value of the inverse cosecant.

• ctg: Another name for cot, used in some countries: the cotangent, equal to $\frac{\cos\theta}{\sin\theta}$.

• ctg-1: Another name for arccot, the inverse cotangent, in the ctg notation used in some countries.

• Ctg-1: Another name for arccot, with a capital letter marking the principal value of the inverse cotangent in ctg notation.

• Cube: A solid with six equal square faces, 12 equal edges and 8 vertices, with volume $s^{3}$ for edge length $s$. To cube a number is to raise it to the third power, as in $2^{3} = 8$.

• Cube Root: The number that, multiplied by itself three times, gives the original number, written $\sqrt[3]{x}$. For example, $\sqrt[3]{27} = 3$ and $\sqrt[3]{-8} = -2$.

• Cubic Polynomial: A polynomial of degree 3, such as $ax^{3} + bx^{2} + cx + d$ with $a \neq 0$. With real coefficients, it has at least one real root and at most three.

• Cuboid: Another name for a box, or rectangular prism: a solid with six rectangular faces, like a brick.

• Curly d: The symbol $\partial$, used for partial derivatives. The expression $\frac{\partial f}{\partial x}$ is the rate of change of $f$ with respect to $x$ while all other variables are held fixed.

• Curve: A continuous line, straight or bending, such as a circle, a parabola or the graph of a function. In many math books, a straight line also counts as a curve.

• Curve Sketching: Drawing the graph of a function from key features found with algebra and calculus, such as intercepts, asymptotes, critical points, intervals of increase and decrease, and concavity.

• Cusp: A sharp point on a curve where the two sides meet while heading in the same direction, like the point of $y = x^{2/3}$ at the origin. The derivative does not exist there.

• Cycloid: The curve traced by a point on the rim of a circle as the circle rolls along a straight line. It looks like a row of arches.

• Cylinder: A solid with two parallel, congruent bases joined by a side surface. The bases are usually circles, and a right circular cylinder with radius $r$ and height $h$ has volume $\pi r^{2} h$.

• Cylindrical Shell Method: A calculus method for finding the volume of a solid of revolution using thin cylindrical shells. For a region under $y = f(x)$ from $x = a$ to $x = b$ (with $a \geq 0$) rotated around the y-axis, $V = \int_{a}^{b} 2\pi x f(x),dx$.

Math Terms Starting With D

• Decagon: A polygon with 10 sides and 10 angles. The interior angles of any decagon add up to $1440^{\circ}$, and each angle of a regular decagon measures $144^{\circ}$.

• Deciles: The nine values that split an ordered data set into 10 equal parts, each holding about 10 percent of the data. The first and ninth deciles are the 10th and 90th percentiles, and the fifth decile is the median.

• Decimal: A number written in base 10 that uses a decimal point to show parts smaller than one. For example, $3.25$ means 3 ones, 2 tenths and 5 hundredths.

• Decreasing Function: A function whose output goes down as the input goes up on an interval, so that $a < b$ gives $f(a) > f(b)$. For example, $f(x) = -2x$ is decreasing everywhere.

• Definite Integral: An integral with lower and upper limits, written $\int_{a}^{b} f(x),dx$, that gives a single number. It measures the signed area between the curve and the x-axis from $a$ to $b$, with area below the axis counted as negative.

• Definite Integral Rules: Properties that make definite integrals easier to work with. Examples are $\int_{a}^{a} f(x),dx = 0$, swapping the limits changes the sign, and the integral of a sum equals the sum of the integrals.

• Degenerate: Describes a limiting case of a figure that has collapsed into a simpler form. For example, a triangle whose three vertices lie on one line is a degenerate triangle.

• Degenerate Conic Sections: The shapes formed when a plane slices a double cone through its vertex: a single point, a single line, or a pair of intersecting lines. Some books also count two parallel lines, which come from certain second-degree equations.

• Degree: In geometry, a unit of angle measure, with $360^{\circ}$ in a full turn. In algebra, a term's degree is the sum of its variables' exponents, and a polynomial's degree is its highest term degree. It is also a unit of temperature.

