√993 at a glance
- Exact value
- √993
- Decimal (10 places)
- 31.5119025132
- Rounded
- 31.5 · 31.51 · 31.512
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.511903
- Prime factorization
- 3 × 331
- Cube root
- 9.976612
How to simplify √993
The prime factorization of 993 is 3 × 331. Every prime appears only once, so there is no pair to bring outside the radical — √993 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 993, 3 and 331 appear an odd number of times, so √993 is irrational and 31.5119025132 is a rounded value.
Where √993 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √993 lies between 31 and 32. 993 is 32 above 961 and 31 below 1,024, so the root is closer to 32.
- Straight line between 961 and 1,024: 31.5079 (0.01% low)
- Tangent from 31, i.e. 31 + 32 ÷ 62: 31.5161 (0.01% high)
- Tangent from 32, i.e. 32 − 31 ÷ 64: 31.5156 (0.01% high)
For √993 the tangent at 32 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 993 is just 31 below 1,024.
Finding √993 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 993: following the tangent line down to zero simplifies to averaging x with 993 ÷ x.
Start from the nearest whole number, 32 (32² = 1,024):
| Step | Guess x | 993 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 32.0000000000 | 31.0312500000 | 31.5156250000 | 2 |
| 2 | 31.5156250000 | 31.5081804660 | 31.5119027330 | 6 |
| 3 | 31.5119027330 | 31.5119022933 | 31.5119025132 | 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 √993 = 31.5119025132 to every decimal shown.
√993 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √993 the pattern is [31; 1, 1, 20, 1, 1, 62] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √993 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 | 5.1 × 10⁻¹ |
| 32/1 | 32.0000000000 | 4.9 × 10⁻¹ |
| 63/2 | 31.5000000000 | 1.2 × 10⁻² |
| 1,292/41 | 31.5121951220 | 2.9 × 10⁻⁴ |
| 1,355/43 | 31.5116279070 | 2.7 × 10⁻⁴ |
| 2,647/84 | 31.5119047619 | 2.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 993y² = 1. Its smallest solution in positive whole numbers is x = 2,647, y = 84.
√993 in geometry and everyday measurements
- 993 square feet is 92.3 m². Laid out as a square — a small house footprint or a lot — it is about 31.51 ft (31 ft 6 in) on a side.
- 993 is not a sum of two whole-number squares — the prime factor 3 and 331 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √993 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 31 box, because 4² + 4² + 31² = 993.
Square roots near √993 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √990 | 3√110 | 31.4643 | No |
| √991 | √991 | 31.4802 | No |
| √992 | 4√62 | 31.4960 | No |
| √993 | √993 | 31.5119 | No |
| √994 | √994 | 31.5278 | No |
| √995 | √995 | 31.5436 | No |
| √996 | 2√249 | 31.5595 | No |
- The cube root of 993 is about 9.976612.
- Squaring undoes the root: (√993)² = 993, while 993² = 986,049 — the number whose square root is 993.
Frequently asked questions
What is the square root of 993?
The square root of 993 is √993, about 31.5119025132. The negative root, −31.511903, also squares to 993.
Is the square root of 993 rational or irrational?
Irrational. 993 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 √993 be simplified?
No. 993 = 3 × 331 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √993 rounded to two decimal places?
√993 ≈ 31.51 to two decimal places (31.5 to one, 31.512 to three). Check: 31.51² = 992.8801, close to 993.