√839 at a glance
- Exact value
- √839
- Decimal (10 places)
- 28.9654967159
- Rounded
- 29.0 · 28.97 · 28.965
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.965497
- Prime factorization
- 839
- Cube root
- 9.431642
How to simplify √839
839 is a prime number, so its only factors are 1 and 839. There is no perfect-square factor to pull out, which means √839 is already in its simplest radical form.
The square root of any prime is irrational. If √839 were a fraction a/b in lowest terms, then a² = 839b², so 839 would divide a — and then 839 would divide b too, contradicting “lowest terms.” That is why the decimal 28.9654967159 is only a rounded value.
Where √839 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √839 lies between 28 and 29. 839 is 55 above 784 and 2 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.9649 (0% low)
- Tangent from 28, i.e. 28 + 55 ÷ 56: 28.9821 (0.06% high)
- Tangent from 29, i.e. 29 − 2 ÷ 58: 28.9655 (0% high)
For √839 the tangent at 29 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 839 is just 2 below 841.
Finding √839 with the Babylonian method
If a guess is too big, 839 ÷ 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√839) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 839 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.9310344828 | 28.9655172414 | 4 |
| 2 | 28.9655172414 | 28.9654761905 | 28.9654967159 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √839 = 28.9654967159 to every decimal shown.
√839 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √839 the pattern is [28; 1, 27, 1, 56] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √839 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 9.7 × 10⁻¹ |
| 29/1 | 29.0000000000 | 3.5 × 10⁻² |
| 811/28 | 28.9642857143 | 1.2 × 10⁻³ |
| 840/29 | 28.9655172414 | 2.1 × 10⁻⁵ |
| 47,851/1,652 | 28.9654963680 | 3.5 × 10⁻⁷ |
| 48,691/1,681 | 28.9654967281 | 1.2 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 839y² = 1. Its smallest solution in positive whole numbers is x = 840, y = 29.
√839 in geometry and everyday measurements
- 839 square feet is 77.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.97 ft (29 ft) on a side.
- 839 is not a sum of two whole-number squares — 839 is itself a prime that is one less than a multiple of 4, which rules that out — so √839 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 √839 as its space diagonal.
Square roots near √839 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √836 | 2√209 | 28.9137 | No |
| √837 | 3√93 | 28.9310 | No |
| √838 | √838 | 28.9482 | No |
| √839 | √839 | 28.9655 | No |
| √840 | 2√210 | 28.9828 | No |
| √841 | 29 | 29.0000 | Yes |
| √842 | √842 | 29.0172 | No |
- The cube root of 839 is about 9.431642.
- Squaring undoes the root: (√839)² = 839, while 839² = 703,921 — the number whose square root is 839.
Frequently asked questions
What is the square root of 839?
The square root of 839 is √839, about 28.9654967159. The negative root, −28.965497, also squares to 839.
Is the square root of 839 rational or irrational?
Irrational. 839 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √839 be simplified?
No. 839 is prime, so there is no perfect square to take out of the radical.
What is √839 rounded to two decimal places?
√839 ≈ 28.97 to two decimal places (29.0 to one, 28.965 to three). Check: 28.97² = 839.2609, close to 839.