Square Root of 587

√587 at a glance

Exact value

√587

Decimal (10 places)

24.2280828792

Rounded

24.2 · 24.23 · 24.228

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.228083

Prime factorization

587

Cube root

8.372967

How to simplify √587

587 is a prime number, so its only factors are 1 and 587. There is no perfect-square factor to pull out, which means √587 is already in its simplest radical form.

The square root of any prime is irrational. If √587 were a fraction a/b in lowest terms, then a² = 587b², so 587 would divide a — and then 587 would divide b too, contradicting “lowest terms.” That is why the decimal 24.2280828792 is only a rounded value.

Where √587 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √587 lies between 24 and 25. 587 is 11 above 576 and 38 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.2245 (0.01% low)
• Tangent from 24, i.e. 24 + 11 ÷ 48: 24.2292 (0% high)
• Tangent from 25, i.e. 25 − 38 ÷ 50: 24.2400 (0.05% high)

For √587 the tangent at 24 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 587 is just 11 above 576.

Finding √587 with the Babylonian method

If a guess is too big, 587 ÷ 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√587) in one step.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √587 = 24.2280828792 to every decimal shown.

√587 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √587 the pattern is [24; 4, 2, 1, 1, 1, 1, 23, 1, 1, 1, 1, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √587 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 − 587y² = 1. Its smallest solution in positive whole numbers is x = 1,907,162, y = 78,717.

√587 in geometry and everyday measurements

• A square garage floor of 587 square feet measures about 24.23 ft (24 ft 3 in) per side, and its corner-to-corner diagonal is √1174 ≈ 34.3 ft.
• 587 is not a sum of two whole-number squares — 587 is itself a prime that is one less than a multiple of 4, which rules that out — so √587 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 15 × 19 box, because 1² + 15² + 19² = 587.

Square roots near √587 and related roots

• The cube root of 587 is about 8.372967.
• Squaring undoes the root: (√587)² = 587, while 587² = 344,569 — the number whose square root is 587.

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