√431 at a glance
- Exact value
- √431
- Decimal (10 places)
- 20.7605394920
- Rounded
- 20.8 · 20.76 · 20.761
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.760539
- Prime factorization
- 431
- Cube root
- 7.553689
How to simplify √431
431 is a prime number, so its only factors are 1 and 431. There is no perfect-square factor to pull out, which means √431 is already in its simplest radical form.
The square root of any prime is irrational. If √431 were a fraction a/b in lowest terms, then a² = 431b², so 431 would divide a — and then 431 would divide b too, contradicting “lowest terms.” That is why the decimal 20.7605394920 is only a rounded value.
Where √431 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √431 lies between 20 and 21. 431 is 31 above 400 and 10 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.7561 (0.02% low)
- Tangent from 20, i.e. 20 + 31 ÷ 40: 20.7750 (0.07% high)
- Tangent from 21, i.e. 21 − 10 ÷ 42: 20.7619 (0.01% high)
For √431 the tangent at 21 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 431 is just 10 below 441.
Finding √431 with the Babylonian method
If a guess is too big, 431 ÷ 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√431) in one step.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 431 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.5238095238 | 20.7619047619 | 2 |
| 2 | 20.7619047619 | 20.7591743119 | 20.7605395369 | 7 |
| 3 | 20.7605395369 | 20.7605394471 | 20.7605394920 | 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 √431 = 20.7605394920 to every decimal shown.
√431 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √431 the pattern is [20; 1, 3, 5, 1, 2, 7, 1, 19, 1, 7, 2, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √431 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 7.6 × 10⁻¹ |
| 21/1 | 21.0000000000 | 2.4 × 10⁻¹ |
| 83/4 | 20.7500000000 | 1.1 × 10⁻² |
| 436/21 | 20.7619047619 | 1.4 × 10⁻³ |
| 519/25 | 20.7600000000 | 5.4 × 10⁻⁴ |
| 1,474/71 | 20.7605633803 | 2.4 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 431y² = 1. Its smallest solution in positive whole numbers is x = 151,560,720, y = 7,300,423.
√431 in geometry and everyday measurements
- A square garage floor of 431 square feet measures about 20.76 ft (20 ft 9 in) per side, and its corner-to-corner diagonal is √862 ≈ 29.4 ft.
- 431 is not a sum of two whole-number squares — 431 is itself a prime that is one less than a multiple of 4, which rules that out — so √431 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 √431 as its space diagonal.
Square roots near √431 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √428 | 2√107 | 20.6882 | No |
| √429 | √429 | 20.7123 | No |
| √430 | √430 | 20.7364 | No |
| √431 | √431 | 20.7605 | No |
| √432 | 12√3 | 20.7846 | No |
| √433 | √433 | 20.8087 | No |
| √434 | √434 | 20.8327 | No |
- The cube root of 431 is about 7.553689.
- Squaring undoes the root: (√431)² = 431, while 431² = 185,761 — the number whose square root is 431.
Frequently asked questions
What is the square root of 431?
The square root of 431 is √431, about 20.7605394920. The negative root, −20.760539, also squares to 431.
Is the square root of 431 rational or irrational?
Irrational. 431 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √431 be simplified?
No. 431 is prime, so there is no perfect square to take out of the radical.
What is √431 rounded to two decimal places?
√431 ≈ 20.76 to two decimal places (20.8 to one, 20.761 to three). Check: 20.76² = 430.9776, close to 431.