• Degree (angle measure): A unit for measuring angles equal to $\frac{1}{360}$ of a full turn. A right angle measures $90^{\circ}$ and a straight angle measures $180^{\circ}$.

• Degree of a Polynomial: The highest degree of any term in a polynomial, found after like terms are combined. For example, $4x^{3} + 2x - 7$ has degree 3.

• Degree of a Term: The sum of the exponents of all the variables in that term. For example, $5x^{2}y^{3}$ has degree 5, and a nonzero constant such as 7 has degree 0.

• Deleted Neighborhood: An interval around a point with the point itself removed, such as all $x$ with $0 < |x - a| < \delta$. It is used in the definition of a limit, where the value at $a$ itself does not matter.

• Del Operator: The symbol $\nabla$, a vector of partial derivative operators used in multivariable calculus. Applied to a function, it gives the gradient, and it also appears in divergence and curl.

• Delta (δ): The fourth Greek letter. Capital $\Delta$ often means "change in," as in $\Delta x$, and lowercase $\delta$ often names a small positive number in limits.

• De Moivre's Theorem: For any real number $\theta$ and integer $n$, $(\cos\theta + i\sin\theta)^{n} = \cos(n\theta) + i\sin(n\theta)$. It is used to find powers and roots of complex numbers.

• Denominator: The bottom number of a fraction, showing how many equal parts make one whole. In $\frac{3}{4}$, the denominator is 4.

• Dependent Variable: The variable whose value depends on another variable, usually the output. In $y = 2x + 1$, $y$ is the dependent variable because its value is set by $x$.

• Derivative: The instantaneous rate of change of a function, equal to the slope of the tangent line at a point. It is written $f'(x)$ or $\frac{dy}{dx}$. For example, the derivative of $x^{2}$ is $2x$.

• Derivative of a Power Series: Inside its open interval of convergence, a power series can be differentiated term by term, and the new series has the same radius of convergence. For example, differentiating $\sum x^{n}$ gives $\sum n x^{n-1}$.

• Derivative Rules: Standard shortcuts for finding derivatives, including the power, constant multiple, sum, product, quotient and chain rules. For example, the power rule gives $\frac{d}{dx}x^{n} = n x^{n-1}$.

• Descartes' Rule of Signs: A way to bound the number of positive real roots of a polynomial: count sign changes between consecutive nonzero coefficients, and the number of positive roots equals that count or is less by an even number. Using $p(-x)$ does the same for negative roots.

• Determinant: A single number computed from a square matrix. For a $2 \times 2$ matrix with rows $a, b$ and $c, d$, it equals $ad - bc$, and a matrix has an inverse only when its determinant is not zero.

• Diagonal: In geometry, a line segment joining two vertices of a polygon that are not next to each other. In a matrix, the main diagonal is the entries running from the top left to the bottom right. See also Diagonal of a Polygon.

• Diagonal Matrix: A square matrix in which every entry off the main diagonal is zero. The identity matrix is one example.

• Diagonal of a Polygon: A line segment joining two vertices of a polygon that are not neighbors. A polygon with $n$ sides has $\frac{n(n-3)}{2}$ diagonals, so a pentagon has 5.

• Diameter: A chord that passes through the center of a circle or sphere, or the length of such a chord. The diameter is twice the radius.

• Diametrically Opposed: Describes two points on a circle or sphere that sit at opposite ends of a diameter. For example, the North and South Poles are diametrically opposed on a globe.

• Difference: The result of subtracting one number from another. The difference of 12 and 5 is 7.

• Difference Identities: Trigonometric identities for the sine, cosine and tangent of a difference of two angles, such as $\sin(A - B) = \sin A\cos B - \cos A\sin B$ and $\cos(A - B) = \cos A\cos B + \sin A\sin B$.

• Difference Quotient: The expression $\frac{f(x + h) - f(x)}{h}$, which gives the slope of the secant line between two points on a graph. Its limit as $h$ approaches 0 is the derivative.

• Differentiable: Describes a function whose derivative exists at a point, so the graph has a single non-vertical tangent line there. Every differentiable function is continuous, but not the reverse: $|x|$ is continuous at 0 yet not differentiable there because of its sharp corner.

