√967 at a glance
- Exact value
- √967
- Decimal (10 places)
- 31.0966236109
- Rounded
- 31.1 · 31.10 · 31.097
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.096624
- Prime factorization
- 967
- Cube root
- 9.888767
How to simplify √967
967 is a prime number, so its only factors are 1 and 967. There is no perfect-square factor to pull out, which means √967 is already in its simplest radical form.
The square root of any prime is irrational. If √967 were a fraction a/b in lowest terms, then a² = 967b², so 967 would divide a — and then 967 would divide b too, contradicting “lowest terms.” That is why the decimal 31.0966236109 is only a rounded value.
Where √967 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √967 lies between 31 and 32. 967 is 6 above 961 and 57 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.0952 (0% low)
- Tangent from 31, i.e. 31 + 6 ÷ 62: 31.0968 (0% high)
- Tangent from 32, i.e. 32 − 57 ÷ 64: 31.1094 (0.04% high)
For √967 the tangent at 31 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 967 is just 6 above 961.
Finding √967 with the Babylonian method
If a guess is too big, 967 ÷ 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√967) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 967 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.1935483871 | 31.0967741935 | 3 |
| 2 | 31.0967741935 | 31.0964730290 | 31.0966236113 | 9 |
| 3 | 31.0966236113 | 31.0966236106 | 31.0966236109 | 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 √967 = 31.0966236109 to every decimal shown.
√967 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √967 the pattern is [31; 10, 2, 1, 6, 4, 3, 2, 2, 1, 1, 8, 3, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √967 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 | 9.7 × 10⁻² |
| 311/10 | 31.1000000000 | 3.4 × 10⁻³ |
| 653/21 | 31.0952380952 | 1.4 × 10⁻³ |
| 964/31 | 31.0967741935 | 1.5 × 10⁻⁴ |
| 6,437/207 | 31.0966183575 | 5.3 × 10⁻⁶ |
| 26,712/859 | 31.0966239814 | 3.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 967y² = 1. Its smallest solution in positive whole numbers is x = 4,649,532,557,817,485,528, y = 149,518,887,194,649,693 — 19 digits for x, even though 967 is small, which is what makes Pell’s equation famous.
√967 in geometry and everyday measurements
- 967 square feet is 89.8 m². Laid out as a square — a small house footprint or a lot — it is about 31.1 ft (31 ft 1 in) on a side.
- 967 is not a sum of two whole-number squares — 967 is itself a prime that is one less than a multiple of 4, which rules that out — so √967 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 √967 as its space diagonal.
Square roots near √967 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √969 | √969 | 31.1288 | No |
| √970 | √970 | 31.1448 | No |
- The cube root of 967 is about 9.888767.
- Squaring undoes the root: (√967)² = 967, while 967² = 935,089 — the number whose square root is 967.
Frequently asked questions
What is the square root of 967?
The square root of 967 is √967, about 31.0966236109. The negative root, −31.096624, also squares to 967.
Is the square root of 967 rational or irrational?
Irrational. 967 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 √967 be simplified?
No. 967 is prime, so there is no perfect square to take out of the radical.
What is √967 rounded to two decimal places?
√967 ≈ 31.10 to two decimal places (31.1 to one, 31.097 to three). Check: 31.10² = 967.21, close to 967.