√439 at a glance
- Exact value
- √439
- Decimal (10 places)
- 20.9523268398
- Rounded
- 21.0 · 20.95 · 20.952
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.952327
- Prime factorization
- 439
- Cube root
- 7.600139
How to simplify √439
439 is a prime number, so its only factors are 1 and 439. There is no perfect-square factor to pull out, which means √439 is already in its simplest radical form.
The square root of any prime is irrational. If √439 were a fraction a/b in lowest terms, then a² = 439b², so 439 would divide a — and then 439 would divide b too, contradicting “lowest terms.” That is why the decimal 20.9523268398 is only a rounded value.
Where √439 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √439 lies between 20 and 21. 439 is 39 above 400 and 2 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.9512 (0.01% low)
- Tangent from 20, i.e. 20 + 39 ÷ 40: 20.9750 (0.11% high)
- Tangent from 21, i.e. 21 − 2 ÷ 42: 20.9524 (0% high)
For √439 the tangent at 21 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 439 is just 2 below 441.
Finding √439 with the Babylonian method
If a guess is too big, 439 ÷ 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√439) in one step.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 439 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.9047619048 | 20.9523809524 | 4 |
| 2 | 20.9523809524 | 20.9522727273 | 20.9523268398 | 10 |
| 3 | 20.9523268398 | 20.9523268397 | 20.9523268398 | 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 √439 = 20.9523268398 to every decimal shown.
√439 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √439 the pattern is [20; 1, 19, 1, 40] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √439 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 | 9.5 × 10⁻¹ |
| 21/1 | 21.0000000000 | 4.8 × 10⁻² |
| 419/20 | 20.9500000000 | 2.3 × 10⁻³ |
| 440/21 | 20.9523809524 | 5.4 × 10⁻⁵ |
| 18,019/860 | 20.9523255814 | 1.3 × 10⁻⁶ |
| 18,459/881 | 20.9523269012 | 6.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 439y² = 1. Its smallest solution in positive whole numbers is x = 440, y = 21.
√439 in geometry and everyday measurements
- A square garage floor of 439 square feet measures about 20.95 ft (20 ft 11 in) per side, and its corner-to-corner diagonal is √878 ≈ 29.6 ft.
- 439 is not a sum of two whole-number squares — 439 is itself a prime that is one less than a multiple of 4, which rules that out — so √439 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 √439 as its space diagonal.
Square roots near √439 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √436 | 2√109 | 20.8806 | No |
| √437 | √437 | 20.9045 | No |
| √438 | √438 | 20.9284 | No |
| √439 | √439 | 20.9523 | No |
| √440 | 2√110 | 20.9762 | No |
| √441 | 21 | 21.0000 | Yes |
| √442 | √442 | 21.0238 | No |
- The cube root of 439 is about 7.600139.
- Squaring undoes the root: (√439)² = 439, while 439² = 192,721 — the number whose square root is 439.
Frequently asked questions
What is the square root of 439?
The square root of 439 is √439, about 20.9523268398. The negative root, −20.952327, also squares to 439.
Is the square root of 439 rational or irrational?
Irrational. 439 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 √439 be simplified?
No. 439 is prime, so there is no perfect square to take out of the radical.
What is √439 rounded to two decimal places?
√439 ≈ 20.95 to two decimal places (21.0 to one, 20.952 to three). Check: 20.95² = 438.9025, close to 439.