Square Root of 259

√259 at a glance

Exact value

√259

Decimal (10 places)

16.0934769394

Rounded

16.1 · 16.09 · 16.093

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.093477

Prime factorization

7 × 37

Cube root

6.374311

How to simplify √259

The prime factorization of 259 is 7 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √259 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 259, 7 and 37 appear an odd number of times, so √259 is irrational and 16.0934769394 is a rounded value.

Where √259 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √259 lies between 16 and 17. 259 is 3 above 256 and 30 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.0909 (0.02% low)
• Tangent from 16, i.e. 16 + 3 ÷ 32: 16.0938 (0% high)
• Tangent from 17, i.e. 17 − 30 ÷ 34: 16.1176 (0.15% high)

For √259 the tangent at 16 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 259 is just 3 above 256.

Finding √259 with the Babylonian method

If a guess is too big, 259 ÷ 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√259) in one step.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √259 = 16.0934769394 to every decimal shown.

√259 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √259 the pattern is [16; 10, 1, 2, 3, 4, 3, 2, 1, 10, 32] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √259 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 − 259y² = 1. Its smallest solution in positive whole numbers is x = 847,225, y = 52,644.

√259 in geometry and everyday measurements

• A square patio or deck of 259 square feet is about 16.09 ft (16 ft 1 in) on each side, so edging all the way around takes 4 × √259 ≈ 64.4 ft.
• 259 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √259 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 15 box, because 3² + 5² + 15² = 259.

Square roots near √259 and related roots

• The cube root of 259 is about 6.374311.
• Squaring undoes the root: (√259)² = 259, while 259² = 67,081 — the number whose square root is 259.

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