√959 at a glance
- Exact value
- √959
- Decimal (10 places)
- 30.9677251344
- Rounded
- 31.0 · 30.97 · 30.968
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.967725
- Prime factorization
- 7 × 137
- Cube root
- 9.861422
How to simplify √959
The prime factorization of 959 is 7 × 137. Every prime appears only once, so there is no pair to bring outside the radical — √959 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 959, 7 and 137 appear an odd number of times, so √959 is irrational and 30.9677251344 is a rounded value.
Where √959 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √959 lies between 30 and 31. 959 is 59 above 900 and 2 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.9672 (0% low)
- Tangent from 30, i.e. 30 + 59 ÷ 60: 30.9833 (0.05% high)
- Tangent from 31, i.e. 31 − 2 ÷ 62: 30.9677 (0% high)
For √959 the tangent at 31 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 959 is just 2 below 961.
Finding √959 with the Babylonian method
If a guess is too big, 959 ÷ 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√959) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 959 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.9354838710 | 30.9677419355 | 4 |
| 2 | 30.9677419355 | 30.9677083333 | 30.9677251344 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √959 = 30.9677251344 to every decimal shown.
√959 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √959 the pattern is [30; 1, 29, 1, 60] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √959 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 9.7 × 10⁻¹ |
| 31/1 | 31.0000000000 | 3.2 × 10⁻² |
| 929/30 | 30.9666666667 | 1.1 × 10⁻³ |
| 960/31 | 30.9677419355 | 1.7 × 10⁻⁵ |
| 58,529/1,890 | 30.9677248677 | 2.7 × 10⁻⁷ |
| 59,489/1,921 | 30.9677251432 | 8.8 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 959y² = 1. Its smallest solution in positive whole numbers is x = 960, y = 31.
√959 in geometry and everyday measurements
- 959 square feet is 89.1 m². Laid out as a square — a small house footprint or a lot — it is about 30.97 ft (31 ft) on a side.
- 959 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 √959 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 √959 as its space diagonal.
Square roots near √959 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √956 | 2√239 | 30.9192 | No |
| √957 | √957 | 30.9354 | No |
| √958 | √958 | 30.9516 | No |
| √959 | √959 | 30.9677 | No |
| √960 | 8√15 | 30.9839 | No |
| √961 | 31 | 31.0000 | Yes |
| √962 | √962 | 31.0161 | No |
- The cube root of 959 is about 9.861422.
- Squaring undoes the root: (√959)² = 959, while 959² = 919,681 — the number whose square root is 959.
Frequently asked questions
What is the square root of 959?
The square root of 959 is √959, about 30.9677251344. The negative root, −30.967725, also squares to 959.
Is the square root of 959 rational or irrational?
Irrational. 959 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √959 be simplified?
No. 959 = 7 × 137 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √959 rounded to two decimal places?
√959 ≈ 30.97 to two decimal places (31.0 to one, 30.968 to three). Check: 30.97² = 959.1409, close to 959.