√239 at a glance
- Exact value
- √239
- Decimal (10 places)
- 15.4596248337
- Rounded
- 15.5 · 15.46 · 15.460
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.459625
- Prime factorization
- 239
- Cube root
- 6.205822
How to simplify √239
239 is a prime number, so its only factors are 1 and 239. There is no perfect-square factor to pull out, which means √239 is already in its simplest radical form.
The square root of any prime is irrational. If √239 were a fraction a/b in lowest terms, then a² = 239b², so 239 would divide a — and then 239 would divide b too, contradicting “lowest terms.” That is why the decimal 15.4596248337 is only a rounded value.
Where √239 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √239 lies between 15 and 16. 239 is 14 above 225 and 17 below 256, so the root is closer to 15.
- Straight line between 225 and 256: 15.4516 (0.05% low)
- Tangent from 15, i.e. 15 + 14 ÷ 30: 15.4667 (0.05% high)
- Tangent from 16, i.e. 16 − 17 ÷ 32: 15.4688 (0.06% high)
For √239 the tangent at 15 wins, missing by only 0.007. Tangent estimates shine when the number sits close to a perfect square — here 239 is just 14 above 225.
Finding √239 with the Babylonian method
If a guess is too big, 239 ÷ 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√239) in one step.
Start from the nearest whole number, 15 (15² = 225):
| Step | Guess x | 239 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 15.9333333333 | 15.4666666667 | 2 |
| 2 | 15.4666666667 | 15.4525862069 | 15.4596264368 | 5 |
| 3 | 15.4596264368 | 15.4596232307 | 15.4596248337 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √239 = 15.4596248337 to every decimal shown.
√239 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √239 the pattern is [15; 2, 5, 1, 2, 4, 15, 4, 2, 1, 5, 2, 30] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √239 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 |
|---|---|---|
| 15/1 | 15.0000000000 | 4.6 × 10⁻¹ |
| 31/2 | 15.5000000000 | 4.0 × 10⁻² |
| 170/11 | 15.4545454545 | 5.1 × 10⁻³ |
| 201/13 | 15.4615384615 | 1.9 × 10⁻³ |
| 572/37 | 15.4594594595 | 1.7 × 10⁻⁴ |
| 2,489/161 | 15.4596273292 | 2.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 239y² = 1. Its smallest solution in positive whole numbers is x = 6,195,120, y = 400,729.
√239 in geometry and everyday measurements
- A square patio or deck of 239 square feet is about 15.46 ft (15 ft 6 in) on each side, so edging all the way around takes 4 × √239 ≈ 61.8 ft.
- 239 is not a sum of two whole-number squares — 239 is itself a prime that is one less than a multiple of 4, which rules that out — so √239 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 √239 as its space diagonal.
Square roots near √239 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √236 | 2√59 | 15.3623 | No |
| √237 | √237 | 15.3948 | No |
| √238 | √238 | 15.4272 | No |
| √239 | √239 | 15.4596 | No |
| √240 | 4√15 | 15.4919 | No |
| √241 | √241 | 15.5242 | No |
| √242 | 11√2 | 15.5563 | No |
- The cube root of 239 is about 6.205822.
- Four times the radicand doubles the root: √956 = 2 × √239 ≈ 30.91925.
Frequently asked questions
What is the square root of 239?
The square root of 239 is √239, about 15.4596248337. The negative root, −15.459625, also squares to 239.
Is the square root of 239 rational or irrational?
Irrational. 239 is not a perfect square — it falls between 225 and 256 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √239 be simplified?
No. 239 is prime, so there is no perfect square to take out of the radical.
What is √239 rounded to two decimal places?
√239 ≈ 15.46 to two decimal places (15.5 to one, 15.460 to three). Check: 15.46² = 239.0116, close to 239.