Square Root of 643

√643 at a glance

Exact value

√643

Decimal (10 places)

25.3574446662

Rounded

25.4 · 25.36 · 25.357

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.357445

Prime factorization

643

Cube root

8.631183

How to simplify √643

643 is a prime number, so its only factors are 1 and 643. There is no perfect-square factor to pull out, which means √643 is already in its simplest radical form.

The square root of any prime is irrational. If √643 were a fraction a/b in lowest terms, then a² = 643b², so 643 would divide a — and then 643 would divide b too, contradicting “lowest terms.” That is why the decimal 25.3574446662 is only a rounded value.

Where √643 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √643 lies between 25 and 26. 643 is 18 above 625 and 33 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.3529 (0.02% low)
• Tangent from 25, i.e. 25 + 18 ÷ 50: 25.3600 (0.01% high)
• Tangent from 26, i.e. 26 − 33 ÷ 52: 25.3654 (0.03% high)

For √643 the tangent at 25 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 643 is just 18 above 625.

Finding √643 with the Babylonian method

If a guess is too big, 643 ÷ 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√643) in one step.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √643 = 25.3574446662 to every decimal shown.

√643 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √643 the pattern is [25; 2, 1, 3, 1, 16, 8, 2, 1, 1, 5, 25, 5, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √643 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 − 643y² = 1. Its smallest solution in positive whole numbers is x = 1,988,960,193,026, y = 78,436,933,185 — 13 digits for x, even though 643 is small, which is what makes Pell’s equation famous.

√643 in geometry and everyday measurements

• A square garage floor of 643 square feet measures about 25.36 ft (25 ft 4 in) per side, and its corner-to-corner diagonal is √1286 ≈ 35.9 ft.
• 643 is not a sum of two whole-number squares — 643 is itself a prime that is one less than a multiple of 4, which rules that out — so √643 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 25 box, because 3² + 3² + 25² = 643.

Square roots near √643 and related roots

• The cube root of 643 is about 8.631183.
• Squaring undoes the root: (√643)² = 643, while 643² = 413,449 — the number whose square root is 643.

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