√559 at a glance
- Exact value
- √559
- Decimal (10 places)
- 23.6431808351
- Rounded
- 23.6 · 23.64 · 23.643
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.643181
- Prime factorization
- 13 × 43
- Cube root
- 8.237661
How to simplify √559
The prime factorization of 559 is 13 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √559 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 559, 13 and 43 appear an odd number of times, so √559 is irrational and 23.6431808351 is a rounded value.
Where √559 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √559 lies between 23 and 24. 559 is 30 above 529 and 17 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.6383 (0.02% low)
- Tangent from 23, i.e. 23 + 30 ÷ 46: 23.6522 (0.04% high)
- Tangent from 24, i.e. 24 − 17 ÷ 48: 23.6458 (0.01% high)
For √559 the tangent at 24 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 559 is just 17 below 576.
Finding √559 with the Babylonian method
If a guess is too big, 559 ÷ 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√559) in one step.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 559 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.2916666667 | 23.6458333333 | 2 |
| 2 | 23.6458333333 | 23.6405286344 | 23.6431809838 | 6 |
| 3 | 23.6431809838 | 23.6431806863 | 23.6431808351 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √559 = 23.6431808351 to every decimal shown.
√559 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √559 the pattern is [23; 1, 1, 1, 4, 15, 1, 1, 4, 1, 2, 1, 4, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √559 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 |
|---|---|---|
| 23/1 | 23.0000000000 | 6.4 × 10⁻¹ |
| 24/1 | 24.0000000000 | 3.6 × 10⁻¹ |
| 47/2 | 23.5000000000 | 1.4 × 10⁻¹ |
| 71/3 | 23.6666666667 | 2.3 × 10⁻² |
| 331/14 | 23.6428571429 | 3.2 × 10⁻⁴ |
| 5,036/213 | 23.6431924883 | 1.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 559y² = 1. Its smallest solution in positive whole numbers is x = 506,568,295, y = 21,425,556.
√559 in geometry and everyday measurements
- A square garage floor of 559 square feet measures about 23.64 ft (23 ft 8 in) per side, and its corner-to-corner diagonal is √1118 ≈ 33.4 ft.
- 559 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √559 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 √559 as its space diagonal.
Square roots near √559 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √556 | 2√139 | 23.5797 | No |
| √557 | √557 | 23.6008 | No |
| √558 | 3√62 | 23.6220 | No |
| √559 | √559 | 23.6432 | No |
| √560 | 4√35 | 23.6643 | No |
| √561 | √561 | 23.6854 | No |
| √562 | √562 | 23.7065 | No |
- The cube root of 559 is about 8.237661.
- Squaring undoes the root: (√559)² = 559, while 559² = 312,481 — the number whose square root is 559.
Frequently asked questions
What is the square root of 559?
The square root of 559 is √559, about 23.6431808351. The negative root, −23.643181, also squares to 559.
Is the square root of 559 rational or irrational?
Irrational. 559 is not a perfect square — it falls between 529 and 576 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √559 be simplified?
No. 559 = 13 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √559 rounded to two decimal places?
√559 ≈ 23.64 to two decimal places (23.6 to one, 23.643 to three). Check: 23.64² = 558.8496, close to 559.