√115 at a glance
- Exact value
- √115
- Decimal (10 places)
- 10.7238052948
- Rounded
- 10.7 · 10.72 · 10.724
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.723805
- Prime factorization
- 5 × 23
- Cube root
- 4.862944
How to simplify √115
The prime factorization of 115 is 5 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √115 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 115, 5 and 23 appear an odd number of times, so √115 is irrational and 10.7238052948 is a rounded value.
Where √115 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √115 lies between 10 and 11. 115 is 15 above 100 and 6 below 121, so the root is closer to 11.
- Straight line between 100 and 121: 10.7143 (0.09% low)
- Tangent from 10, i.e. 10 + 15 ÷ 20: 10.7500 (0.24% high)
- Tangent from 11, i.e. 11 − 6 ÷ 22: 10.7273 (0.03% high)
For √115 the tangent at 11 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 115 is just 6 below 121.
Finding √115 with the Babylonian method
If a guess is too big, 115 ÷ 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√115) in one step.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 115 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 10.4545454545 | 10.7272727273 | 2 |
| 2 | 10.7272727273 | 10.7203389831 | 10.7238058552 | 6 |
| 3 | 10.7238058552 | 10.7238047344 | 10.7238052948 | 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 √115 = 10.7238052948 to every decimal shown.
√115 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √115 the pattern is [10; 1, 2, 1, 1, 1, 1, 1, 2, 1, 20] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √115 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 |
|---|---|---|
| 10/1 | 10.0000000000 | 7.2 × 10⁻¹ |
| 11/1 | 11.0000000000 | 2.8 × 10⁻¹ |
| 32/3 | 10.6666666667 | 5.7 × 10⁻² |
| 43/4 | 10.7500000000 | 2.6 × 10⁻² |
| 75/7 | 10.7142857143 | 9.5 × 10⁻³ |
| 118/11 | 10.7272727273 | 3.5 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 115y² = 1. Its smallest solution in positive whole numbers is x = 1,126, y = 105.
√115 in geometry and everyday measurements
- A square room or garden bed covering 115 square feet measures about 10.72 ft (10 ft 9 in) along each wall.
- 115 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √115 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 9 box, because 3² + 5² + 9² = 115.
Square roots near √115 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √112 | 4√7 | 10.5830 | No |
| √113 | √113 | 10.6301 | No |
| √114 | √114 | 10.6771 | No |
| √115 | √115 | 10.7238 | No |
| √116 | 2√29 | 10.7703 | No |
| √117 | 3√13 | 10.8167 | No |
| √118 | √118 | 10.8628 | No |
- The cube root of 115 is about 4.862944.
- Four times the radicand doubles the root: √460 = 2 × √115 ≈ 21.447611.
Frequently asked questions
What is the square root of 115?
The square root of 115 is √115, about 10.7238052948. The negative root, −10.723805, also squares to 115.
Is the square root of 115 rational or irrational?
Irrational. 115 is not a perfect square — it falls between 100 and 121 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √115 be simplified?
No. 115 = 5 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √115 rounded to two decimal places?
√115 ≈ 10.72 to two decimal places (10.7 to one, 10.724 to three). Check: 10.72² = 114.9184, close to 115.