√31 at a glance
- Exact value
- √31
- Decimal (10 places)
- 5.5677643628
- Rounded
- 5.6 · 5.57 · 5.568
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.567764
- Prime factorization
- 31
- Cube root
- 3.141381
How to simplify √31
31 is a prime number, so its only factors are 1 and 31. There is no perfect-square factor to pull out, which means √31 is already in its simplest radical form.
The square root of any prime is irrational. If √31 were a fraction a/b in lowest terms, then a² = 31b², so 31 would divide a — and then 31 would divide b too, contradicting “lowest terms.” That is why the decimal 5.5677643628 is only a rounded value.
Where √31 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √31 lies between 5 and 6. 31 is 6 above 25 and 5 below 36, so the root is closer to 6.
- Straight line between 25 and 36: 5.5455 (0.4% low)
- Tangent from 5, i.e. 5 + 6 ÷ 10: 5.6000 (0.58% high)
- Tangent from 6, i.e. 6 − 5 ÷ 12: 5.5833 (0.28% high)
For √31 the tangent at 6 wins, missing by only 0.0156. Tangent estimates shine when the number sits close to a perfect square — here 31 is just 5 below 36.
Finding √31 with the Babylonian method
If a guess is too big, 31 ÷ 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√31) in one step.
Start from the nearest whole number, 6 (6² = 36):
| Step | Guess x | 31 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 5.1666666667 | 5.5833333333 | 1 |
| 2 | 5.5833333333 | 5.5522388060 | 5.5677860697 | 4 |
| 3 | 5.5677860697 | 5.5677426561 | 5.5677643629 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √31 = 5.5677643628 to every decimal shown.
√31 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √31 the pattern is [5; 1, 1, 3, 5, 3, 1, 1, 10] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √31 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 5.7 × 10⁻¹ |
| 6/1 | 6.0000000000 | 4.3 × 10⁻¹ |
| 11/2 | 5.5000000000 | 6.8 × 10⁻² |
| 39/7 | 5.5714285714 | 3.7 × 10⁻³ |
| 206/37 | 5.5675675676 | 2.0 × 10⁻⁴ |
| 657/118 | 5.5677966102 | 3.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 31y² = 1. Its smallest solution in positive whole numbers is x = 1,520, y = 273.
√31 in geometry and everyday measurements
- A square room or garden bed covering 31 square feet measures about 5.57 ft (5 ft 7 in) along each wall.
- 31 is not a sum of two whole-number squares — 31 is itself a prime that is one less than a multiple of 4, which rules that out — so √31 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 √31 as its space diagonal.
Square roots near √31 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √28 | 2√7 | 5.2915 | No |
| √29 | √29 | 5.3852 | No |
| √30 | √30 | 5.4772 | No |
| √31 | √31 | 5.5678 | No |
| √32 | 4√2 | 5.6569 | No |
| √33 | √33 | 5.7446 | No |
| √34 | √34 | 5.8310 | No |
- The cube root of 31 is about 3.141381.
- Four times the radicand doubles the root: √124 = 2 × √31 ≈ 11.135529.
Frequently asked questions
What is the square root of 31?
The square root of 31 is √31, about 5.5677643628. The negative root, −5.567764, also squares to 31.
Is the square root of 31 rational or irrational?
Irrational. 31 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √31 be simplified?
No. 31 is prime, so there is no perfect square to take out of the radical.
What is √31 rounded to two decimal places?
√31 ≈ 5.57 to two decimal places (5.6 to one, 5.568 to three). Check: 5.57² = 31.0249, close to 31.