√111 at a glance
- Exact value
- √111
- Decimal (10 places)
- 10.5356537529
- Rounded
- 10.5 · 10.54 · 10.536
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.535654
- Prime factorization
- 3 × 37
- Cube root
- 4.805896
How to simplify √111
The prime factorization of 111 is 3 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √111 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 111, 3 and 37 appear an odd number of times, so √111 is irrational and 10.5356537529 is a rounded value.
Where √111 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √111 lies between 10 and 11. 111 is 11 above 100 and 10 below 121, so the root is closer to 11.
- Straight line between 100 and 121: 10.5238 (0.11% low)
- Tangent from 10, i.e. 10 + 11 ÷ 20: 10.5500 (0.14% high)
- Tangent from 11, i.e. 11 − 10 ÷ 22: 10.5455 (0.09% high)
For √111 the tangent at 11 wins, missing by only 0.0098. Tangent estimates shine when the number sits close to a perfect square — here 111 is just 10 below 121.
Finding √111 with the Babylonian method
If a guess is too big, 111 ÷ 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√111) in one step.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 111 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 10.0909090909 | 10.5454545455 | 2 |
| 2 | 10.5454545455 | 10.5258620690 | 10.5356583072 | 5 |
| 3 | 10.5356583072 | 10.5356491985 | 10.5356537529 | 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 √111 = 10.5356537529 to every decimal shown.
√111 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √111 the pattern is [10; 1, 1, 6, 1, 1, 20] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √111 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 | 5.4 × 10⁻¹ |
| 11/1 | 11.0000000000 | 4.6 × 10⁻¹ |
| 21/2 | 10.5000000000 | 3.6 × 10⁻² |
| 137/13 | 10.5384615385 | 2.8 × 10⁻³ |
| 158/15 | 10.5333333333 | 2.3 × 10⁻³ |
| 295/28 | 10.5357142857 | 6.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 111y² = 1. Its smallest solution in positive whole numbers is x = 295, y = 28.
√111 in geometry and everyday measurements
- A square room or garden bed covering 111 square feet measures about 10.54 ft (10 ft 6 in) along each wall.
- 111 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √111 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 √111 as its space diagonal.
Square roots near √111 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √114 | √114 | 10.6771 | No |
- The cube root of 111 is about 4.805896.
- Four times the radicand doubles the root: √444 = 2 × √111 ≈ 21.071308.
Frequently asked questions
What is the square root of 111?
The square root of 111 is √111, about 10.5356537529. The negative root, −10.535654, also squares to 111.
Is the square root of 111 rational or irrational?
Irrational. 111 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 √111 be simplified?
No. 111 = 3 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √111 rounded to two decimal places?
√111 ≈ 10.54 to two decimal places (10.5 to one, 10.536 to three). Check: 10.54² = 111.0916, close to 111.