√113 at a glance
- Exact value
- √113
- Decimal (10 places)
- 10.6301458127
- Rounded
- 10.6 · 10.63 · 10.630
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.630146
- Prime factorization
- 113
- Cube root
- 4.834588
How to simplify √113
113 is a prime number, so its only factors are 1 and 113. There is no perfect-square factor to pull out, which means √113 is already in its simplest radical form.
The square root of any prime is irrational. If √113 were a fraction a/b in lowest terms, then a² = 113b², so 113 would divide a — and then 113 would divide b too, contradicting “lowest terms.” That is why the decimal 10.6301458127 is only a rounded value.
Where √113 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √113 lies between 10 and 11. 113 is 13 above 100 and 8 below 121, so the root is closer to 11.
- Straight line between 100 and 121: 10.6190 (0.1% low)
- Tangent from 10, i.e. 10 + 13 ÷ 20: 10.6500 (0.19% high)
- Tangent from 11, i.e. 11 − 8 ÷ 22: 10.6364 (0.06% high)
For √113 the tangent at 11 wins, missing by only 0.0062. Tangent estimates shine when the number sits close to a perfect square — here 113 is just 8 below 121.
Finding √113 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 113: following the tangent line down to zero simplifies to averaging x with 113 ÷ x.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 113 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 10.2727272727 | 10.6363636364 | 2 |
| 2 | 10.6363636364 | 10.6239316239 | 10.6301476301 | 5 |
| 3 | 10.6301476301 | 10.6301439953 | 10.6301458127 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √113 = 10.6301458127 to every decimal shown.
√113 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √113 the pattern is [10; 1, 1, 1, 2, 2, 1, 1, 1, 20] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √113 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 | 6.3 × 10⁻¹ |
| 11/1 | 11.0000000000 | 3.7 × 10⁻¹ |
| 21/2 | 10.5000000000 | 1.3 × 10⁻¹ |
| 32/3 | 10.6666666667 | 3.7 × 10⁻² |
| 85/8 | 10.6250000000 | 5.1 × 10⁻³ |
| 202/19 | 10.6315789474 | 1.4 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 113y² = 1. Its smallest solution in positive whole numbers is x = 1,204,353, y = 113,296. Because the period is odd, the equation with −1 on the right also has a solution: 776² − 113 × 73² = −1.
√113 in geometry and everyday measurements
- A square room or garden bed covering 113 square feet measures about 10.63 ft (10 ft 8 in) along each wall.
- 113 = 7² + 8², so by the Pythagorean theorem √113 is the diagonal of a 7 × 8 rectangle — and the distance between the points (0, 0) and (7, 8) on a grid.
Square roots near √113 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √110 | √110 | 10.4881 | No |
| √111 | √111 | 10.5357 | No |
| √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 |
- The cube root of 113 is about 4.834588.
- Four times the radicand doubles the root: √452 = 2 × √113 ≈ 21.260292.
Frequently asked questions
What is the square root of 113?
The square root of 113 is √113, about 10.6301458127. The negative root, −10.630146, also squares to 113.
Is the square root of 113 rational or irrational?
Irrational. 113 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 √113 be simplified?
No. 113 is prime, so there is no perfect square to take out of the radical.
What is √113 rounded to two decimal places?
√113 ≈ 10.63 to two decimal places (10.6 to one, 10.630 to three). Check: 10.63² = 112.9969, close to 113.