√403 at a glance
- Exact value
- √403
- Decimal (10 places)
- 20.0748598999
- Rounded
- 20.1 · 20.07 · 20.075
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.074860
- Prime factorization
- 13 × 31
- Cube root
- 7.386437
How to simplify √403
The prime factorization of 403 is 13 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √403 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 403, 13 and 31 appear an odd number of times, so √403 is irrational and 20.0748598999 is a rounded value.
Where √403 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √403 lies between 20 and 21. 403 is 3 above 400 and 38 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.0732 (0.01% low)
- Tangent from 20, i.e. 20 + 3 ÷ 40: 20.0750 (0% high)
- Tangent from 21, i.e. 21 − 38 ÷ 42: 20.0952 (0.1% high)
For √403 the tangent at 20 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 403 is just 3 above 400.
Finding √403 with the Babylonian method
If a guess is too big, 403 ÷ 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√403) in one step.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 403 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.1500000000 | 20.0750000000 | 3 |
| 2 | 20.0750000000 | 20.0747198007 | 20.0748599004 | 9 |
| 3 | 20.0748599004 | 20.0748598994 | 20.0748598999 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √403 = 20.0748598999 to every decimal shown.
√403 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √403 the pattern is [20; 13, 2, 1, 3, 1, 3, 1, 2, 13, 40] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √403 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 | 7.5 × 10⁻² |
| 261/13 | 20.0769230769 | 2.1 × 10⁻³ |
| 542/27 | 20.0740740741 | 7.9 × 10⁻⁴ |
| 803/40 | 20.0750000000 | 1.4 × 10⁻⁴ |
| 2,951/147 | 20.0748299320 | 3.0 × 10⁻⁵ |
| 3,754/187 | 20.0748663102 | 6.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 403y² = 1. Its smallest solution in positive whole numbers is x = 669,878, y = 33,369.
√403 in geometry and everyday measurements
- A square garage floor of 403 square feet measures about 20.07 ft (20 ft 1 in) per side, and its corner-to-corner diagonal is √806 ≈ 28.4 ft.
- 403 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √403 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 13 × 15 box, because 3² + 13² + 15² = 403.
Square roots near √403 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √406 | √406 | 20.1494 | No |
- The cube root of 403 is about 7.386437.
- Squaring undoes the root: (√403)² = 403, while 403² = 162,409 — the number whose square root is 403.
Frequently asked questions
What is the square root of 403?
The square root of 403 is √403, about 20.0748598999. The negative root, −20.074860, also squares to 403.
Is the square root of 403 rational or irrational?
Irrational. 403 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 √403 be simplified?
No. 403 = 13 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √403 rounded to two decimal places?
√403 ≈ 20.07 to two decimal places (20.1 to one, 20.075 to three). Check: 20.07² = 402.8049, close to 403.