√231 at a glance
- Exact value
- √231
- Decimal (10 places)
- 15.1986841536
- Rounded
- 15.2 · 15.20 · 15.199
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.198684
- Prime factorization
- 3 × 7 × 11
- Cube root
- 6.135792
How to simplify √231
The prime factorization of 231 is 3 × 7 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √231 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 231, 3, 7 and 11 appear an odd number of times, so √231 is irrational and 15.1986841536 is a rounded value.
Where √231 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √231 lies between 15 and 16. 231 is 6 above 225 and 25 below 256, so the root is closer to 15.
- Straight line between 225 and 256: 15.1935 (0.03% low)
- Tangent from 15, i.e. 15 + 6 ÷ 30: 15.2000 (0.01% high)
- Tangent from 16, i.e. 16 − 25 ÷ 32: 15.2188 (0.13% high)
For √231 the tangent at 15 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 231 is just 6 above 225.
Finding √231 with the Babylonian method
If a guess is too big, 231 ÷ 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√231) in one step.
Start from the nearest whole number, 15 (15² = 225):
| Step | Guess x | 231 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 15.4000000000 | 15.2000000000 | 2 |
| 2 | 15.2000000000 | 15.1973684211 | 15.1986842105 | 7 |
| 3 | 15.1986842105 | 15.1986840966 | 15.1986841536 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √231 = 15.1986841536 to every decimal shown.
√231 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √231 the pattern is [15; 5, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √231 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 | 2.0 × 10⁻¹ |
| 76/5 | 15.2000000000 | 1.3 × 10⁻³ |
| 2,295/151 | 15.1986754967 | 8.7 × 10⁻⁶ |
| 11,551/760 | 15.1986842105 | 5.7 × 10⁻⁸ |
| 348,825/22,951 | 15.1986841532 | 3.7 × 10⁻¹⁰ |
| 1,755,676/115,515 | 15.1986841536 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 231y² = 1. Its smallest solution in positive whole numbers is x = 76, y = 5.
√231 in geometry and everyday measurements
- A square patio or deck of 231 square feet is about 15.2 ft (15 ft 2 in) on each side, so edging all the way around takes 4 × √231 ≈ 60.8 ft.
- 231 is not a sum of two whole-number squares — the prime factor 3, 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √231 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 √231 as its space diagonal.
Square roots near √231 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √228 | 2√57 | 15.0997 | No |
| √229 | √229 | 15.1327 | No |
| √230 | √230 | 15.1658 | No |
| √231 | √231 | 15.1987 | No |
| √232 | 2√58 | 15.2315 | No |
| √233 | √233 | 15.2643 | No |
| √234 | 3√26 | 15.2971 | No |
- The cube root of 231 is about 6.135792.
- Four times the radicand doubles the root: √924 = 2 × √231 ≈ 30.397368.
Frequently asked questions
What is the square root of 231?
The square root of 231 is √231, about 15.1986841536. The negative root, −15.198684, also squares to 231.
Is the square root of 231 rational or irrational?
Irrational. 231 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 √231 be simplified?
No. 231 = 3 × 7 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √231 rounded to two decimal places?
√231 ≈ 15.20 to two decimal places (15.2 to one, 15.199 to three). Check: 15.20² = 231.04, close to 231.