Square Root of 2

The square root of 2 is 1.41421356, rounded to eight decimal places, and it is the most famous irrational number in mathematics. It is the length of the diagonal of a square with sides of 1, by Pythagoras: 1² + 1² = 2, so the diagonal is √2. The square root calculator above gives the value to as many digits as you need, and this page explains why the digits never end, how to approximate the value by hand, and where it appears in real mathematics.

The ancient Greeks proved that the square root of 2 cannot be written as a fraction, which was a shock: it meant some lengths cannot be measured exactly with any rational number. The proof is short and elegant, and it is still taught today as the first proof most students meet. This page works through the proof, shows the approximations used through history, and lists the places where the square root of 2 turns up, from paper sizes to engineering.

The Square Root: What It Is and How to Find It

The square root of a number is the value that, multiplied by itself, gives the original number. The square root of 144 is 12, because 12 × 12 = 144, and the square root of 2 is about 1.41421, because 1.41421 × 1.41421 is very close to 2. The symbol is √, so √144 = 12 and √2 ≈ 1.41421. Every positive number has two square roots, one positive and one negative, but the symbol √ on its own always means the positive one, called the principal square root. The square root of a negative number is not a real number at all, which is why the calculator reports an error for negative inputs.

Most square roots are irrational numbers: their decimal expansion never ends and never repeats. The square root of 2 is the famous example — its digits run 1.41421356 and continue forever without a pattern, a fact proved by the ancient Greeks. That is why we round: √50 ≈ 7.07 is accurate to two decimal places, and √50 ≈ 7.071 is accurate to three. The calculator above also handles cube roots and other roots: the cube root of 27 is 3, because 3 × 3 × 3 = 27, and the fourth root of 81 is 3, because 3⁴ = 81.

Square roots appear wherever a square is involved: the side of a square from its area, the diagonal of a rectangle, the standard deviation in statistics, and the quadratic formula in algebra. The National Institute of Standards and Technology lists the square roots of small integers among its standard mathematical constants, and the same values turn up in geometry, physics, and engineering problems of every kind.

Step 1 — Check Whether the Number Is a Perfect Square

Try the squares you know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. If your number is on that list, its square root is exact: √144 = 12, √225 = 15. If it is not, the root is irrational and you will estimate or use the calculator.

Step 2 — Estimate Between Two Whole Numbers

For √50, note that 49 is 7² and 64 is 8², so √50 sits just above 7. For √125, note that 121 is 11² and 144 is 12², so the answer sits just above 11, and indeed √125 ≈ 11.18. A good estimate tells you at a glance whether a calculator answer is plausible.

Step 3 — Refine and Check by Squaring

Refine the estimate by squaring your guess: 7.07² = 49.9849, just under 50, so √50 ≈ 7.07 is right to two decimal places. Always check by squaring the answer you report; if the square does not return your original number, something went wrong.

Worked Examples: The Square Root of 2 and Its Relatives

These values all follow from the square root of 2, and each one is checked by squaring.

The third row shows the simplification rule at work: 50 = 25 × 2, and 25 is a perfect square, so √50 = 5√2, which is 5 × 1.41421 = 7.07107. The last row is the value behind the 45-degree angle: sin(45°) = cos(45°) = 1/√2 ≈ 0.70711, which appears in every right triangle with two equal sides. Try 2, 8, and 50 in the calculator above and confirm each value.

How to Use the Square Root Calculator

Enter any number, choose the root, and the result appears instantly, with the exact simplified form where one exists. Use it for homework checks, geometry, and any calculation where a root is one step of a longer problem.

1. Enter the number. Type the value whose root you want, such as 72. The default of 72 shows √72 ≈ 8.49, or 6√2 in exact form.
2. Choose the root. Square root is the default; switch to cube root, fourth root, or any other root for problems such as ∛27 = 3.
3. Read the decimal result. The calculator gives the root to several decimal places, which is the form you need for measurement and engineering work.
4. Read the simplified form. Where the number has a square factor, the calculator shows the exact form: √72 = 6√2, √50 = 5√2, √125 = 5√5.
5. Square the answer to check. Multiply the result by itself and confirm you get back to the number you entered; that one check catches almost every input error.

