√831 at a glance
- Exact value
- √831
- Decimal (10 places)
- 28.8270706108
- Rounded
- 28.8 · 28.83 · 28.827
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.827071
- Prime factorization
- 3 × 277
- Cube root
- 9.401569
How to simplify √831
The prime factorization of 831 is 3 × 277. Every prime appears only once, so there is no pair to bring outside the radical — √831 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 831, 3 and 277 appear an odd number of times, so √831 is irrational and 28.8270706108 is a rounded value.
Where √831 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √831 lies between 28 and 29. 831 is 47 above 784 and 10 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.8246 (0.01% low)
- Tangent from 28, i.e. 28 + 47 ÷ 56: 28.8393 (0.04% high)
- Tangent from 29, i.e. 29 − 10 ÷ 58: 28.8276 (0% high)
For √831 the tangent at 29 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 831 is just 10 below 841.
Finding √831 with the Babylonian method
If a guess is too big, 831 ÷ 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√831) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 831 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.6551724138 | 28.8275862069 | 3 |
| 2 | 28.8275862069 | 28.8265550239 | 28.8270706154 | 8 |
| 3 | 28.8270706154 | 28.8270706062 | 28.8270706108 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √831 = 28.8270706108 to every decimal shown.
√831 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √831 the pattern is [28; 1, 4, 1, 3, 1, 1, 1, 1, 18, 1, 1, 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 √831 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 8.3 × 10⁻¹ |
| 29/1 | 29.0000000000 | 1.7 × 10⁻¹ |
| 144/5 | 28.8000000000 | 2.7 × 10⁻² |
| 173/6 | 28.8333333333 | 6.3 × 10⁻³ |
| 663/23 | 28.8260869565 | 9.8 × 10⁻⁴ |
| 836/29 | 28.8275862069 | 5.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 831y² = 1. Its smallest solution in positive whole numbers is x = 9,799,705, y = 339,948.
√831 in geometry and everyday measurements
- 831 square feet is 77.2 m². Laid out as a square — a small house footprint or a lot — it is about 28.83 ft (28 ft 10 in) on a side.
- 831 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √831 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 √831 as its space diagonal.
Square roots near √831 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √828 | 6√23 | 28.7750 | No |
| √829 | √829 | 28.7924 | No |
| √830 | √830 | 28.8097 | No |
| √831 | √831 | 28.8271 | No |
| √832 | 8√13 | 28.8444 | No |
| √833 | 7√17 | 28.8617 | No |
| √834 | √834 | 28.8791 | No |
- The cube root of 831 is about 9.401569.
- Squaring undoes the root: (√831)² = 831, while 831² = 690,561 — the number whose square root is 831.
Frequently asked questions
What is the square root of 831?
The square root of 831 is √831, about 28.8270706108. The negative root, −28.827071, also squares to 831.
Is the square root of 831 rational or irrational?
Irrational. 831 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √831 be simplified?
No. 831 = 3 × 277 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √831 rounded to two decimal places?
√831 ≈ 28.83 to two decimal places (28.8 to one, 28.827 to three). Check: 28.83² = 831.1689, close to 831.