Square Root of 659

√659 at a glance

Exact value

√659

Decimal (10 places)

25.6709953060

Rounded

25.7 · 25.67 · 25.671

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.670995

Prime factorization

659

Cube root

8.702188

How to simplify √659

659 is a prime number, so its only factors are 1 and 659. There is no perfect-square factor to pull out, which means √659 is already in its simplest radical form.

The square root of any prime is irrational. If √659 were a fraction a/b in lowest terms, then a² = 659b², so 659 would divide a — and then 659 would divide b too, contradicting “lowest terms.” That is why the decimal 25.6709953060 is only a rounded value.

Where √659 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √659 lies between 25 and 26. 659 is 34 above 625 and 17 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.6667 (0.02% low)
• Tangent from 25, i.e. 25 + 34 ÷ 50: 25.6800 (0.04% high)
• Tangent from 26, i.e. 26 − 17 ÷ 52: 25.6731 (0.01% high)

For √659 the tangent at 26 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 659 is just 17 below 676.

Finding √659 with the Babylonian method

If a guess is too big, 659 ÷ 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√659) in one step.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √659 = 25.6709953060 to every decimal shown.

√659 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √659 the pattern is [25; 1, 2, 25, 2, 1, 50] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √659 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 − 659y² = 1. Its smallest solution in positive whole numbers is x = 5,930, y = 231.

√659 in geometry and everyday measurements

• 659 square feet is 61.2 m². Laid out as a square — a small house footprint or a lot — it is about 25.67 ft (25 ft 8 in) on a side.
• 659 is not a sum of two whole-number squares — 659 is itself a prime that is one less than a multiple of 4, which rules that out — so √659 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 25 box, because 3² + 5² + 25² = 659.

Square roots near √659 and related roots

• The cube root of 659 is about 8.702188.
• Squaring undoes the root: (√659)² = 659, while 659² = 434,281 — the number whose square root is 659.

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