Square Root of 2

√2 at a glance

Exact value

√2

Decimal (10 places)

1.4142135624

Rounded

1.4 · 1.41 · 1.414

Perfect square?

No — between 1² and 2²

Rational?

Irrational

Both square roots

±1.414214

Prime factorization

2

Cube root

1.259921

How to simplify √2

2 is a prime number, so its only factors are 1 and 2. There is no perfect-square factor to pull out, which means √2 is already in its simplest radical form.

The square root of any prime is irrational. If √2 were a fraction a/b in lowest terms, then a² = 2b², so 2 would divide a — and then 2 would divide b too, contradicting “lowest terms.” That is why the decimal 1.4142135624 is only a rounded value.

Where √2 sits between perfect squares

1 = 1² and 4 = 2² are the nearest perfect squares, so √2 lies between 1 and 2. 2 is 1 above 1 and 2 below 4, so the root is closer to 1.

• Straight line between 1 and 4: 1.3333 (5.72% low)
• Tangent from 1, i.e. 1 + 1 ÷ 2: 1.5000 (6.07% high)
• Tangent from 2, i.e. 2 − 2 ÷ 4: 1.5000 (6.07% high)

For √2 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √2 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 1 (1² = 1):

The count of correct decimals went 1, 2, 5 and all 10 over 4 steps — roughly doubling each time — until the guess matched √2 = 1.4142135624 to every decimal shown.

√2 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √2 the pattern is [1; 2] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 2 is one more than a perfect square (1² + 1). A pattern that never ends is one more proof that √2 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 2y² = 1. Its smallest solution in positive whole numbers is x = 3, y = 2. Because the period is odd, the equation with −1 on the right also has a solution: 1² − 2 × 1² = −1.

√2 in geometry and everyday measurements

• A square tile with an area of 2 square inches has sides about 1.414 in long.
• 2 = 1² + 1², so by the Pythagorean theorem √2 is the diagonal of a 1 × 1 rectangle — and the distance between the points (0, 0) and (1, 1) on a grid.

√2 is the length of the diagonal of a 1 × 1 square, and the long side of every ISO 216 sheet (A4, A3, A5 …) is √2 times its short side, which is why folding an A4 sheet in half gives an A5 sheet of exactly the same shape.

Square roots near √2 and related roots

• The cube root of 2 is about 1.259921.
• Multiplying the radicand by 100 multiplies the root by 10: √200 = 10 × √2 ≈ 14.142136.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.