Square Root of 631

√631 at a glance

Exact value

√631

Decimal (10 places)

25.1197133742

Rounded

25.1 · 25.12 · 25.120

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.119713

Prime factorization

631

Cube root

8.577152

How to simplify √631

631 is a prime number, so its only factors are 1 and 631. There is no perfect-square factor to pull out, which means √631 is already in its simplest radical form.

The square root of any prime is irrational. If √631 were a fraction a/b in lowest terms, then a² = 631b², so 631 would divide a — and then 631 would divide b too, contradicting “lowest terms.” That is why the decimal 25.1197133742 is only a rounded value.

Where √631 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √631 lies between 25 and 26. 631 is 6 above 625 and 45 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.1176 (0.01% low)
• Tangent from 25, i.e. 25 + 6 ÷ 50: 25.1200 (0% high)
• Tangent from 26, i.e. 26 − 45 ÷ 52: 25.1346 (0.06% high)

For √631 the tangent at 25 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 631 is just 6 above 625.

Finding √631 with the Babylonian method

If a guess is too big, 631 ÷ 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√631) in one step.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √631 = 25.1197133742 to every decimal shown.

√631 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √631 the pattern is [25; 8, 2, 1, 4, 1, 9, 4, 2, 6, 1, 2, 1, …] with the block of 48 terms after the semicolon repeating forever (only the first 12 of the 48 are shown). A pattern that never ends is one more proof that √631 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 − 631y² = 1. Its smallest solution in positive whole numbers is x = 48,961,575,312,998,650,035,560, y = 1,949,129,537,575,151,036,427 — 23 digits for x, even though 631 is small, which is what makes Pell’s equation famous.

√631 in geometry and everyday measurements

• A square garage floor of 631 square feet measures about 25.12 ft (25 ft 1 in) per side, and its corner-to-corner diagonal is √1262 ≈ 35.5 ft.
• 631 is not a sum of two whole-number squares — 631 is itself a prime that is one less than a multiple of 4, which rules that out — so √631 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 √631 as its space diagonal.

Square roots near √631 and related roots

• The cube root of 631 is about 8.577152.
• Squaring undoes the root: (√631)² = 631, while 631² = 398,161 — the number whose square root is 631.

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