Square Root of 593

√593 at a glance

Exact value

√593

Decimal (10 places)

24.3515913238

Rounded

24.4 · 24.35 · 24.352

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.351591

Prime factorization

593

Cube root

8.401398

How to simplify √593

593 is a prime number, so its only factors are 1 and 593. There is no perfect-square factor to pull out, which means √593 is already in its simplest radical form.

The square root of any prime is irrational. If √593 were a fraction a/b in lowest terms, then a² = 593b², so 593 would divide a — and then 593 would divide b too, contradicting “lowest terms.” That is why the decimal 24.3515913238 is only a rounded value.

Where √593 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √593 lies between 24 and 25. 593 is 17 above 576 and 32 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.3469 (0.02% low)
• Tangent from 24, i.e. 24 + 17 ÷ 48: 24.3542 (0.01% high)
• Tangent from 25, i.e. 25 − 32 ÷ 50: 24.3600 (0.03% high)

For √593 the tangent at 24 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 593 is just 17 above 576.

Finding √593 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 593: following the tangent line down to zero simplifies to averaging x with 593 ÷ x.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √593 = 24.3515913238 to every decimal shown.

√593 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √593 the pattern is [24; 2, 1, 5, 2, 2, 1, 1, 2, 2, 5, 1, 2, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √593 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 − 593y² = 1. Its smallest solution in positive whole numbers is x = 721,517,598,849, y = 29,629,176,560. Because the period is odd, the equation with −1 on the right also has a solution: 600,632² − 593 × 24,665² = −1.

√593 in geometry and everyday measurements

• A square garage floor of 593 square feet measures about 24.35 ft (24 ft 4 in) per side, and its corner-to-corner diagonal is √1186 ≈ 34.4 ft.
• 593 = 8² + 23², so by the Pythagorean theorem √593 is the diagonal of a 8 × 23 rectangle — and the distance between the points (0, 0) and (8, 23) on a grid.

Square roots near √593 and related roots

• The cube root of 593 is about 8.401398.
• Squaring undoes the root: (√593)² = 593, while 593² = 351,649 — the number whose square root is 593.

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