Square Root Of 2: Value 1.4142, Proof And Methods

What Is A Square Root?

A square root of a number $n$ is a number $r$ with $r^2 = n$. Both $\sqrt{2}$ and $-\sqrt{2}$ square to 2, so the equation $x^2 = 2$ has two solutions.

The radical sign alone means the principal (non-negative) square root. So $\sqrt{2} \approx 1.4142$, never $-1.4142$.

Does The Square Root Of 2 Really Exist If It Never Ends?

Yes. Draw a unit square and its diagonal is a real, measurable length whose square is 2.

Squaring also traps it. Because $1.4^2 = 1.96$ and $1.5^2 = 2.25$, the number lies between 1.4 and 1.5, and each extra decimal place narrows the gap further without ever closing it.

Where Does $\sqrt{2}$ Show Up, And Why Does It Matter?

The square root of 2 is the length of the diagonal of any square with side 1. The Pythagoras theorem gives it directly: $1^2 + 1^2 = 2$, so the diagonal is $\sqrt{2}$.

Scale the square and the diagonal scales with it. A square of side $s$ has a diagonal of $s\sqrt{2}$, which is the diagonal of square formula used in tiling, carpentry and screen sizes.

The same number sits inside every isosceles right triangle, the 45-45-90 shape. That is why sin 45 degrees equals $\frac{\sqrt{2}}{2} \approx 0.7071$.

Paper uses it too. When Germany standardised the A series of paper sizes, it chose the ratio $\sqrt{2} : 1$ (Wikipedia, Square root of 2), so folding an A4 sheet in half gives an A5 sheet with exactly the same shape.

Quick Reference Table

The square roots of 1 to 10 show how rare whole-number answers are. Only 1, 4 and 9 give rational roots in this range.

Notice that $\sqrt{8} = 2\sqrt{2}$: the square root of 2 keeps reappearing inside other roots. The square root 1 to 25 table extends this list.

Is The Square Root Of 2 Rational Or Irrational?

The square root of 2 is irrational. It cannot be written as $\frac{p}{q}$ with whole numbers $p$ and $q$, so it is not a rational number, and its decimal never ends or repeats.

This was probably the first number ever shown to be an irrational number. The proof below is short enough to check line by line.

How Do You Prove The Square Root Of 2 Is Irrational By Contradiction?

Assume the opposite and show it breaks.

1. Suppose $\sqrt{2} = \frac{p}{q}$, where $p$ and $q$ are whole numbers with no common factor.

2. Square both sides: $2 = \frac{p^2}{q^2}$, so $p^2 = 2q^2$.

3. Then $p^2$ is even, which forces $p$ to be even (an odd number squared is odd).

4. Write $p = 2k$. Then $4k^2 = 2q^2$, so $q^2 = 2k^2$.

5. Now $q^2$ is even, so $q$ is even too.

6. Both $p$ and $q$ are even, so they share the factor 2. That contradicts step 1.

No such fraction exists, so $\sqrt{2}$ is irrational. The same argument, with 3 in place of 2, appears in the proof that root 3 is irrational.

Is There A Geometric Proof That $\sqrt{2}$ Is Irrational?

There is, and it needs only squares. Suppose a whole-number square of side $p$ had exactly the same area as two whole-number squares of side $q$, so $p^2 = 2q^2$ with $p$ as small as possible.

Place the two $q$-squares in opposite corners of the $p$-square. They overlap in a small square of side $2q - p$, and they leave two uncovered corner squares of side $p - q$.

The overlap must equal the uncovered area, so $(2q - p)^2 = 2(p - q)^2$. That is a smaller whole-number solution of the same equation, which contradicts "as small as possible".

How Do You Find The Square Root Of 2?

Four hand methods give the decimal, and each one shows a different side of the number. None of them ever finishes, which is the irrationality showing up in practice.

Method 1: Long Division

Write 2 as $2.00,00,00,00$ and work one pair of zeros at a time. At each step, double the root found so far, then choose the largest digit $d$ so that (doubled root with $d$ attached) $\times, d$ fits.

