√983 at a glance
- Exact value
- √983
- Decimal (10 places)
- 31.3528308132
- Rounded
- 31.4 · 31.35 · 31.353
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.352831
- Prime factorization
- 983
- Cube root
- 9.943009
How to simplify √983
983 is a prime number, so its only factors are 1 and 983. There is no perfect-square factor to pull out, which means √983 is already in its simplest radical form.
The square root of any prime is irrational. If √983 were a fraction a/b in lowest terms, then a² = 983b², so 983 would divide a — and then 983 would divide b too, contradicting “lowest terms.” That is why the decimal 31.3528308132 is only a rounded value.
Where √983 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √983 lies between 31 and 32. 983 is 22 above 961 and 41 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.3492 (0.01% low)
- Tangent from 31, i.e. 31 + 22 ÷ 62: 31.3548 (0.01% high)
- Tangent from 32, i.e. 32 − 41 ÷ 64: 31.3594 (0.02% high)
For √983 the tangent at 31 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 983 is just 22 above 961.
Finding √983 with the Babylonian method
If a guess is too big, 983 ÷ 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√983) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 983 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.7096774194 | 31.3548387097 | 2 |
| 2 | 31.3548387097 | 31.3508230453 | 31.3528308775 | 7 |
| 3 | 31.3528308775 | 31.3528307489 | 31.3528308132 | 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 √983 = 31.3528308132 to every decimal shown.
√983 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √983 the pattern is [31; 2, 1, 5, 31, 5, 1, 2, 62] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √983 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 | 3.5 × 10⁻¹ |
| 63/2 | 31.5000000000 | 1.5 × 10⁻¹ |
| 94/3 | 31.3333333333 | 1.9 × 10⁻² |
| 533/17 | 31.3529411765 | 1.1 × 10⁻⁴ |
| 16,617/530 | 31.3528301887 | 6.2 × 10⁻⁷ |
| 83,618/2,667 | 31.3528308961 | 8.3 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 983y² = 1. Its smallest solution in positive whole numbers is x = 284,088, y = 9,061.
√983 in geometry and everyday measurements
- 983 square feet is 91.3 m². Laid out as a square — a small house footprint or a lot — it is about 31.35 ft (31 ft 4 in) on a side.
- 983 is not a sum of two whole-number squares — 983 is itself a prime that is one less than a multiple of 4, which rules that out — so √983 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 √983 as its space diagonal.
Square roots near √983 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √980 | 14√5 | 31.3050 | No |
| √981 | 3√109 | 31.3209 | No |
| √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 |
- The cube root of 983 is about 9.943009.
- Squaring undoes the root: (√983)² = 983, while 983² = 966,289 — the number whose square root is 983.
Frequently asked questions
What is the square root of 983?
The square root of 983 is √983, about 31.3528308132. The negative root, −31.352831, also squares to 983.
Is the square root of 983 rational or irrational?
Irrational. 983 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 √983 be simplified?
No. 983 is prime, so there is no perfect square to take out of the radical.
What is √983 rounded to two decimal places?
√983 ≈ 31.35 to two decimal places (31.4 to one, 31.353 to three). Check: 31.35² = 982.8225, close to 983.