Square Root of 1

√1 at a glance

Exact value

1

Decimal (10 places)

1

Rounded

1

Perfect square?

Yes — 1²

Rational?

Rational (whole number)

Both square roots

±1

Prime factorization

1 (no prime factors)

Cube root

1

How to simplify √1

1 is the first perfect square: 1 × 1 = 1. Its principal square root is 1, and together with 0 it is one of only two numbers that equal their own square root.

Where √1 sits between perfect squares

1 is 1². The perfect squares on either side are 0 (0²) and 4 (2²), so the gaps are 1 below and 3 above — consecutive squares always differ by consecutive odd numbers.

Finding √1 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 1: following the tangent line down to zero simplifies to averaging x with 1 ÷ x.

To show the method at work on a perfect square, start one above the answer, at 2:

The count of correct decimals went 0, 1, 3, 7 and all 10 over 5 steps — roughly doubling each time — until the guess matched √1 = 1.0000000000 to every decimal shown.

√1 in geometry and everyday measurements

• A square tile with an area of 1 square inch has sides exactly 1 in long.
• 1 is not a sum of two whole-number squares — no pair of squares adds up to it — so √1 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √1 as its space diagonal.

Square roots near √1 and related roots

• The cube root of 1 is exactly 1 — 1 is a perfect cube as well (1³), which makes it a perfect sixth power (1⁶).
• Multiplying the radicand by 100 multiplies the root by 10: √100 = 10 × √1 ≈ 10.

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