√955 at a glance
- Exact value
- √955
- Decimal (10 places)
- 30.9030742807
- Rounded
- 30.9 · 30.90 · 30.903
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.903074
- Prime factorization
- 5 × 191
- Cube root
- 9.847692
How to simplify √955
The prime factorization of 955 is 5 × 191. Every prime appears only once, so there is no pair to bring outside the radical — √955 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 955, 5 and 191 appear an odd number of times, so √955 is irrational and 30.9030742807 is a rounded value.
Where √955 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √955 lies between 30 and 31. 955 is 55 above 900 and 6 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.9016 (0% low)
- Tangent from 30, i.e. 30 + 55 ÷ 60: 30.9167 (0.04% high)
- Tangent from 31, i.e. 31 − 6 ÷ 62: 30.9032 (0% high)
For √955 the tangent at 31 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 955 is just 6 below 961.
Finding √955 with the Babylonian method
If a guess is too big, 955 ÷ 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√955) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 955 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.8064516129 | 30.9032258065 | 3 |
| 2 | 30.9032258065 | 30.9029227557 | 30.9030742811 | 9 |
| 3 | 30.9030742811 | 30.9030742804 | 30.9030742807 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √955 = 30.9030742807 to every decimal shown.
√955 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √955 the pattern is [30; 1, 9, 3, 6, 1, 1, 5, 12, 5, 1, 1, 6, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √955 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 | 9.0 × 10⁻¹ |
| 31/1 | 31.0000000000 | 9.7 × 10⁻² |
| 309/10 | 30.9000000000 | 3.1 × 10⁻³ |
| 958/31 | 30.9032258065 | 1.5 × 10⁻⁴ |
| 6,057/196 | 30.9030612245 | 1.3 × 10⁻⁵ |
| 7,015/227 | 30.9030837004 | 9.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 955y² = 1. Its smallest solution in positive whole numbers is x = 2,095,256,249, y = 67,800,900.
√955 in geometry and everyday measurements
- 955 square feet is 88.7 m². Laid out as a square — a small house footprint or a lot — it is about 30.9 ft (30 ft 11 in) on a side.
- 955 is not a sum of two whole-number squares — the prime factor 191 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √955 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 15 × 27 box, because 1² + 15² + 27² = 955.
Square roots near √955 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √952 | 2√238 | 30.8545 | No |
| √953 | √953 | 30.8707 | No |
| √954 | 3√106 | 30.8869 | No |
| √955 | √955 | 30.9031 | No |
| √956 | 2√239 | 30.9192 | No |
| √957 | √957 | 30.9354 | No |
| √958 | √958 | 30.9516 | No |
- The cube root of 955 is about 9.847692.
- Squaring undoes the root: (√955)² = 955, while 955² = 912,025 — the number whose square root is 955.
Frequently asked questions
What is the square root of 955?
The square root of 955 is √955, about 30.9030742807. The negative root, −30.903074, also squares to 955.
Is the square root of 955 rational or irrational?
Irrational. 955 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 √955 be simplified?
No. 955 = 5 × 191 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √955 rounded to two decimal places?
√955 ≈ 30.90 to two decimal places (30.9 to one, 30.903 to three). Check: 30.90² = 954.81, close to 955.