√964 at a glance
- Exact value
- 2√241
- Decimal (10 places)
- 31.0483493925
- Rounded
- 31.0 · 31.05 · 31.048
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.048349
- Prime factorization
- 2² × 241
- Cube root
- 9.878530
How to simplify √964
Look for the largest perfect square that divides 964. Here it is 4 (2²), because 964 = 4 × 241 and 241 has no square factor left:
The prime factorization tells the same story: 964 = 2² × 241. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 241 stays inside.
Check: (2√241)² = 2² × 241 = 4 × 241 = 964. As a decimal, 2√241 = 2 × 15.5241746963 ≈ 31.0483493925.
Where √964 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √964 lies between 31 and 32. 964 is 3 above 961 and 60 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.0476 (0% low)
- Tangent from 31, i.e. 31 + 3 ÷ 62: 31.0484 (0% high)
- Tangent from 32, i.e. 32 − 60 ÷ 64: 31.0625 (0.05% high)
For √964 the tangent at 31 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 964 is just 3 above 961.
Finding √964 with the Babylonian method
Picture a rectangle with an area of 964 and one side x; the other side must be 964 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √964.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 964 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.0967741935 | 31.0483870968 | 4 |
| 2 | 31.0483870968 | 31.0483116883 | 31.0483493925 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √964 = 31.0483493925 to every decimal shown.
√964 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √964 the pattern is [31; 20, 1, 2, 6, 1, 1, 3, 1, 1, 1, 1, 11, …] with the block of 34 terms after the semicolon repeating forever (only the first 12 of the 34 are shown). A pattern that never ends is one more proof that √964 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 | 4.8 × 10⁻² |
| 621/20 | 31.0500000000 | 1.7 × 10⁻³ |
| 652/21 | 31.0476190476 | 7.3 × 10⁻⁴ |
| 1,925/62 | 31.0483870968 | 3.8 × 10⁻⁵ |
| 12,202/393 | 31.0483460560 | 3.3 × 10⁻⁶ |
| 14,127/455 | 31.0483516484 | 2.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 964y² = 1. Its smallest solution in positive whole numbers is x = 10,085,143,557,001,249, y = 324,820,602,522,300 — 17 digits for x, even though 964 is small, which is what makes Pell’s equation famous.
√964 in geometry and everyday measurements
- 964 square feet is 89.6 m². Laid out as a square — a small house footprint or a lot — it is about 31.05 ft (31 ft 1 in) on a side.
- 964 = 8² + 30², so by the Pythagorean theorem √964 is the diagonal of a 8 × 30 rectangle — and the distance between the points (0, 0) and (8, 30) on a grid.
- Since √964 = 2√241, a length of √964 is exactly 2 copies of the length √241 laid end to end.
Square roots near √964 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √961 | 31 | 31.0000 | Yes |
| √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 |
- The cube root of 964 is about 9.878530.
- Because 964 = 4 × 241, the root is twice √241: 2 × 15.524175 ≈ 31.048349.
Frequently asked questions
What is the square root of 964?
The square root of 964 is 2√241 in simplest radical form, which is about 31.0483493925. The negative root, −31.048349, also squares to 964.
Is the square root of 964 rational or irrational?
Irrational. 964 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 √964 be simplified?
Yes. The largest perfect square dividing 964 is 4, so √964 = √4 × √241 = 2√241.
What is √964 rounded to two decimal places?
√964 ≈ 31.05 to two decimal places (31.0 to one, 31.048 to three). Check: 31.05² = 964.1025, close to 964.