√989 at a glance
- Exact value
- √989
- Decimal (10 places)
- 31.4483703870
- Rounded
- 31.4 · 31.45 · 31.448
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.448370
- Prime factorization
- 23 × 43
- Cube root
- 9.963198
How to simplify √989
The prime factorization of 989 is 23 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √989 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 989, 23 and 43 appear an odd number of times, so √989 is irrational and 31.4483703870 is a rounded value.
Where √989 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √989 lies between 31 and 32. 989 is 28 above 961 and 35 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.4444 (0.01% low)
- Tangent from 31, i.e. 31 + 28 ÷ 62: 31.4516 (0.01% high)
- Tangent from 32, i.e. 32 − 35 ÷ 64: 31.4531 (0.02% high)
For √989 the tangent at 31 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 989 is just 28 above 961.
Finding √989 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 989: following the tangent line down to zero simplifies to averaging x with 989 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 989 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.9032258065 | 31.4516129032 | 2 |
| 2 | 31.4516129032 | 31.4451282051 | 31.4483705542 | 6 |
| 3 | 31.4483705542 | 31.4483702199 | 31.4483703870 | 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 √989 = 31.4483703870 to every decimal shown.
√989 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √989 the pattern is [31; 2, 4, 2, 1, 11, 1, 8, 15, 1, 1, 1, 1, …] 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 √989 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 | 4.5 × 10⁻¹ |
| 63/2 | 31.5000000000 | 5.2 × 10⁻² |
| 283/9 | 31.4444444444 | 3.9 × 10⁻³ |
| 629/20 | 31.4500000000 | 1.6 × 10⁻³ |
| 912/29 | 31.4482758621 | 9.5 × 10⁻⁵ |
| 10,661/339 | 31.4483775811 | 7.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 989y² = 1. Its smallest solution in positive whole numbers is x = 550,271,588,560,695, y = 17,497,618,534,396 — 15 digits for x, even though 989 is small, which is what makes Pell’s equation famous.
√989 in geometry and everyday measurements
- 989 square feet is 91.9 m². Laid out as a square — a small house footprint or a lot — it is about 31.45 ft (31 ft 5 in) on a side.
- 989 is not a sum of two whole-number squares — the prime factor 23 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √989 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 12 × 29 box, because 2² + 12² + 29² = 989.
Square roots near √989 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √986 | √986 | 31.4006 | No |
| √987 | √987 | 31.4166 | No |
| √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 |
- The cube root of 989 is about 9.963198.
- Squaring undoes the root: (√989)² = 989, while 989² = 978,121 — the number whose square root is 989.
Frequently asked questions
What is the square root of 989?
The square root of 989 is √989, about 31.4483703870. The negative root, −31.448370, also squares to 989.
Is the square root of 989 rational or irrational?
Irrational. 989 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 √989 be simplified?
No. 989 = 23 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √989 rounded to two decimal places?
√989 ≈ 31.45 to two decimal places (31.4 to one, 31.448 to three). Check: 31.45² = 989.1025, close to 989.