• Differential: A small change in a variable, written $dx$ or $dy$. For $y = f(x)$, the differential is $dy = f'(x),dx$, which estimates how much $y$ changes when $x$ changes slightly.

• Differential Equation: An equation that relates a function to one or more of its derivatives, such as $\frac{dy}{dx} = 3y$. Solving it means finding the function, and such equations model growth, motion and cooling.

• Differentiation: The process of finding a derivative. For example, differentiating $x^{3}$ gives $3x^{2}$.

• Differentiation Rules: Another name for the derivative rules: shortcuts such as the power, product, quotient and chain rules for finding derivatives.

• Digit: Any of the ten symbols 0 to 9 used to write numbers in base 10. The number 507 has three digits: 5, 0 and 7.

• Dihedral Angle: The angle between two planes that meet along a line, such as two faces of a polyhedron sharing an edge. It is measured in a plane perpendicular to that line, and every dihedral angle of a cube is $90^{\circ}$.

• Dilation: A transformation that enlarges or shrinks a figure or graph by a scale factor. A factor greater than 1 stretches it, and a factor between 0 and 1 shrinks it.

• Dilation of a Geometric Figure: A transformation that resizes a figure from a fixed center by a scale factor $k$. Each point moves along a line through the center to $k$ times its original distance, so the image is similar to the original.

• Dilation of a Graph: A stretch or compression of a graph in one direction. For $k > 1$, the graph of $y = k f(x)$ is $k$ times taller than $y = f(x)$, and the graph of $y = f(kx)$ is $k$ times narrower.

• Dimensions: The measurements of an object, such as length, width and height, or the number of independent directions in a space. A line is one-dimensional, a square two-dimensional and a cube three-dimensional.

• Dimensions of a Matrix: The number of rows and columns in a matrix, written rows by columns. A matrix with 2 rows and 3 columns is a $2 \times 3$ matrix.

• Directly Proportional: Another name for direct proportion: two quantities are directly proportional when one is always a fixed multiple of the other, as in $y = kx$.

• Direct Proportion: A relationship in which one quantity is a constant multiple of another, written $y = kx$ with $k \neq 0$. If $x$ doubles, $y$ doubles, as when total cost grows with the number of items bought at a fixed price.

• Directrices of a Hyperbola: The two fixed lines, one for each focus, used to define a hyperbola. For every point on the curve, its distance to a focus divided by its distance to the matching directrix equals the eccentricity, which is greater than 1.

• Directrices of an Ellipse: The two fixed lines outside an ellipse, one for each focus. For every point on the ellipse, its distance to a focus divided by its distance to the matching directrix equals the eccentricity, which is less than 1.

• Directrix of a Parabola: The fixed line used to define a parabola: every point on the parabola is the same distance from the focus as from the directrix. For $x^{2} = 4py$, the directrix is $y = -p$.

• Direct Variation: Another name for direct proportion: $y$ varies directly with $x$ when $y = kx$ for some nonzero constant $k$.

• Discontinuity: A point where a function is not continuous, for example where its graph has a hole, a jump or a vertical asymptote. The graph of $f(x) = \frac{1}{x}$ breaks at $x = 0$.

• Discontinuous Function: A function whose graph has at least one break, such as a hole, a jump or a vertical asymptote, so it cannot be drawn without lifting the pencil. The floor function, which jumps at every integer, is an example.

• Discrete: Made of separate, countable values with gaps between them rather than a continuous range. The number of students in a class is discrete data, while height is continuous.

• Discriminant of a Quadratic: The value $b^{2} - 4ac$ for the quadratic $ax^{2} + bx + c = 0$. If it is positive there are two real roots, if it is zero there is one repeated real root, and if it is negative there are no real roots.

• Disjoint Events: Events that cannot happen at the same time, also called mutually exclusive events. Rolling a 2 and rolling a 5 on one die are disjoint, so $P(A \text{ or } B) = P(A) + P(B)$.

• Disjoint Sets: Sets that have no elements in common, so their intersection is the empty set. For example, ${1, 2}$ and ${3, 4}$ are disjoint.

