√479 at a glance
- Exact value
- √479
- Decimal (10 places)
- 21.8860686282
- Rounded
- 21.9 · 21.89 · 21.886
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.886069
- Prime factorization
- 479
- Cube root
- 7.824294
How to simplify √479
479 is a prime number, so its only factors are 1 and 479. There is no perfect-square factor to pull out, which means √479 is already in its simplest radical form.
The square root of any prime is irrational. If √479 were a fraction a/b in lowest terms, then a² = 479b², so 479 would divide a — and then 479 would divide b too, contradicting “lowest terms.” That is why the decimal 21.8860686282 is only a rounded value.
Where √479 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √479 lies between 21 and 22. 479 is 38 above 441 and 5 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.8837 (0.01% low)
- Tangent from 21, i.e. 21 + 38 ÷ 42: 21.9048 (0.09% high)
- Tangent from 22, i.e. 22 − 5 ÷ 44: 21.8864 (0% high)
For √479 the tangent at 22 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 479 is just 5 below 484.
Finding √479 with the Babylonian method
If a guess is too big, 479 ÷ 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√479) in one step.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 479 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.7727272727 | 21.8863636364 | 3 |
| 2 | 21.8863636364 | 21.8857736241 | 21.8860686302 | 8 |
| 3 | 21.8860686302 | 21.8860686263 | 21.8860686282 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √479 = 21.8860686282 to every decimal shown.
√479 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √479 the pattern is [21; 1, 7, 1, 3, 2, 21, 2, 3, 1, 7, 1, 42] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √479 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 | 8.9 × 10⁻¹ |
| 22/1 | 22.0000000000 | 1.1 × 10⁻¹ |
| 175/8 | 21.8750000000 | 1.1 × 10⁻² |
| 197/9 | 21.8888888889 | 2.8 × 10⁻³ |
| 766/35 | 21.8857142857 | 3.5 × 10⁻⁴ |
| 1,729/79 | 21.8860759494 | 7.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 479y² = 1. Its smallest solution in positive whole numbers is x = 2,989,440, y = 136,591.
√479 in geometry and everyday measurements
- A square garage floor of 479 square feet measures about 21.89 ft (21 ft 11 in) per side, and its corner-to-corner diagonal is √958 ≈ 31 ft.
- 479 is not a sum of two whole-number squares — 479 is itself a prime that is one less than a multiple of 4, which rules that out — so √479 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 √479 as its space diagonal.
Square roots near √479 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √476 | 2√119 | 21.8174 | No |
| √477 | 3√53 | 21.8403 | No |
| √478 | √478 | 21.8632 | No |
| √479 | √479 | 21.8861 | No |
| √480 | 4√30 | 21.9089 | No |
| √481 | √481 | 21.9317 | No |
| √482 | √482 | 21.9545 | No |
- The cube root of 479 is about 7.824294.
- Squaring undoes the root: (√479)² = 479, while 479² = 229,441 — the number whose square root is 479.
Frequently asked questions
What is the square root of 479?
The square root of 479 is √479, about 21.8860686282. The negative root, −21.886069, also squares to 479.
Is the square root of 479 rational or irrational?
Irrational. 479 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 √479 be simplified?
No. 479 is prime, so there is no perfect square to take out of the radical.
What is √479 rounded to two decimal places?
√479 ≈ 21.89 to two decimal places (21.9 to one, 21.886 to three). Check: 21.89² = 479.1721, close to 479.