√402 at a glance
- Exact value
- √402
- Decimal (10 places)
- 20.0499376558
- Rounded
- 20.0 · 20.05 · 20.050
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.049938
- Prime factorization
- 2 × 3 × 67
- Cube root
- 7.380323
How to simplify √402
The prime factorization of 402 is 2 × 3 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √402 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 402, 2, 3 and 67 appear an odd number of times, so √402 is irrational and 20.0499376558 is a rounded value.
Where √402 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √402 lies between 20 and 21. 402 is 2 above 400 and 39 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.0488 (0.01% low)
- Tangent from 20, i.e. 20 + 2 ÷ 40: 20.0500 (0% high)
- Tangent from 21, i.e. 21 − 39 ÷ 42: 20.0714 (0.11% high)
For √402 the tangent at 20 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 402 is just 2 above 400.
Finding √402 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, 20 (20² = 400):
| Step | Guess x | 402 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.1000000000 | 20.0500000000 | 4 |
| 2 | 20.0500000000 | 20.0498753117 | 20.0499376559 | 10 |
| 3 | 20.0499376559 | 20.0499376557 | 20.0499376558 | all 10 shown |
The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √402 = 20.0499376558 to every decimal shown.
√402 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √402 the pattern is [20; 20, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √402 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 | 5.0 × 10⁻² |
| 401/20 | 20.0500000000 | 6.2 × 10⁻⁵ |
| 16,060/801 | 20.0499375780 | 7.8 × 10⁻⁸ |
| 321,601/16,040 | 20.0499376559 | 9.7 × 10⁻¹¹ |
| 12,880,100/642,401 | 20.0499376558 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 402y² = 1. Its smallest solution in positive whole numbers is x = 401, y = 20.
√402 in geometry and everyday measurements
- A square garage floor of 402 square feet measures about 20.05 ft (20 ft 1 in) per side, and its corner-to-corner diagonal is √804 ≈ 28.4 ft.
- 402 is not a sum of two whole-number squares — the prime factor 3 and 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √402 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 20 box, because 1² + 1² + 20² = 402.
Square roots near √402 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √399 | √399 | 19.9750 | No |
| √400 | 20 | 20.0000 | Yes |
| √401 | √401 | 20.0250 | No |
| √402 | √402 | 20.0499 | No |
| √403 | √403 | 20.0749 | No |
| √404 | 2√101 | 20.0998 | No |
| √405 | 9√5 | 20.1246 | No |
- The cube root of 402 is about 7.380323.
- Squaring undoes the root: (√402)² = 402, while 402² = 161,604 — the number whose square root is 402.
Frequently asked questions
What is the square root of 402?
The square root of 402 is √402, about 20.0499376558. The negative root, −20.049938, also squares to 402.
Is the square root of 402 rational or irrational?
Irrational. 402 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 √402 be simplified?
No. 402 = 2 × 3 × 67 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √402 rounded to two decimal places?
√402 ≈ 20.05 to two decimal places (20.0 to one, 20.050 to three). Check: 20.05² = 402.0025, close to 402.