Square Root of 641

√641 at a glance

Exact value

√641

Decimal (10 places)

25.3179778023

Rounded

25.3 · 25.32 · 25.318

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.317978

Prime factorization

641

Cube root

8.622225

How to simplify √641

641 is a prime number, so its only factors are 1 and 641. There is no perfect-square factor to pull out, which means √641 is already in its simplest radical form.

The square root of any prime is irrational. If √641 were a fraction a/b in lowest terms, then a² = 641b², so 641 would divide a — and then 641 would divide b too, contradicting “lowest terms.” That is why the decimal 25.3179778023 is only a rounded value.

Where √641 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √641 lies between 25 and 26. 641 is 16 above 625 and 35 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.3137 (0.02% low)
• Tangent from 25, i.e. 25 + 16 ÷ 50: 25.3200 (0.01% high)
• Tangent from 26, i.e. 26 − 35 ÷ 52: 25.3269 (0.04% high)

For √641 the tangent at 25 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 641 is just 16 above 625.

Finding √641 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 641: following the tangent line down to zero simplifies to averaging x with 641 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √641 = 25.3179778023 to every decimal shown.

√641 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √641 the pattern is [25; 3, 6, 1, 9, 3, 1, 3, 1, 5, 1, 1, 5, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √641 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 − 641y² = 1. Its smallest solution in positive whole numbers is x = 2,609,429,220,845,977,814,049, y = 103,066,257,550,962,737,720 — 22 digits for x, even though 641 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 36,120,833,468² − 641 × 1,426,687,145² = −1.

√641 in geometry and everyday measurements

• A square garage floor of 641 square feet measures about 25.32 ft (25 ft 4 in) per side, and its corner-to-corner diagonal is √1282 ≈ 35.8 ft.
• 641 = 4² + 25², so by the Pythagorean theorem √641 is the diagonal of a 4 × 25 rectangle — and the distance between the points (0, 0) and (4, 25) on a grid.

Square roots near √641 and related roots

• The cube root of 641 is about 8.622225.
• Squaring undoes the root: (√641)² = 641, while 641² = 410,881 — the number whose square root is 641.

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