√581 at a glance
- Exact value
- √581
- Decimal (10 places)
- 24.1039415864
- Rounded
- 24.1 · 24.10 · 24.104
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.103942
- Prime factorization
- 7 × 83
- Cube root
- 8.344341
How to simplify √581
The prime factorization of 581 is 7 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √581 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 581, 7 and 83 appear an odd number of times, so √581 is irrational and 24.1039415864 is a rounded value.
Where √581 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √581 lies between 24 and 25. 581 is 5 above 576 and 44 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.1020 (0.01% low)
- Tangent from 24, i.e. 24 + 5 ÷ 48: 24.1042 (0% high)
- Tangent from 25, i.e. 25 − 44 ÷ 50: 24.1200 (0.07% high)
For √581 the tangent at 24 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 581 is just 5 above 576.
Finding √581 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 581: following the tangent line down to zero simplifies to averaging x with 581 ÷ x.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 581 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.2083333333 | 24.1041666667 | 3 |
| 2 | 24.1041666667 | 24.1037165082 | 24.1039415874 | 8 |
| 3 | 24.1039415874 | 24.1039415853 | 24.1039415864 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √581 = 24.1039415864 to every decimal shown.
√581 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √581 the pattern is [24; 9, 1, 1, 1, 1, 1, 3, 11, 1, 3, 2, 6, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √581 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 | 1.0 × 10⁻¹ |
| 217/9 | 24.1111111111 | 7.2 × 10⁻³ |
| 241/10 | 24.1000000000 | 3.9 × 10⁻³ |
| 458/19 | 24.1052631579 | 1.3 × 10⁻³ |
| 699/29 | 24.1034482759 | 4.9 × 10⁻⁴ |
| 1,157/48 | 24.1041666667 | 2.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 581y² = 1. Its smallest solution in positive whole numbers is x = 152,071,153,975, y = 6,308,974,548.
√581 in geometry and everyday measurements
- A square garage floor of 581 square feet measures about 24.1 ft (24 ft 1 in) per side, and its corner-to-corner diagonal is √1162 ≈ 34.1 ft.
- 581 is not a sum of two whole-number squares — the prime factor 7 and 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √581 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 24 box, because 1² + 2² + 24² = 581.
Square roots near √581 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √578 | 17√2 | 24.0416 | No |
| √579 | √579 | 24.0624 | No |
| √580 | 2√145 | 24.0832 | No |
| √581 | √581 | 24.1039 | No |
| √582 | √582 | 24.1247 | No |
| √583 | √583 | 24.1454 | No |
| √584 | 2√146 | 24.1661 | No |
- The cube root of 581 is about 8.344341.
- Squaring undoes the root: (√581)² = 581, while 581² = 337,561 — the number whose square root is 581.
Frequently asked questions
What is the square root of 581?
The square root of 581 is √581, about 24.1039415864. The negative root, −24.103942, also squares to 581.
Is the square root of 581 rational or irrational?
Irrational. 581 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 √581 be simplified?
No. 581 = 7 × 83 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √581 rounded to two decimal places?
√581 ≈ 24.10 to two decimal places (24.1 to one, 24.104 to three). Check: 24.10² = 580.81, close to 581.