√483 at a glance
- Exact value
- √483
- Decimal (10 places)
- 21.9772609758
- Rounded
- 22.0 · 21.98 · 21.977
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.977261
- Prime factorization
- 3 × 7 × 23
- Cube root
- 7.846013
How to simplify √483
The prime factorization of 483 is 3 × 7 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √483 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 483, 3, 7 and 23 appear an odd number of times, so √483 is irrational and 21.9772609758 is a rounded value.
Where √483 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √483 lies between 21 and 22. 483 is 42 above 441 and 1 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.9767 (0% low)
- Tangent from 21, i.e. 21 + 42 ÷ 42: 22.0000 (0.1% high)
- Tangent from 22, i.e. 22 − 1 ÷ 44: 21.9773 (0% high)
For √483 the tangent at 22 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 483 is just 1 below 484.
Finding √483 with the Babylonian method
If a guess is too big, 483 ÷ 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√483) in one step.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 483 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.9545454545 | 21.9772727273 | 4 |
| 2 | 21.9772727273 | 21.9772492244 | 21.9772609758 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √483 = 21.9772609758 to every decimal shown.
√483 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √483 the pattern is [21; 1, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √483 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 9.8 × 10⁻¹ |
| 22/1 | 22.0000000000 | 2.3 × 10⁻² |
| 945/43 | 21.9767441860 | 5.2 × 10⁻⁴ |
| 967/44 | 21.9772727273 | 1.2 × 10⁻⁵ |
| 41,559/1,891 | 21.9772607086 | 2.7 × 10⁻⁷ |
| 42,526/1,935 | 21.9772609819 | 6.1 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 483y² = 1. Its smallest solution in positive whole numbers is x = 22, y = 1.
√483 in geometry and everyday measurements
- A square garage floor of 483 square feet measures about 21.98 ft (22 ft) per side, and its corner-to-corner diagonal is √966 ≈ 31.1 ft.
- 483 is not a sum of two whole-number squares — the prime factor 3, 7 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √483 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 19 box, because 1² + 11² + 19² = 483.
Square roots near √483 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √480 | 4√30 | 21.9089 | No |
| √481 | √481 | 21.9317 | No |
| √482 | √482 | 21.9545 | No |
| √483 | √483 | 21.9773 | No |
| √484 | 22 | 22.0000 | Yes |
| √485 | √485 | 22.0227 | No |
| √486 | 9√6 | 22.0454 | No |
- The cube root of 483 is about 7.846013.
- Squaring undoes the root: (√483)² = 483, while 483² = 233,289 — the number whose square root is 483.
Frequently asked questions
What is the square root of 483?
The square root of 483 is √483, about 21.9772609758. The negative root, −21.977261, also squares to 483.
Is the square root of 483 rational or irrational?
Irrational. 483 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √483 be simplified?
No. 483 = 3 × 7 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √483 rounded to two decimal places?
√483 ≈ 21.98 to two decimal places (22.0 to one, 21.977 to three). Check: 21.98² = 483.1204, close to 483.