At step 2, the next digit could not be 5, since $25 \times 5 = 125$ is more than 100. The remainder never reaches zero, so the digits keep coming: $1.414213\ldots$

Method 2: Squaring Estimates

Guess, square, and adjust. This is slower than long division but needs nothing beyond multiplication.

1. $1^2 = 1$ and $2^2 = 4$, so the root is between 1 and 2.

2. $1.4^2 = 1.96$ and $1.5^2 = 2.25$, so it is between 1.4 and 1.5.

3. $1.41^2 = 1.9881$ and $1.42^2 = 2.0164$, so it is between 1.41 and 1.42.

4. $1.4142^2 = 1.99996164$, just under 2, so $\sqrt{2} \approx 1.4142$.

Method 3: The Babylonian Method

Start with a guess $x$, then replace it with the average of $x$ and $\frac{2}{x}$. If $x$ is too big, $\frac{2}{x}$ is too small, so their average lands closer.

$$x_{\text{new}} = \frac{1}{2}\left(x + \frac{2}{x}\right)$$

The number of correct digits roughly doubles each round. Calculators use refinements of this same idea, which is also known as Newton's method.

Method 4: Prime Factorization

Prime factorization simplifies a root by pulling out pairs of equal factors. Since 2 is itself prime, there is no pair to pull out.

$$\sqrt{2} = \sqrt{2}$$

The radical stays as it is. This method does not give a decimal for $\sqrt{2}$, but it is the method that turns $\sqrt{8}$ into $2\sqrt{2}$ or $\sqrt{50}$ into $5\sqrt{2}$, which the simplifying radical expressions page covers in full.

Can The Square Root Of 2 Be Written As A Fraction?

Not exactly, because it is irrational. A fraction can only approximate it, and some fractions do that remarkably well.

Each fraction misses the equation $p^2 = 2q^2$ by exactly 1, which is why none can ever hit $\sqrt{2}$ itself. The ancient Indian Sulba Sutras already used $\frac{577}{408}$, and a Babylonian clay tablet, YBC 7289, records $1;24,51,10$ in base 60, which is $1.414212963$ in decimals (MacTutor).

Why Is The Square Root Of 2 Divided By 2 The Same As 1 Over Root 2?

They are the same number written two ways. Multiply the top and bottom of $\frac{1}{\sqrt{2}}$ by $\sqrt{2}$:

$$\frac{1}{\sqrt{2}} = \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$$

$$= \frac{\sqrt{2}}{2}$$

$$\approx 0.7071$$

This step is called rationalising the denominator, and $\frac{\sqrt{2}}{2}$ is the standard form. Halve it again and $\frac{\sqrt{2}}{4} \approx 0.3536$.

Examples Of Square Root Of 2

Example 1

A square tile has sides of 5 cm. How long is its diagonal?

$$d = 5\sqrt{2}$$

$$\approx 5 \times 1.41421$$

$$\approx 7.0711 \text{ cm}$$

Example 2

Simplify $\sqrt{2} + \sqrt{2}$.

It is tempting to combine the numbers under one root: $\sqrt{2} + \sqrt{2} = \sqrt{4} = 2$.

Check with decimals. Each $\sqrt{2}$ is about 1.414, so the sum is about 2.828, not 2.

Treat $\sqrt{2}$ like a unit, the way $x + x = 2x$:

$$\sqrt{2} + \sqrt{2} = 2\sqrt{2}$$

$$\approx 2.8284$$

Example 3

Simplify $\sqrt{50} + \sqrt{18}$.

$$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$

$$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$

$$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$$

$$\approx 11.3137$$

The square root of 50 and the square root of 8 both simplify to a multiple of $\sqrt{2}$ in the same way.

Example 4

Rationalise $\frac{4}{\sqrt{2}}$.

$$\frac{4}{\sqrt{2}} = \frac{4\sqrt{2}}{2}$$

$$= 2\sqrt{2}$$

$$\approx 2.8284$$

Example 5

Given $\sqrt{2} \approx 1.4142$, find $\sqrt{0.02}$ and $\sqrt{200}$.

$$\sqrt{0.02} = \sqrt{\frac{2}{100}} = \frac{\sqrt{2}}{10} \approx 0.1414$$

$$\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \approx 14.142$$

The first instinct is often to match $\sqrt{0.02}$ with $\sqrt{0.2}$, but the decimal must be written over a perfect square such as 100 before the root comes out cleanly.

