√991 at a glance
- Exact value
- √991
- Decimal (10 places)
- 31.4801524774
- Rounded
- 31.5 · 31.48 · 31.480
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.480152
- Prime factorization
- 991
- Cube root
- 9.969910
How to simplify √991
991 is a prime number, so its only factors are 1 and 991. There is no perfect-square factor to pull out, which means √991 is already in its simplest radical form.
The square root of any prime is irrational. If √991 were a fraction a/b in lowest terms, then a² = 991b², so 991 would divide a — and then 991 would divide b too, contradicting “lowest terms.” That is why the decimal 31.4801524774 is only a rounded value.
Where √991 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √991 lies between 31 and 32. 991 is 30 above 961 and 33 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.4762 (0.01% low)
- Tangent from 31, i.e. 31 + 30 ÷ 62: 31.4839 (0.01% high)
- Tangent from 32, i.e. 32 − 33 ÷ 64: 31.4844 (0.01% high)
For √991 the tangent at 31 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 991 is just 30 above 961.
Finding √991 with the Babylonian method
If a guess is too big, 991 ÷ 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√991) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 991 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.9677419355 | 31.4838709677 | 2 |
| 2 | 31.4838709677 | 31.4764344262 | 31.4801526970 | 6 |
| 3 | 31.4801526970 | 31.4801522578 | 31.4801524774 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √991 = 31.4801524774 to every decimal shown.
√991 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √991 the pattern is [31; 2, 12, 10, 2, 2, 2, 1, 1, 2, 6, 1, 1, …] with the block of 60 terms after the semicolon repeating forever (only the first 12 of the 60 are shown). A pattern that never ends is one more proof that √991 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 × 10⁻¹ |
| 63/2 | 31.5000000000 | 2.0 × 10⁻² |
| 787/25 | 31.4800000000 | 1.5 × 10⁻⁴ |
| 7,933/252 | 31.4801587302 | 6.3 × 10⁻⁶ |
| 16,653/529 | 31.4801512287 | 1.2 × 10⁻⁶ |
| 41,239/1,310 | 31.4801526718 | 1.9 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 991y² = 1. Its smallest solution in positive whole numbers is x = 379,516,400,906,811,930,638,014,896,080, y = 12,055,735,790,331,359,447,442,538,767 — 30 digits for x, even though 991 is small, which is what makes Pell’s equation famous.
√991 in geometry and everyday measurements
- 991 square feet is 92.1 m². Laid out as a square — a small house footprint or a lot — it is about 31.48 ft (31 ft 6 in) on a side.
- 991 is not a sum of two whole-number squares — 991 is itself a prime that is one less than a multiple of 4, which rules that out — so √991 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 √991 as its space diagonal.
Square roots near √991 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √988 | 2√247 | 31.4325 | No |
| √989 | √989 | 31.4484 | No |
| √990 | 3√110 | 31.4643 | No |
| √991 | √991 | 31.4802 | No |
| √992 | 4√62 | 31.4960 | No |
| √993 | √993 | 31.5119 | No |
| √994 | √994 | 31.5278 | No |
- The cube root of 991 is about 9.969910.
- Squaring undoes the root: (√991)² = 991, while 991² = 982,081 — the number whose square root is 991.
Frequently asked questions
What is the square root of 991?
The square root of 991 is √991, about 31.4801524774. The negative root, −31.480152, also squares to 991.
Is the square root of 991 rational or irrational?
Irrational. 991 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 √991 be simplified?
No. 991 is prime, so there is no perfect square to take out of the radical.
What is √991 rounded to two decimal places?
√991 ≈ 31.48 to two decimal places (31.5 to one, 31.480 to three). Check: 31.48² = 990.9904, close to 991.