√979 at a glance
- Exact value
- √979
- Decimal (10 places)
- 31.2889756943
- Rounded
- 31.3 · 31.29 · 31.289
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.288976
- Prime factorization
- 11 × 89
- Cube root
- 9.929504
How to simplify √979
The prime factorization of 979 is 11 × 89. Every prime appears only once, so there is no pair to bring outside the radical — √979 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 979, 11 and 89 appear an odd number of times, so √979 is irrational and 31.2889756943 is a rounded value.
Where √979 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √979 lies between 31 and 32. 979 is 18 above 961 and 45 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.2857 (0.01% low)
- Tangent from 31, i.e. 31 + 18 ÷ 62: 31.2903 (0% high)
- Tangent from 32, i.e. 32 − 45 ÷ 64: 31.2969 (0.03% high)
For √979 the tangent at 31 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 979 is just 18 above 961.
Finding √979 with the Babylonian method
If a guess is too big, 979 ÷ 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√979) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 979 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.5806451613 | 31.2903225806 | 2 |
| 2 | 31.2903225806 | 31.2876288660 | 31.2889757233 | 7 |
| 3 | 31.2889757233 | 31.2889756653 | 31.2889756943 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √979 = 31.2889756943 to every decimal shown.
√979 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √979 the pattern is [31; 3, 2, 5, 1, 4, 1, 5, 2, 3, 62] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √979 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 | 2.9 × 10⁻¹ |
| 94/3 | 31.3333333333 | 4.4 × 10⁻² |
| 219/7 | 31.2857142857 | 3.3 × 10⁻³ |
| 1,189/38 | 31.2894736842 | 5.0 × 10⁻⁴ |
| 1,408/45 | 31.2888888889 | 8.7 × 10⁻⁵ |
| 6,821/218 | 31.2889908257 | 1.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 979y² = 1. Its smallest solution in positive whole numbers is x = 360,449, y = 11,520.
√979 in geometry and everyday measurements
- 979 square feet is 91 m². Laid out as a square — a small house footprint or a lot — it is about 31.29 ft (31 ft 3 in) on a side.
- 979 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √979 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 31 box, because 3² + 3² + 31² = 979.
Square roots near √979 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √976 | 4√61 | 31.2410 | No |
| √977 | √977 | 31.2570 | No |
| √978 | √978 | 31.2730 | No |
| √979 | √979 | 31.2890 | No |
| √980 | 14√5 | 31.3050 | No |
| √981 | 3√109 | 31.3209 | No |
| √982 | √982 | 31.3369 | No |
- The cube root of 979 is about 9.929504.
- Squaring undoes the root: (√979)² = 979, while 979² = 958,441 — the number whose square root is 979.
Frequently asked questions
What is the square root of 979?
The square root of 979 is √979, about 31.2889756943. The negative root, −31.288976, also squares to 979.
Is the square root of 979 rational or irrational?
Irrational. 979 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 √979 be simplified?
No. 979 = 11 × 89 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √979 rounded to two decimal places?
√979 ≈ 31.29 to two decimal places (31.3 to one, 31.289 to three). Check: 31.29² = 979.0641, close to 979.