Square Root of 123

√123 at a glance

Exact value

√123

Decimal (10 places)

11.0905365064

Rounded

11.1 · 11.09 · 11.091

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.090537

Prime factorization

3 × 41

Cube root

4.973190

How to simplify √123

The prime factorization of 123 is 3 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √123 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 123, 3 and 41 appear an odd number of times, so √123 is irrational and 11.0905365064 is a rounded value.

Where √123 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √123 lies between 11 and 12. 123 is 2 above 121 and 21 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.0870 (0.03% low)
• Tangent from 11, i.e. 11 + 2 ÷ 22: 11.0909 (0% high)
• Tangent from 12, i.e. 12 − 21 ÷ 24: 11.1250 (0.31% high)

For √123 the tangent at 11 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 123 is just 2 above 121.

Finding √123 with the Babylonian method

If a guess is too big, 123 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√123) in one step.

Start from the nearest whole number, 11 (11² = 121):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √123 = 11.0905365064 to every decimal shown.

√123 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √123 the pattern is [11; 11, 22] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √123 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 − 123y² = 1. Its smallest solution in positive whole numbers is x = 122, y = 11.

√123 in geometry and everyday measurements

• A square room or garden bed covering 123 square feet measures about 11.09 ft (11 ft 1 in) along each wall.
• 123 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √123 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 11 box, because 1² + 1² + 11² = 123.

Square roots near √123 and related roots

• The cube root of 123 is about 4.973190.
• Four times the radicand doubles the root: √492 = 2 × √123 ≈ 22.181073.

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