• Disjunction: In logic, a compound statement that joins two statements with "or," written $p \lor q$. It is true when at least one of the two statements is true.

• Disk: The region made of a circle together with all the points inside it. A closed disk includes the boundary circle, and an open disk does not.

• Disk Method: A way to find the volume of a solid of revolution by slicing it into thin circular disks and adding their volumes with an integral. Rotating $y = f(x)$ about the x-axis from $a$ to $b$ gives $V = \pi\int_{a}^{b} [f(x)]^{2},dx$.

• Distance Formula: A formula for the distance between points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$: $d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$. It comes from the Pythagorean theorem.

• Distance from a Point to a Line: The length of the perpendicular segment from the point to the line. For the line $ax + by + c = 0$ and the point $(x_{1}, y_{1})$, it equals $\frac{|ax_{1} + by_{1} + c|}{\sqrt{a^{2} + b^{2}}}$.

• Distinct: Different from one another. For example, a quadratic with two distinct roots has two roots that are not equal.

• Distribute: To multiply a factor across each term inside parentheses, also called expanding. For example, $3(x + 4)$ becomes $3x + 12$.

• Distributing Rules: The patterns used to multiply out expressions, such as $a(b + c) = ab + ac$ and $(a + b)(c + d) = ac + ad + bc + bd$. They show how to remove parentheses correctly.

• Distributive Property: The rule that multiplying a sum by a number gives the same result as multiplying each addend and then adding: $a(b + c) = ab + ac$. For example, $5 \times 12 = 5 \times 10 + 5 \times 2 = 60$.

• Diverge: To fail to approach a single finite value. A sequence or series diverges if its terms or sums grow without bound, swing back and forth, or otherwise settle on no limit.

• Divergent Sequence: A sequence that does not have a finite limit. For example, $1, 2, 3, 4, \ldots$ and $1, -1, 1, -1, \ldots$ both diverge.

• Divergent Series: An infinite series whose partial sums do not approach a finite limit. The harmonic series $1 + \frac{1}{2} + \frac{1}{3} + \cdots$ diverges even though its terms shrink toward 0.

• Dividend: The number being divided in a division problem. In $20 \div 4 = 5$, the dividend is 20.

• Divisor: The number you divide by. In $20 \div 4 = 5$, the divisor is 4. The word also means a number that divides another exactly, so the positive divisors of 6 are 1, 2, 3 and 6.

• Dodecagon: A polygon with 12 sides and 12 angles. Its interior angles add up to $1800^{\circ}$, and each angle of a regular dodecagon measures $150^{\circ}$.

• Dodecahedron: A polyhedron with 12 faces. The regular dodecahedron, one of the five Platonic solids, has 12 regular pentagon faces, 20 vertices and 30 edges.

• Domain: The set of all input values for which a function is defined. For example, the domain of $f(x) = \sqrt{x}$ in the real numbers is all $x \geq 0$.

• Domain of Definition: The largest set of inputs for which a function's formula makes sense, used when no domain is stated. For $f(x) = \frac{1}{x - 2}$, it is all real numbers except 2.

• Dot Product: A way of multiplying two vectors that gives a single number, found by adding the products of matching components. For example, $(1, 2) \cdot (3, 4) = 3 + 8 = 11$. Two nonzero vectors with a dot product of 0 are perpendicular.

• Double Angle Identities: Trigonometric identities that express functions of $2\theta$ using functions of $\theta$, such as $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = \cos^{2}\theta - \sin^{2}\theta$.

• Double Cone: Two cones placed tip to tip along the same axis, like an hourglass. Slicing a double cone with a plane produces the conic sections.

• Double Number Identities: Another name for the double angle identities: formulas such as $\sin 2\theta = 2\sin\theta\cos\theta$ that rewrite functions of a doubled angle.

• Double Root: A root that appears twice, so its factor is squared. For example, $x^{2} - 6x + 9 = (x - 3)^{2}$ has a double root at $x = 3$, where the graph touches the x-axis without crossing it.

• Doubling Time: The time it takes a quantity that grows exponentially to double in size. For continuous growth $A = A_{0}e^{rt}$, it equals $\frac{\ln 2}{r}$, no matter how large the starting amount is.