√998 at a glance
- Exact value
- √998
- Decimal (10 places)
- 31.5911379979
- Rounded
- 31.6 · 31.59 · 31.591
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.591138
- Prime factorization
- 2 × 499
- Cube root
- 9.993329
How to simplify √998
The prime factorization of 998 is 2 × 499. Every prime appears only once, so there is no pair to bring outside the radical — √998 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 998, 2 and 499 appear an odd number of times, so √998 is irrational and 31.5911379979 is a rounded value.
Where √998 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √998 lies between 31 and 32. 998 is 37 above 961 and 26 below 1,024, so the root is closer to 32.
- Straight line between 961 and 1,024: 31.5873 (0.01% low)
- Tangent from 31, i.e. 31 + 37 ÷ 62: 31.5968 (0.02% high)
- Tangent from 32, i.e. 32 − 26 ÷ 64: 31.5938 (0.01% high)
For √998 the tangent at 32 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 998 is just 26 below 1,024.
Finding √998 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 32 (32² = 1,024):
| Step | Guess x | 998 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 32.0000000000 | 31.1875000000 | 31.5937500000 | 2 |
| 2 | 31.5937500000 | 31.5885262117 | 31.5911381058 | 6 |
| 3 | 31.5911381058 | 31.5911378899 | 31.5911379979 | 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 √998 = 31.5911379979 to every decimal shown.
√998 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √998 the pattern is [31; 1, 1, 2, 4, 8, 1, 3, 1, 30, 1, 3, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √998 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 | 5.9 × 10⁻¹ |
| 32/1 | 32.0000000000 | 4.1 × 10⁻¹ |
| 63/2 | 31.5000000000 | 9.1 × 10⁻² |
| 158/5 | 31.6000000000 | 8.9 × 10⁻³ |
| 695/22 | 31.5909090909 | 2.3 × 10⁻⁴ |
| 5,718/181 | 31.5911602210 | 2.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 998y² = 1. Its smallest solution in positive whole numbers is x = 984,076,901, y = 31,150,410.
√998 in geometry and everyday measurements
- 998 square feet is 92.7 m². Laid out as a square — a small house footprint or a lot — it is about 31.59 ft (31 ft 7 in) on a side.
- 998 is not a sum of two whole-number squares — the prime factor 499 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √998 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 31 box, because 1² + 6² + 31² = 998.
Square roots near √998 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √994 | √994 | 31.5278 | No |
| √995 | √995 | 31.5436 | No |
| √996 | 2√249 | 31.5595 | No |
| √997 | √997 | 31.5753 | No |
| √998 | √998 | 31.5911 | No |
| √999 | 3√111 | 31.6070 | No |
| √1000 | 10√10 | 31.6228 | No |
- The cube root of 998 is about 9.993329.
- Squaring undoes the root: (√998)² = 998, while 998² = 996,004 — the number whose square root is 998.
Frequently asked questions
What is the square root of 998?
The square root of 998 is √998, about 31.5911379979. The negative root, −31.591138, also squares to 998.
Is the square root of 998 rational or irrational?
Irrational. 998 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 √998 be simplified?
No. 998 = 2 × 499 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √998 rounded to two decimal places?
√998 ≈ 31.59 to two decimal places (31.6 to one, 31.591 to three). Check: 31.59² = 997.9281, close to 998.