√142 at a glance
- Exact value
- √142
- Decimal (10 places)
- 11.9163752878
- Rounded
- 11.9 · 11.92 · 11.916
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.916375
- Prime factorization
- 2 × 71
- Cube root
- 5.217103
How to simplify √142
The prime factorization of 142 is 2 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √142 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 142, 2 and 71 appear an odd number of times, so √142 is irrational and 11.9163752878 is a rounded value.
Where √142 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √142 lies between 11 and 12. 142 is 21 above 121 and 2 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.9130 (0.03% low)
- Tangent from 11, i.e. 11 + 21 ÷ 22: 11.9545 (0.32% high)
- Tangent from 12, i.e. 12 − 2 ÷ 24: 11.9167 (0% high)
For √142 the tangent at 12 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 142 is just 2 below 144.
Finding √142 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, 12 (12² = 144):
| Step | Guess x | 142 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.8333333333 | 11.9166666667 | 3 |
| 2 | 11.9166666667 | 11.9160839161 | 11.9163752914 | 8 |
| 3 | 11.9163752914 | 11.9163752843 | 11.9163752878 | 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 √142 = 11.9163752878 to every decimal shown.
√142 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √142 the pattern is [11; 1, 10, 1, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √142 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 |
|---|---|---|
| 11/1 | 11.0000000000 | 9.2 × 10⁻¹ |
| 12/1 | 12.0000000000 | 8.4 × 10⁻² |
| 131/11 | 11.9090909091 | 7.3 × 10⁻³ |
| 143/12 | 11.9166666667 | 2.9 × 10⁻⁴ |
| 3,277/275 | 11.9163636364 | 1.2 × 10⁻⁵ |
| 3,420/287 | 11.9163763066 | 1.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 142y² = 1. Its smallest solution in positive whole numbers is x = 143, y = 12.
√142 in geometry and everyday measurements
- A square room or garden bed covering 142 square feet measures about 11.92 ft (11 ft 11 in) along each wall.
- 142 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √142 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 6 × 9 box, because 5² + 6² + 9² = 142.
Square roots near √142 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √139 | √139 | 11.7898 | No |
| √140 | 2√35 | 11.8322 | No |
| √141 | √141 | 11.8743 | No |
| √142 | √142 | 11.9164 | No |
| √143 | √143 | 11.9583 | No |
| √144 | 12 | 12.0000 | Yes |
| √145 | √145 | 12.0416 | No |
- The cube root of 142 is about 5.217103.
- Four times the radicand doubles the root: √568 = 2 × √142 ≈ 23.832751.
Frequently asked questions
What is the square root of 142?
The square root of 142 is √142, about 11.9163752878. The negative root, −11.916375, also squares to 142.
Is the square root of 142 rational or irrational?
Irrational. 142 is not a perfect square — it falls between 121 and 144 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √142 be simplified?
No. 142 = 2 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √142 rounded to two decimal places?
√142 ≈ 11.92 to two decimal places (11.9 to one, 11.916 to three). Check: 11.92² = 142.0864, close to 142.