Approximations of the Square Root of 2 Through History

Mathematicians have approximated this number for thousands of years, and each approximation shows the tools of its time.

Common Mistakes With Square Roots

Square roots are a classic source of small, stubborn errors. These are the ones to watch for.

• Adding roots as if they were numbers. √2 + √3 is not √5. Roots add only when the radicand matches: √2 + 3√2 = 4√2.
• Forgetting the negative root. The equation x² = 49 has two solutions, x = 7 and x = −7, even though √49 on its own means 7.
• Taking the root of a negative number. There is no real number whose square is negative, so √(−4) has no real answer and the calculator correctly refuses it.
• Rounding too early. Rounding √2 to 1.41 and then squaring gives 1.9881, not 2. Keep extra digits until the last step of the problem.
• Confusing square and square root. Squaring makes numbers bigger than 1 larger; the square root makes them smaller. √144 = 12, while 144² = 20,736.

Where You Will Use Square Roots

Square roots are not just a school topic. They appear in practical work wherever area, distance, or variation is measured.

• Geometry. The side of a square garden with an area of 144 square metres is √144 = 12 metres, and the diagonal of a 3 by 4 rectangle is √(9 + 16) = 5.
• Statistics. The standard deviation is the square root of the variance, so every report of spread in test scores or measurements runs through a square root.
• Physics. The speed of a falling object, the period of a pendulum, and the energy in a spring all involve square roots of the quantities you measure.
• Construction. The 3-4-5 triangle, checked with √(3² + 4²) = 5, is how builders square up corners on site.
• Finance. The square root of time rule scales risk and volatility: over four times the period, typical movement roughly doubles, because √4 = 2.

Why the Square Root of 2 Cannot Be a Fraction

The proof assumes the opposite and finds a contradiction. Suppose √2 = p/q in lowest terms, with p and q whole numbers sharing no factor. Then p² = 2q², so p² is even, which means p is even, so p = 2k. Substituting gives 4k² = 2q², so q² = 2k², which means q is even too. But p and q were supposed to share no factor, and both being even contradicts that, so the assumption is impossible. No fraction equals the square root of 2, which is why its decimal expansion never ends and never repeats.

Tips for Accurate Results

A little care with roots pays off, because a root error hides inside every later step of a problem.

1. Memorise the squares up to 15². Knowing 12² = 144 and 15² = 225 lets you spot perfect squares instantly and estimate everything else.
2. Simplify before approximating. Reducing √72 to 6√2 keeps the answer exact, and the decimal 8.49 follows from it without extra rounding error.
3. Keep two extra digits. Carry two more decimal places than your final answer needs, and round only at the end.
4. Check with a square. Squaring your answer is the fastest possible check, and it takes two seconds.
5. Watch the domain. The input must be zero or positive for a real square root; a negative input means the problem needs a different approach.

Square Root Quick Answers

These are the root values people look up most often, each one verified against the calculator above.

Related tools on this site: the main square root calculator, scientific calculator, basic calculator, fraction calculator, and the square root calculator again whenever you need a fresh check.

Frequently Asked Questions

What is the square root of 2?

The square root of 2 is about 1.41421356. It is irrational, so its decimal expansion never ends and never repeats, and any value you write down is a rounded approximation. Squaring 1.41421 gives 1.99999, just under 2.

Why is the square root of 2 irrational?

Because it cannot be written as a fraction of two whole numbers. The classical proof shows that if √2 = p/q in lowest terms, then both p and q must be even, which contradicts the claim that the fraction is in lowest terms. So no fraction equals it.

What is the square root of 8?

The square root of 8 is about 2.82843. It simplifies to 2√2, because 8 = 4 × 2 and 4 is a perfect square, so √8 = 2 × 1.41421 = 2.82843.

What is 1 divided by the square root of 2?

It is about 0.70711, which is √2/2. This value is sin(45°) and cos(45°), and it appears in every isosceles right triangle and in the geometry of squares.

Where is the square root of 2 used?

It is the diagonal of a unit square, the ratio behind A-series paper sizes, and a constant in trigonometry, engineering, and computer graphics, anywhere a 45-degree angle or a square diagonal appears.

Disclaimer: This calculator is for general information only. Results are rounded for display; for engineering, construction, or safety-critical work, confirm measurements with the applicable standards and a qualified professional.