√523 at a glance
- Exact value
- √523
- Decimal (10 places)
- 22.8691932521
- Rounded
- 22.9 · 22.87 · 22.869
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.869193
- Prime factorization
- 523
- Cube root
- 8.056886
How to simplify √523
523 is a prime number, so its only factors are 1 and 523. There is no perfect-square factor to pull out, which means √523 is already in its simplest radical form.
The square root of any prime is irrational. If √523 were a fraction a/b in lowest terms, then a² = 523b², so 523 would divide a — and then 523 would divide b too, contradicting “lowest terms.” That is why the decimal 22.8691932521 is only a rounded value.
Where √523 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √523 lies between 22 and 23. 523 is 39 above 484 and 6 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.8667 (0.01% low)
- Tangent from 22, i.e. 22 + 39 ÷ 44: 22.8864 (0.08% high)
- Tangent from 23, i.e. 23 − 6 ÷ 46: 22.8696 (0% high)
For √523 the tangent at 23 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 523 is just 6 below 529.
Finding √523 with the Babylonian method
If a guess is too big, 523 ÷ 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√523) in one step.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 523 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.7391304348 | 22.8695652174 | 3 |
| 2 | 22.8695652174 | 22.8688212928 | 22.8691932551 | 8 |
| 3 | 22.8691932551 | 22.8691932490 | 22.8691932521 | 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 √523 = 22.8691932521 to every decimal shown.
√523 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √523 the pattern is [22; 1, 6, 1, 1, 1, 4, 2, 3, 14, 1, 21, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √523 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 8.7 × 10⁻¹ |
| 23/1 | 23.0000000000 | 1.3 × 10⁻¹ |
| 160/7 | 22.8571428571 | 1.2 × 10⁻² |
| 183/8 | 22.8750000000 | 5.8 × 10⁻³ |
| 343/15 | 22.8666666667 | 2.5 × 10⁻³ |
| 526/23 | 22.8695652174 | 3.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 523y² = 1. Its smallest solution in positive whole numbers is x = 81,810,300,626, y = 3,577,314,675.
√523 in geometry and everyday measurements
- A square garage floor of 523 square feet measures about 22.87 ft (22 ft 10 in) per side, and its corner-to-corner diagonal is √1046 ≈ 32.3 ft.
- 523 is not a sum of two whole-number squares — 523 is itself a prime that is one less than a multiple of 4, which rules that out — so √523 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 21 box, because 1² + 9² + 21² = 523.
Square roots near √523 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √520 | 2√130 | 22.8035 | No |
| √521 | √521 | 22.8254 | No |
| √522 | 3√58 | 22.8473 | No |
| √523 | √523 | 22.8692 | No |
| √524 | 2√131 | 22.8910 | No |
| √525 | 5√21 | 22.9129 | No |
| √526 | √526 | 22.9347 | No |
- The cube root of 523 is about 8.056886.
- Squaring undoes the root: (√523)² = 523, while 523² = 273,529 — the number whose square root is 523.
Frequently asked questions
What is the square root of 523?
The square root of 523 is √523, about 22.8691932521. The negative root, −22.869193, also squares to 523.
Is the square root of 523 rational or irrational?
Irrational. 523 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √523 be simplified?
No. 523 is prime, so there is no perfect square to take out of the radical.
What is √523 rounded to two decimal places?
√523 ≈ 22.87 to two decimal places (22.9 to one, 22.869 to three). Check: 22.87² = 523.0369, close to 523.