√119 at a glance
- Exact value
- √119
- Decimal (10 places)
- 10.9087121146
- Rounded
- 10.9 · 10.91 · 10.909
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.908712
- Prime factorization
- 7 × 17
- Cube root
- 4.918685
How to simplify √119
The prime factorization of 119 is 7 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √119 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 119, 7 and 17 appear an odd number of times, so √119 is irrational and 10.9087121146 is a rounded value.
Where √119 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √119 lies between 10 and 11. 119 is 19 above 100 and 2 below 121, so the root is closer to 11.
- Straight line between 100 and 121: 10.9048 (0.04% low)
- Tangent from 10, i.e. 10 + 19 ÷ 20: 10.9500 (0.38% high)
- Tangent from 11, i.e. 11 − 2 ÷ 22: 10.9091 (0% high)
For √119 the tangent at 11 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 119 is just 2 below 121.
Finding √119 with the Babylonian method
If a guess is too big, 119 ÷ 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√119) in one step.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 119 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 10.8181818182 | 10.9090909091 | 3 |
| 2 | 10.9090909091 | 10.9083333333 | 10.9087121212 | 8 |
| 3 | 10.9087121212 | 10.9087121081 | 10.9087121146 | 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 √119 = 10.9087121146 to every decimal shown.
√119 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √119 the pattern is [10; 1, 9, 1, 20] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √119 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 | 9.1 × 10⁻¹ |
| 11/1 | 11.0000000000 | 9.1 × 10⁻² |
| 109/10 | 10.9000000000 | 8.7 × 10⁻³ |
| 120/11 | 10.9090909091 | 3.8 × 10⁻⁴ |
| 2,509/230 | 10.9086956522 | 1.6 × 10⁻⁵ |
| 2,629/241 | 10.9087136929 | 1.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 119y² = 1. Its smallest solution in positive whole numbers is x = 120, y = 11.
√119 in geometry and everyday measurements
- A square room or garden bed covering 119 square feet measures about 10.91 ft (10 ft 11 in) along each wall.
- 119 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √119 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 √119 as its space diagonal.
Square roots near √119 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √116 | 2√29 | 10.7703 | No |
| √117 | 3√13 | 10.8167 | No |
| √118 | √118 | 10.8628 | No |
| √119 | √119 | 10.9087 | No |
| √120 | 2√30 | 10.9545 | No |
| √121 | 11 | 11.0000 | Yes |
| √122 | √122 | 11.0454 | No |
- The cube root of 119 is about 4.918685.
- Four times the radicand doubles the root: √476 = 2 × √119 ≈ 21.817424.
Frequently asked questions
What is the square root of 119?
The square root of 119 is √119, about 10.9087121146. The negative root, −10.908712, also squares to 119.
Is the square root of 119 rational or irrational?
Irrational. 119 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 √119 be simplified?
No. 119 = 7 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √119 rounded to two decimal places?
√119 ≈ 10.91 to two decimal places (10.9 to one, 10.909 to three). Check: 10.91² = 119.0281, close to 119.