√965 at a glance
- Exact value
- √965
- Decimal (10 places)
- 31.0644491340
- Rounded
- 31.1 · 31.06 · 31.064
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.064449
- Prime factorization
- 5 × 193
- Cube root
- 9.881945
How to simplify √965
The prime factorization of 965 is 5 × 193. Every prime appears only once, so there is no pair to bring outside the radical — √965 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 965, 5 and 193 appear an odd number of times, so √965 is irrational and 31.0644491340 is a rounded value.
Where √965 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √965 lies between 31 and 32. 965 is 4 above 961 and 59 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.0635 (0% low)
- Tangent from 31, i.e. 31 + 4 ÷ 62: 31.0645 (0% high)
- Tangent from 32, i.e. 32 − 59 ÷ 64: 31.0781 (0.04% high)
For √965 the tangent at 31 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 965 is just 4 above 961.
Finding √965 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 965: following the tangent line down to zero simplifies to averaging x with 965 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 965 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.1290322581 | 31.0645161290 | 4 |
| 2 | 31.0645161290 | 31.0643821391 | 31.0644491341 | 10 |
| 3 | 31.0644491341 | 31.0644491339 | 31.0644491340 | all 10 shown |
The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √965 = 31.0644491340 to every decimal shown.
√965 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √965 the pattern is [31; 15, 1, 1, 15, 62] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √965 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 |
|---|---|---|
| 31/1 | 31.0000000000 | 6.4 × 10⁻² |
| 466/15 | 31.0666666667 | 2.2 × 10⁻³ |
| 497/16 | 31.0625000000 | 1.9 × 10⁻³ |
| 963/31 | 31.0645161290 | 6.7 × 10⁻⁵ |
| 14,942/481 | 31.0644490644 | 7.0 × 10⁻⁸ |
| 927,367/29,853 | 31.0644491341 | 7.2 × 10⁻¹¹ |
The same fractions solve Pell’s equation, x² − 965y² = 1. Its smallest solution in positive whole numbers is x = 446,526,729, y = 14,374,204. Because the period is odd, the equation with −1 on the right also has a solution: 14,942² − 965 × 481² = −1.
√965 in geometry and everyday measurements
- 965 square feet is 89.7 m². Laid out as a square — a small house footprint or a lot — it is about 31.06 ft (31 ft 1 in) on a side.
- 965 = 2² + 31² = 17² + 26², so by the Pythagorean theorem √965 is the diagonal of rectangles measuring 2 × 31 and 17 × 26 — and the distance between the points (0, 0) and (2, 31) on a grid.
Square roots near √965 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √962 | √962 | 31.0161 | No |
| √963 | 3√107 | 31.0322 | No |
| √964 | 2√241 | 31.0483 | No |
| √965 | √965 | 31.0644 | No |
| √966 | √966 | 31.0805 | No |
| √967 | √967 | 31.0966 | No |
| √968 | 22√2 | 31.1127 | No |
- The cube root of 965 is about 9.881945.
- Squaring undoes the root: (√965)² = 965, while 965² = 931,225 — the number whose square root is 965.
Frequently asked questions
What is the square root of 965?
The square root of 965 is √965, about 31.0644491340. The negative root, −31.064449, also squares to 965.
Is the square root of 965 rational or irrational?
Irrational. 965 is not a perfect square — it falls between 961 and 1024 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √965 be simplified?
No. 965 = 5 × 193 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √965 rounded to two decimal places?
√965 ≈ 31.06 to two decimal places (31.1 to one, 31.064 to three). Check: 31.06² = 964.7236, close to 965.