√995 at a glance
- Exact value
- √995
- Decimal (10 places)
- 31.5436205912
- Rounded
- 31.5 · 31.54 · 31.544
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.543621
- Prime factorization
- 5 × 199
- Cube root
- 9.983305
How to simplify √995
The prime factorization of 995 is 5 × 199. Every prime appears only once, so there is no pair to bring outside the radical — √995 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 995, 5 and 199 appear an odd number of times, so √995 is irrational and 31.5436205912 is a rounded value.
Where √995 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √995 lies between 31 and 32. 995 is 34 above 961 and 29 below 1,024, so the root is closer to 32.
- Straight line between 961 and 1,024: 31.5397 (0.01% low)
- Tangent from 31, i.e. 31 + 34 ÷ 62: 31.5484 (0.02% high)
- Tangent from 32, i.e. 32 − 29 ÷ 64: 31.5469 (0.01% high)
For √995 the tangent at 32 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 995 is just 29 below 1,024.
Finding √995 with the Babylonian method
If a guess is too big, 995 ÷ 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√995) in one step.
Start from the nearest whole number, 32 (32² = 1,024):
| Step | Guess x | 995 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 32.0000000000 | 31.0937500000 | 31.5468750000 | 2 |
| 2 | 31.5468750000 | 31.5403665181 | 31.5436207590 | 6 |
| 3 | 31.5436207590 | 31.5436204233 | 31.5436205912 | 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 √995 = 31.5436205912 to every decimal shown.
√995 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √995 the pattern is [31; 1, 1, 5, 4, 3, 12, 3, 4, 5, 1, 1, 62] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √995 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 | 5.4 × 10⁻¹ |
| 32/1 | 32.0000000000 | 4.6 × 10⁻¹ |
| 63/2 | 31.5000000000 | 4.4 × 10⁻² |
| 347/11 | 31.5454545455 | 1.8 × 10⁻³ |
| 1,451/46 | 31.5434782609 | 1.4 × 10⁻⁴ |
| 4,700/149 | 31.5436241611 | 3.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 995y² = 1. Its smallest solution in positive whole numbers is x = 8,835,999, y = 280,120.
√995 in geometry and everyday measurements
- 995 square feet is 92.4 m². Laid out as a square — a small house footprint or a lot — it is about 31.54 ft (31 ft 7 in) on a side.
- 995 is not a sum of two whole-number squares — the prime factor 199 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √995 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 31 box, because 3² + 5² + 31² = 995.
Square roots near √995 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √997 | √997 | 31.5753 | No |
| √998 | √998 | 31.5911 | No |
- The cube root of 995 is about 9.983305.
- Squaring undoes the root: (√995)² = 995, while 995² = 990,025 — the number whose square root is 995.
Frequently asked questions
What is the square root of 995?
The square root of 995 is √995, about 31.5436205912. The negative root, −31.543621, also squares to 995.
Is the square root of 995 rational or irrational?
Irrational. 995 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 √995 be simplified?
No. 995 = 5 × 199 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √995 rounded to two decimal places?
√995 ≈ 31.54 to two decimal places (31.5 to one, 31.544 to three). Check: 31.54² = 994.7716, close to 995.