√143 at a glance
- Exact value
- √143
- Decimal (10 places)
- 11.9582607431
- Rounded
- 12.0 · 11.96 · 11.958
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.958261
- Prime factorization
- 11 × 13
- Cube root
- 5.229322
How to simplify √143
The prime factorization of 143 is 11 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √143 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 143, 11 and 13 appear an odd number of times, so √143 is irrational and 11.9582607431 is a rounded value.
Where √143 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √143 lies between 11 and 12. 143 is 22 above 121 and 1 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.9565 (0.01% low)
- Tangent from 11, i.e. 11 + 22 ÷ 22: 12.0000 (0.35% high)
- Tangent from 12, i.e. 12 − 1 ÷ 24: 11.9583 (0% high)
For √143 the tangent at 12 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 143 is just 1 below 144.
Finding √143 with the Babylonian method
If a guess is too big, 143 ÷ 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√143) in one step.
Start from the nearest whole number, 12 (12² = 144):
| Step | Guess x | 143 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.9166666667 | 11.9583333333 | 4 |
| 2 | 11.9583333333 | 11.9581881533 | 11.9582607433 | 9 |
| 3 | 11.9582607433 | 11.9582607429 | 11.9582607431 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √143 = 11.9582607431 to every decimal shown.
√143 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √143 the pattern is [11; 1, 22] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √143 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 | 9.6 × 10⁻¹ |
| 12/1 | 12.0000000000 | 4.2 × 10⁻² |
| 275/23 | 11.9565217391 | 1.7 × 10⁻³ |
| 287/24 | 11.9583333333 | 7.3 × 10⁻⁵ |
| 6,589/551 | 11.9582577132 | 3.0 × 10⁻⁶ |
| 6,876/575 | 11.9582608696 | 1.3 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 143y² = 1. Its smallest solution in positive whole numbers is x = 12, y = 1.
√143 in geometry and everyday measurements
- A square room or garden bed covering 143 square feet measures about 11.96 ft (11 ft 11 in) along each wall.
- 143 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 √143 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 √143 as its space diagonal.
Square roots near √143 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √146 | √146 | 12.0830 | No |
- The cube root of 143 is about 5.229322.
- Four times the radicand doubles the root: √572 = 2 × √143 ≈ 23.916521.
Frequently asked questions
What is the square root of 143?
The square root of 143 is √143, about 11.9582607431. The negative root, −11.958261, also squares to 143.
Is the square root of 143 rational or irrational?
Irrational. 143 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 √143 be simplified?
No. 143 = 11 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √143 rounded to two decimal places?
√143 ≈ 11.96 to two decimal places (12.0 to one, 11.958 to three). Check: 11.96² = 143.0416, close to 143.