√502 at a glance
- Exact value
- √502
- Decimal (10 places)
- 22.4053565024
- Rounded
- 22.4 · 22.41 · 22.405
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.405357
- Prime factorization
- 2 × 251
- Cube root
- 7.947574
How to simplify √502
The prime factorization of 502 is 2 × 251. Every prime appears only once, so there is no pair to bring outside the radical — √502 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 502, 2 and 251 appear an odd number of times, so √502 is irrational and 22.4053565024 is a rounded value.
Where √502 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √502 lies between 22 and 23. 502 is 18 above 484 and 27 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.4000 (0.02% low)
- Tangent from 22, i.e. 22 + 18 ÷ 44: 22.4091 (0.02% high)
- Tangent from 23, i.e. 23 − 27 ÷ 46: 22.4130 (0.03% high)
For √502 the tangent at 22 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 502 is just 18 above 484.
Finding √502 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 502 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.8181818182 | 22.4090909091 | 2 |
| 2 | 22.4090909091 | 22.4016227181 | 22.4053568136 | 6 |
| 3 | 22.4053568136 | 22.4053561912 | 22.4053565024 | 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 √502 = 22.4053565024 to every decimal shown.
√502 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √502 the pattern is [22; 2, 2, 7, 14, 1, 4, 22, 4, 1, 14, 7, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √502 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 | 4.1 × 10⁻¹ |
| 45/2 | 22.5000000000 | 9.5 × 10⁻² |
| 112/5 | 22.4000000000 | 5.4 × 10⁻³ |
| 829/37 | 22.4054054054 | 4.9 × 10⁻⁵ |
| 11,718/523 | 22.4053537285 | 2.8 × 10⁻⁶ |
| 12,547/560 | 22.4053571429 | 6.4 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 502y² = 1. Its smallest solution in positive whole numbers is x = 3,832,352,837, y = 171,046,278.
√502 in geometry and everyday measurements
- A square garage floor of 502 square feet measures about 22.41 ft (22 ft 5 in) per side, and its corner-to-corner diagonal is √1004 ≈ 31.7 ft.
- 502 is not a sum of two whole-number squares — the prime factor 251 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √502 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 22 box, because 3² + 3² + 22² = 502.
Square roots near √502 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √499 | √499 | 22.3383 | No |
| √500 | 10√5 | 22.3607 | No |
| √501 | √501 | 22.3830 | No |
| √502 | √502 | 22.4054 | No |
| √503 | √503 | 22.4277 | No |
| √504 | 6√14 | 22.4499 | No |
| √505 | √505 | 22.4722 | No |
- The cube root of 502 is about 7.947574.
- Squaring undoes the root: (√502)² = 502, while 502² = 252,004 — the number whose square root is 502.
Frequently asked questions
What is the square root of 502?
The square root of 502 is √502, about 22.4053565024. The negative root, −22.405357, also squares to 502.
Is the square root of 502 rational or irrational?
Irrational. 502 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 √502 be simplified?
No. 502 = 2 × 251 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √502 rounded to two decimal places?
√502 ≈ 22.41 to two decimal places (22.4 to one, 22.405 to three). Check: 22.41² = 502.2081, close to 502.