√985 at a glance
- Exact value
- √985
- Decimal (10 places)
- 31.3847096530
- Rounded
- 31.4 · 31.38 · 31.385
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.384710
- Prime factorization
- 5 × 197
- Cube root
- 9.949748
How to simplify √985
The prime factorization of 985 is 5 × 197. Every prime appears only once, so there is no pair to bring outside the radical — √985 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 985, 5 and 197 appear an odd number of times, so √985 is irrational and 31.3847096530 is a rounded value.
Where √985 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √985 lies between 31 and 32. 985 is 24 above 961 and 39 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.3810 (0.01% low)
- Tangent from 31, i.e. 31 + 24 ÷ 62: 31.3871 (0.01% high)
- Tangent from 32, i.e. 32 − 39 ÷ 64: 31.3906 (0.02% high)
For √985 the tangent at 31 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 985 is just 24 above 961.
Finding √985 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 985: following the tangent line down to zero simplifies to averaging x with 985 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 985 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.7741935484 | 31.3870967742 | 2 |
| 2 | 31.3870967742 | 31.3823227133 | 31.3847097437 | 7 |
| 3 | 31.3847097437 | 31.3847095622 | 31.3847096530 | 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 √985 = 31.3847096530 to every decimal shown.
√985 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √985 the pattern is [31; 2, 1, 1, 2, 62] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √985 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 | 3.8 × 10⁻¹ |
| 63/2 | 31.5000000000 | 1.2 × 10⁻¹ |
| 94/3 | 31.3333333333 | 5.1 × 10⁻² |
| 157/5 | 31.4000000000 | 1.5 × 10⁻² |
| 408/13 | 31.3846153846 | 9.4 × 10⁻⁵ |
| 25,453/811 | 31.3847102343 | 5.8 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 985y² = 1. Its smallest solution in positive whole numbers is x = 332,929, y = 10,608. Because the period is odd, the equation with −1 on the right also has a solution: 408² − 985 × 13² = −1.
√985 in geometry and everyday measurements
- 985 square feet is 91.5 m². Laid out as a square — a small house footprint or a lot — it is about 31.38 ft (31 ft 5 in) on a side.
- 985 = 12² + 29² = 16² + 27², so by the Pythagorean theorem √985 is the diagonal of rectangles measuring 12 × 29 and 16 × 27 — and the distance between the points (0, 0) and (12, 29) on a grid.
Square roots near √985 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √982 | √982 | 31.3369 | No |
| √983 | √983 | 31.3528 | No |
| √984 | 2√246 | 31.3688 | No |
| √985 | √985 | 31.3847 | No |
| √986 | √986 | 31.4006 | No |
| √987 | √987 | 31.4166 | No |
| √988 | 2√247 | 31.4325 | No |
- The cube root of 985 is about 9.949748.
- Squaring undoes the root: (√985)² = 985, while 985² = 970,225 — the number whose square root is 985.
Frequently asked questions
What is the square root of 985?
The square root of 985 is √985, about 31.3847096530. The negative root, −31.384710, also squares to 985.
Is the square root of 985 rational or irrational?
Irrational. 985 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 √985 be simplified?
No. 985 = 5 × 197 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √985 rounded to two decimal places?
√985 ≈ 31.38 to two decimal places (31.4 to one, 31.385 to three). Check: 31.38² = 984.7044, close to 985.