Example 6

Expand $(1 + \sqrt{2})^2$ and find $(\sqrt{2} + 1)(\sqrt{2} - 1)$.

$$(1 + \sqrt{2})^2 = 1 + 2\sqrt{2} + 2$$

$$= 3 + 2\sqrt{2} \approx 5.8284$$

$$(\sqrt{2} + 1)(\sqrt{2} - 1) = 2 - 1$$

$$= 1$$

Two irrational numbers can multiply to a whole number, which is exactly the trick behind rationalising.

Where Do Students Go Wrong With The Square Root Of 2?

Mistake 1: Treating 1.414 As The Exact Value

Where it slips in: Any problem that asks for an exact answer, or a long calculation that reuses the decimal.

Don't do this: Writing $\sqrt{2} = 1.414$ and carrying it through. In fact $1.414^2 = 1.999396$, not 2.

The correct way: Keep $\sqrt{2}$ as a symbol until the last line, then round once. Students who round at the start see small errors grow with every multiplication, and the fix is a habit, not a formula.

Mistake 2: Adding Under One Root

Where it slips in: Sums of different roots, such as $\sqrt{2} + \sqrt{3}$.

Don't do this: Writing $\sqrt{2} + \sqrt{3} = \sqrt{5}$. The left side is about 3.1463 and $\sqrt{5}$ is about 2.2361.

The correct way: Only like roots combine ($3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$). Products do merge, so $\sqrt{2} \times \sqrt{3} = \sqrt{6}$; the memoriser who learned "roots combine" without the word "multiply" is the one who writes $\sqrt{5}$ here.

Mistake 3: Dropping The Negative Solution

Where it slips in: Solving $x^2 = 2$.

Don't do this: Answering only $x = \sqrt{2}$.

The correct way: Write $x = \sqrt{2}$ or $x = -\sqrt{2}$, since $(-\sqrt{2})^2 = 2$ as well. The radical sign gives the positive root, but the equation asks for every number that squares to 2.

Where Can You Get Help With Irrational Numbers And Radicals?

Irrational numbers, radicals and proofs by contradiction are where many students first meet real mathematical reasoning. Working through them with a teacher who explains each step makes the ideas stick.

You can work one-to-one with an algebra tutor, get support from a high school math tutor for proofs and radicals, or arrange regular math tutoring through the school year.

Conclusion

• The square root of 2 is about 1.4142, the diagonal of a square with side 1.

• It is irrational: assuming $\sqrt{2} = \frac{p}{q}$ makes both $p$ and $q$ even, a contradiction.

• Long division, squaring estimates and the Babylonian method all approach 1.41421356 without ever finishing.

• Fractions such as $\frac{99}{70}$ and $\frac{577}{408}$ come close, but no fraction is exact.

• $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.7071$, and only like roots combine when you add.

Practice These Before Moving On

1. Find $\sqrt{2} \times \sqrt{8}$.

2. Simplify $\sqrt{32}$ and give its decimal to 4 places.

3. Rationalise $\frac{3}{\sqrt{2}}$.

4. A square has sides of 3 m. Find its diagonal to 4 decimal places.

5. Square the fraction $\frac{99}{70}$. How far is the result from 2?

Answers:

• Answer to Question 1: $4$

• Answer to Question 2: $4\sqrt{2} \approx 5.6569$

• Answer to Question 3: $\frac{3\sqrt{2}}{2} \approx 2.1213$

• Answer to Question 4: $3\sqrt{2} \approx 4.2426$ m

• Answer to Question 5: $\frac{9801}{4900} \approx 2.000204$, about 0.0002 above 2

If Question 2 stalls, return to Method 4 and look for the largest perfect square inside 32. Want a teacher to check your working? Book a free demo class.

Read More

• Square root of 3 and why it is irrational

• Square root of 5 with decimal and simplest form

• Squares and square roots explained from the start

• Perfect squares and how to recognise them

• Square root tricks for quick mental estimates