√590 at a glance
- Exact value
- √590
- Decimal (10 places)
- 24.2899156030
- Rounded
- 24.3 · 24.29 · 24.290
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.289916
- Prime factorization
- 2 × 5 × 59
- Cube root
- 8.387207
How to simplify √590
The prime factorization of 590 is 2 × 5 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √590 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 590, 2, 5 and 59 appear an odd number of times, so √590 is irrational and 24.2899156030 is a rounded value.
Where √590 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √590 lies between 24 and 25. 590 is 14 above 576 and 35 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.2857 (0.02% low)
- Tangent from 24, i.e. 24 + 14 ÷ 48: 24.2917 (0.01% high)
- Tangent from 25, i.e. 25 − 35 ÷ 50: 24.3000 (0.04% high)
For √590 the tangent at 24 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 590 is just 14 above 576.
Finding √590 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 590 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.5833333333 | 24.2916666667 | 2 |
| 2 | 24.2916666667 | 24.2881646655 | 24.2899156661 | 7 |
| 3 | 24.2899156661 | 24.2899155399 | 24.2899156030 | 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 √590 = 24.2899156030 to every decimal shown.
√590 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √590 the pattern is [24; 3, 2, 4, 2, 3, 48] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √590 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 | 2.9 × 10⁻¹ |
| 73/3 | 24.3333333333 | 4.3 × 10⁻² |
| 170/7 | 24.2857142857 | 4.2 × 10⁻³ |
| 753/31 | 24.2903225806 | 4.1 × 10⁻⁴ |
| 1,676/69 | 24.2898550725 | 6.1 × 10⁻⁵ |
| 5,781/238 | 24.2899159664 | 3.6 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 590y² = 1. Its smallest solution in positive whole numbers is x = 5,781, y = 238.
√590 in geometry and everyday measurements
- A square garage floor of 590 square feet measures about 24.29 ft (24 ft 3 in) per side, and its corner-to-corner diagonal is √1180 ≈ 34.4 ft.
- 590 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √590 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 15 × 19 box, because 2² + 15² + 19² = 590.
Square roots near √590 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √587 | √587 | 24.2281 | No |
| √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 |
- The cube root of 590 is about 8.387207.
- Squaring undoes the root: (√590)² = 590, while 590² = 348,100 — the number whose square root is 590.
Frequently asked questions
What is the square root of 590?
The square root of 590 is √590, about 24.2899156030. The negative root, −24.289916, also squares to 590.
Is the square root of 590 rational or irrational?
Irrational. 590 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 √590 be simplified?
No. 590 = 2 × 5 × 59 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √590 rounded to two decimal places?
√590 ≈ 24.29 to two decimal places (24.3 to one, 24.290 to three). Check: 24.29² = 590.0041, close to 590.