Exponents & roots

Square Root of 2

The square root of 2 is approximately 1.41421356. It is the positive number that, multiplied by itself, gives exactly 2. Its decimal goes on forever without repeating, because √2 is irrational: it cannot be written as a fraction of two whole numbers. This guide gives √2 to 100 decimal places, proves it is irrational in a few lines, and shows three ways to calculate it by hand.

What is the square root of 2?

√2 is the number whose square is 2. Since 1² = 1 and 2² = 4, it must lie between 1 and 2, and much closer to 1. To 8 decimal places, rounded:

Every positive number has two square roots, one positive and one negative. The other square root of 2 is −√2 ≈ −1.41421356, because a negative times a negative is also positive. When people write √2 or say “the square root of 2,” they mean the positive one, called the principal square root.

√2 to 100 decimal places

Here are the first 100 digits after the decimal point, in groups of ten. These are truncated, not rounded, so each digit shown is exactly right.

Common roundings

The third column shows why every decimal is only an approximation: squaring any rounded value misses 2, by less each time but never reaching it. For everyday work, 1.414 is plenty; for engineering, 1.41421356 covers almost any need. The guide to rounding to decimal places explains how to cut a long decimal to the precision you need, and the square root calculator gives √x for any number as a decimal and in simplest radical form.

Why the square root of 2 is irrational

A rational number is one that can be written as a fraction a/b of two whole numbers. The proof that √2 is not one of them is more than two thousand years old and needs nothing beyond even and odd numbers. It works by contradiction: assume √2 is a fraction, and show that the assumption breaks.

1. Assume √2 = a/b, where the fraction is in lowest terms, so a and b share no common factor. Any fraction can be simplified to lowest terms, so this is allowed.
2. Square both sides: 2 = a²/b², so a² = 2b².
3. So a² is even, because it equals 2 times something. If a were odd, a² would be odd too (odd × odd = odd), so a must be even. Write a = 2k.
4. Substitute: (2k)² = 2b² gives 4k² = 2b², so b² = 2k².
5. So b² is even, and by the same reasoning b is even.
6. Contradiction. Both a and b are even, so they share a factor of 2. But we chose a/b in lowest terms. The assumption must be false.

So no fraction squares to exactly 2, and √2 is irrational. That is also why its decimal never ends or repeats: every terminating or repeating decimal can be turned back into a fraction, as the decimal to fraction calculator shows, and √2 has no fraction to turn into.

Method 1: the long-hand (digit-by-digit) method

The long-hand square root method finds one digit at a time, much like long division. It is slow but exact, and each digit it produces is final.

1. Write 2 as 2.00 00 00 00, splitting the digits into pairs from the decimal point.
2. Find the largest whole number whose square fits into the first group. For 2 that is 1 (1² = 1). Write 1 as the first digit and subtract: remainder 1.
3. Bring down the next pair to make the new working number.
4. Double the root found so far and write it with a blank digit on the end: 2_. Find the largest digit d such that (2d) × d fits into the working number. That d is the next digit of the root.
5. Subtract, bring down the next pair, and repeat.

In the second row, the root so far is 1, doubled to 2, so the trial divisor is “2d.” A digit of 4 gives 24 × 4 = 96, which fits into 100, while 5 would give 25 × 5 = 125, which does not. Each later row doubles the digits found so far in the same way: 14 doubled is 28, 141 doubled is 282, 1414 doubled is 2828.

Why does it work? If the root so far is r and the next digit is d, the new root is 10r + d, and (10r + d)² = 100r² + (20r + d) × d. The (20r + d) × d term is exactly the “double the root, append d, multiply by d” trial. The method peels off the square one digit at a time.

Method 2: the Babylonian method (average and divide)

The Babylonian method is much faster. Start with any guess, then repeatedly average the guess with 2 divided by the guess.

The averaging step needs only division and addition, so you can run it with the division calculator or even on paper. The fraction 577/408 from round 3 is accurate enough for most practical work.

Method 3: Newton’s method

Newton’s method finds where a function equals zero, using the tangent line at each guess. To find √2, solve x² − 2 = 0.

Simplify the right-hand side and something familiar appears:

That is exactly the Babylonian formula. The ancient average-and-divide method is Newton’s method applied to x² − 2, which explains why it converges so fast. The general version, x → (x + N/x) ÷ 2, finds √N for any positive N.

A slower method: squeeze it between squares

If you only have multiplication, trap √2 between two decimals whose squares straddle 2, then narrow the gap one decimal place at a time.

This gains only one digit per round, against Babylon’s doubling, but it shows clearly why the answer starts 1.414.

Fractions that approximate √2

No fraction equals √2, but some come remarkably close. The best ones follow a simple pattern: from p/q, the next is (p + 2q)/(q + p).

The last column shows why these are so good: the numerator squared is always within 1 of twice the denominator squared, so p²/q² is always within 1/q² of 2. The estimates alternate above and below √2. You can check any of them with the fraction to decimal calculator.

Where √2 shows up, and how to simplify radicals with it

• The diagonal of a square. By the Pythagorean theorem, a square with side 1 has diagonal √(1² + 1²) = √2. A square with side 10 inches has a diagonal of 10√2 ≈ 14.14 inches.
• The 45-45-90 triangle. Its sides are in the ratio 1 : 1 : √2, which is why sin 45° = cos 45° = √2/2 ≈ 0.70710678.
• Paper sizes. ISO paper such as A4 (210 × 297 mm) has sides in the ratio 1 : √2, so cutting a sheet in half gives a smaller sheet with the same shape. 297 ÷ 210 ≈ 1.4143.

√2 also appears whenever you simplify the square root of 2 times a perfect square. Factor out the largest perfect square, which the prime factorization calculator makes easy to spot.

The last row is called rationalizing the denominator: multiply top and bottom by √2 to get √2/2. And since a square root is a power of one half, √2 = 2^(1/2); the exponent rules cover fractional exponents.

Common mistakes with √2

• Treating 1.414 as exact. 1.414² = 1.999396, not 2. Keep √2 as a symbol until the last step and round only the final answer.
• Adding under the root. √2 + √2 = 2√2 ≈ 2.828, not √4 = 2. Like terms add their coefficients; the radical stays.
• Forgetting that √2 × √2 = 2. Multiplying √2 by itself gives exactly 2, with no rounding error. √2 × √8 = √16 = 4 works the same way.
• Calling 99/70 or 577/408 the square root of 2. They are excellent approximations, but the proof above shows no fraction is exact.
• Dropping the negative root when solving equations. x² = 2 has two solutions, x = √2 and x = −√2.

Frequently asked questions

Calculators for this

Every one of these shows the working, so you can check the method as well as the answer.

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