√110 at a glance
- Exact value
- √110
- Decimal (10 places)
- 10.4880884817
- Rounded
- 10.5 · 10.49 · 10.488
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.488088
- Prime factorization
- 2 × 5 × 11
- Cube root
- 4.791420
How to simplify √110
The prime factorization of 110 is 2 × 5 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √110 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 110, 2, 5 and 11 appear an odd number of times, so √110 is irrational and 10.4880884817 is a rounded value.
Where √110 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √110 lies between 10 and 11. 110 is 10 above 100 and 11 below 121, so the root is closer to 10.
- Straight line between 100 and 121: 10.4762 (0.11% low)
- Tangent from 10, i.e. 10 + 10 ÷ 20: 10.5000 (0.11% high)
- Tangent from 11, i.e. 11 − 11 ÷ 22: 10.5000 (0.11% high)
For √110 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √110 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 10 (10² = 100):
| Step | Guess x | 110 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 11.0000000000 | 10.5000000000 | 1 |
| 2 | 10.5000000000 | 10.4761904762 | 10.4880952381 | 5 |
| 3 | 10.4880952381 | 10.4880817253 | 10.4880884817 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √110 = 10.4880884817 to every decimal shown.
√110 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √110 the pattern is [10; 2, 20] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √110 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 | 4.9 × 10⁻¹ |
| 21/2 | 10.5000000000 | 1.2 × 10⁻² |
| 430/41 | 10.4878048780 | 2.8 × 10⁻⁴ |
| 881/84 | 10.4880952381 | 6.8 × 10⁻⁶ |
| 18,050/1,721 | 10.4880883207 | 1.6 × 10⁻⁷ |
| 36,981/3,526 | 10.4880884855 | 3.8 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 110y² = 1. Its smallest solution in positive whole numbers is x = 21, y = 2.
√110 in geometry and everyday measurements
- A square room or garden bed covering 110 square feet measures about 10.49 ft (10 ft 6 in) along each wall.
- 110 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √110 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 10 box, because 1² + 3² + 10² = 110.
Square roots near √110 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √107 | √107 | 10.3441 | No |
| √108 | 6√3 | 10.3923 | No |
| √109 | √109 | 10.4403 | No |
| √110 | √110 | 10.4881 | No |
| √111 | √111 | 10.5357 | No |
| √112 | 4√7 | 10.5830 | No |
| √113 | √113 | 10.6301 | No |
- The cube root of 110 is about 4.791420.
- Four times the radicand doubles the root: √440 = 2 × √110 ≈ 20.976177.
Frequently asked questions
What is the square root of 110?
The square root of 110 is √110, about 10.4880884817. The negative root, −10.488088, also squares to 110.
Is the square root of 110 rational or irrational?
Irrational. 110 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 √110 be simplified?
No. 110 = 2 × 5 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √110 rounded to two decimal places?
√110 ≈ 10.49 to two decimal places (10.5 to one, 10.488 to three). Check: 10.49² = 110.0401, close to 110.