√511 at a glance
- Exact value
- √511
- Decimal (10 places)
- 22.6053091109
- Rounded
- 22.6 · 22.61 · 22.605
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.605309
- Prime factorization
- 7 × 73
- Cube root
- 7.994788
How to simplify √511
The prime factorization of 511 is 7 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √511 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 511, 7 and 73 appear an odd number of times, so √511 is irrational and 22.6053091109 is a rounded value.
Where √511 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √511 lies between 22 and 23. 511 is 27 above 484 and 18 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.6000 (0.02% low)
- Tangent from 22, i.e. 22 + 27 ÷ 44: 22.6136 (0.04% high)
- Tangent from 23, i.e. 23 − 18 ÷ 46: 22.6087 (0.01% high)
For √511 the tangent at 23 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 511 is just 18 below 529.
Finding √511 with the Babylonian method
If a guess is too big, 511 ÷ 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√511) in one step.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 511 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.2173913043 | 22.6086956522 | 2 |
| 2 | 22.6086956522 | 22.6019230769 | 22.6053093645 | 6 |
| 3 | 22.6053093645 | 22.6053088573 | 22.6053091109 | 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 √511 = 22.6053091109 to every decimal shown.
√511 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √511 the pattern is [22; 1, 1, 1, 1, 6, 1, 14, 4, 1, 21, 1, 4, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √511 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 6.1 × 10⁻¹ |
| 23/1 | 23.0000000000 | 3.9 × 10⁻¹ |
| 45/2 | 22.5000000000 | 1.1 × 10⁻¹ |
| 68/3 | 22.6666666667 | 6.1 × 10⁻² |
| 113/5 | 22.6000000000 | 5.3 × 10⁻³ |
| 746/33 | 22.6060606061 | 7.5 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 511y² = 1. Its smallest solution in positive whole numbers is x = 4,188,548,960, y = 185,290,497.
√511 in geometry and everyday measurements
- A square garage floor of 511 square feet measures about 22.61 ft (22 ft 7 in) per side, and its corner-to-corner diagonal is √1022 ≈ 32 ft.
- 511 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √511 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 √511 as its space diagonal.
Square roots near √511 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √508 | 2√127 | 22.5389 | No |
| √509 | √509 | 22.5610 | No |
| √510 | √510 | 22.5832 | No |
| √511 | √511 | 22.6053 | No |
| √512 | 16√2 | 22.6274 | No |
| √513 | 3√57 | 22.6495 | No |
| √514 | √514 | 22.6716 | No |
- The cube root of 511 is about 7.994788.
- Squaring undoes the root: (√511)² = 511, while 511² = 261,121 — the number whose square root is 511.
Frequently asked questions
What is the square root of 511?
The square root of 511 is √511, about 22.6053091109. The negative root, −22.605309, also squares to 511.
Is the square root of 511 rational or irrational?
Irrational. 511 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √511 be simplified?
No. 511 = 7 × 73 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √511 rounded to two decimal places?
√511 ≈ 22.61 to two decimal places (22.6 to one, 22.605 to three). Check: 22.61² = 511.2121, close to 511.