√591 at a glance
- Exact value
- √591
- Decimal (10 places)
- 24.3104915623
- Rounded
- 24.3 · 24.31 · 24.310
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.310492
- Prime factorization
- 3 × 197
- Cube root
- 8.391942
How to simplify √591
The prime factorization of 591 is 3 × 197. Every prime appears only once, so there is no pair to bring outside the radical — √591 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 591, 3 and 197 appear an odd number of times, so √591 is irrational and 24.3104915623 is a rounded value.
Where √591 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √591 lies between 24 and 25. 591 is 15 above 576 and 34 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.3061 (0.02% low)
- Tangent from 24, i.e. 24 + 15 ÷ 48: 24.3125 (0.01% high)
- Tangent from 25, i.e. 25 − 34 ÷ 50: 24.3200 (0.04% high)
For √591 the tangent at 24 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 591 is just 15 above 576.
Finding √591 with the Babylonian method
If a guess is too big, 591 ÷ 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√591) in one step.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 591 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.6250000000 | 24.3125000000 | 2 |
| 2 | 24.3125000000 | 24.3084832905 | 24.3104916452 | 7 |
| 3 | 24.3104916452 | 24.3104914793 | 24.3104915623 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √591 = 24.3104915623 to every decimal shown.
√591 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √591 the pattern is [24; 3, 4, 1, 1, 7, 1, 1, 4, 3, 48] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √591 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 24/1 | 24.0000000000 | 3.1 × 10⁻¹ |
| 73/3 | 24.3333333333 | 2.3 × 10⁻² |
| 316/13 | 24.3076923077 | 2.8 × 10⁻³ |
| 389/16 | 24.3125000000 | 2.0 × 10⁻³ |
| 705/29 | 24.3103448276 | 1.5 × 10⁻⁴ |
| 5,324/219 | 24.3105022831 | 1.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 591y² = 1. Its smallest solution in positive whole numbers is x = 165,676, y = 6,815.
√591 in geometry and everyday measurements
- A square garage floor of 591 square feet measures about 24.31 ft (24 ft 4 in) per side, and its corner-to-corner diagonal is √1182 ≈ 34.4 ft.
- 591 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √591 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 √591 as its space diagonal.
Square roots near √591 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √588 | 14√3 | 24.2487 | No |
| √589 | √589 | 24.2693 | No |
| √590 | √590 | 24.2899 | No |
| √591 | √591 | 24.3105 | No |
| √592 | 4√37 | 24.3311 | No |
| √593 | √593 | 24.3516 | No |
| √594 | 3√66 | 24.3721 | No |
- The cube root of 591 is about 8.391942.
- Squaring undoes the root: (√591)² = 591, while 591² = 349,281 — the number whose square root is 591.
Frequently asked questions
What is the square root of 591?
The square root of 591 is √591, about 24.3104915623. The negative root, −24.310492, also squares to 591.
Is the square root of 591 rational or irrational?
Irrational. 591 is not a perfect square — it falls between 576 and 625 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √591 be simplified?
No. 591 = 3 × 197 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √591 rounded to two decimal places?
√591 ≈ 24.31 to two decimal places (24.3 to one, 24.310 to three). Check: 24.31² = 590.9761, close to 591.