√253 at a glance
- Exact value
- √253
- Decimal (10 places)
- 15.9059737206
- Rounded
- 15.9 · 15.91 · 15.906
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.905974
- Prime factorization
- 11 × 23
- Cube root
- 6.324704
How to simplify √253
The prime factorization of 253 is 11 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √253 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 253, 11 and 23 appear an odd number of times, so √253 is irrational and 15.9059737206 is a rounded value.
Where √253 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √253 lies between 15 and 16. 253 is 28 above 225 and 3 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.9032 (0.02% low)
- Tangent from 15, i.e. 15 + 28 ÷ 30: 15.9333 (0.17% high)
- Tangent from 16, i.e. 16 − 3 ÷ 32: 15.9063 (0% high)
For √253 the tangent at 16 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 253 is just 3 below 256.
Finding √253 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 253: following the tangent line down to zero simplifies to averaging x with 253 ÷ x.
Start from the nearest whole number, 16 (16² = 256):
| Step | Guess x | 253 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.8125000000 | 15.9062500000 | 3 |
| 2 | 15.9062500000 | 15.9056974460 | 15.9059737230 | 8 |
| 3 | 15.9059737230 | 15.9059737182 | 15.9059737206 | 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 √253 = 15.9059737206 to every decimal shown.
√253 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √253 the pattern is [15; 1, 9, 1, 1, 1, 2, 1, 7, 4, 2, 2, 2, …] 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 √253 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 |
|---|---|---|
| 15/1 | 15.0000000000 | 9.1 × 10⁻¹ |
| 16/1 | 16.0000000000 | 9.4 × 10⁻² |
| 159/10 | 15.9000000000 | 6.0 × 10⁻³ |
| 175/11 | 15.9090909091 | 3.1 × 10⁻³ |
| 334/21 | 15.9047619048 | 1.2 × 10⁻³ |
| 509/32 | 15.9062500000 | 2.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 253y² = 1. Its smallest solution in positive whole numbers is x = 3,222,617,399, y = 202,604,220.
√253 in geometry and everyday measurements
- A square patio or deck of 253 square feet is about 15.91 ft (15 ft 11 in) on each side, so edging all the way around takes 4 × √253 ≈ 63.6 ft.
- 253 is not a sum of two whole-number squares — the prime factor 11 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √253 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 10 × 12 box, because 3² + 10² + 12² = 253.
Square roots near √253 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √250 | 5√10 | 15.8114 | No |
| √251 | √251 | 15.8430 | No |
| √252 | 6√7 | 15.8745 | No |
| √253 | √253 | 15.9060 | No |
| √254 | √254 | 15.9374 | No |
| √255 | √255 | 15.9687 | No |
| √256 | 16 | 16.0000 | Yes |
- The cube root of 253 is about 6.324704.
- Squaring undoes the root: (√253)² = 253, while 253² = 64,009 — the number whose square root is 253.
Frequently asked questions
What is the square root of 253?
The square root of 253 is √253, about 15.9059737206. The negative root, −15.905974, also squares to 253.
Is the square root of 253 rational or irrational?
Irrational. 253 is not a perfect square — it falls between 225 and 256 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √253 be simplified?
No. 253 = 11 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √253 rounded to two decimal places?
√253 ≈ 15.91 to two decimal places (15.9 to one, 15.906 to three). Check: 15.91² = 253.1281, close to 253.