√583 at a glance
- Exact value
- √583
- Decimal (10 places)
- 24.1453929353
- Rounded
- 24.1 · 24.15 · 24.145
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.145393
- Prime factorization
- 11 × 53
- Cube root
- 8.353905
How to simplify √583
The prime factorization of 583 is 11 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √583 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 583, 11 and 53 appear an odd number of times, so √583 is irrational and 24.1453929353 is a rounded value.
Where √583 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √583 lies between 24 and 25. 583 is 7 above 576 and 42 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.1429 (0.01% low)
- Tangent from 24, i.e. 24 + 7 ÷ 48: 24.1458 (0% high)
- Tangent from 25, i.e. 25 − 42 ÷ 50: 24.1600 (0.06% high)
For √583 the tangent at 24 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 583 is just 7 above 576.
Finding √583 with the Babylonian method
If a guess is too big, 583 ÷ 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√583) in one step.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 583 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.2916666667 | 24.1458333333 | 3 |
| 2 | 24.1458333333 | 24.1449525453 | 24.1453929393 | 8 |
| 3 | 24.1453929393 | 24.1453929313 | 24.1453929353 | 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 √583 = 24.1453929353 to every decimal shown.
√583 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √583 the pattern is [24; 6, 1, 7, 5, 4, 5, 7, 1, 6, 48] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √583 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 | 1.5 × 10⁻¹ |
| 145/6 | 24.1666666667 | 2.1 × 10⁻² |
| 169/7 | 24.1428571429 | 2.5 × 10⁻³ |
| 1,328/55 | 24.1454545455 | 6.2 × 10⁻⁵ |
| 6,809/282 | 24.1453900709 | 2.9 × 10⁻⁶ |
| 28,564/1,183 | 24.1453930685 | 1.3 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 583y² = 1. Its smallest solution in positive whole numbers is x = 8,429,543, y = 349,116.
√583 in geometry and everyday measurements
- A square garage floor of 583 square feet measures about 24.15 ft (24 ft 2 in) per side, and its corner-to-corner diagonal is √1166 ≈ 34.1 ft.
- 583 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √583 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 √583 as its space diagonal.
Square roots near √583 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √585 | 3√65 | 24.1868 | No |
| √586 | √586 | 24.2074 | No |
- The cube root of 583 is about 8.353905.
- Squaring undoes the root: (√583)² = 583, while 583² = 339,889 — the number whose square root is 583.
Frequently asked questions
What is the square root of 583?
The square root of 583 is √583, about 24.1453929353. The negative root, −24.145393, also squares to 583.
Is the square root of 583 rational or irrational?
Irrational. 583 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 √583 be simplified?
No. 583 = 11 × 53 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √583 rounded to two decimal places?
√583 ≈ 24.15 to two decimal places (24.1 to one, 24.145 to three). Check: 24.15² = 583.2225, close to 583.