√939 at a glance
- Exact value
- √939
- Decimal (10 places)
- 30.6431068921
- Rounded
- 30.6 · 30.64 · 30.643
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.643107
- Prime factorization
- 3 × 313
- Cube root
- 9.792386
How to simplify √939
The prime factorization of 939 is 3 × 313. Every prime appears only once, so there is no pair to bring outside the radical — √939 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 939, 3 and 313 appear an odd number of times, so √939 is irrational and 30.6431068921 is a rounded value.
Where √939 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √939 lies between 30 and 31. 939 is 39 above 900 and 22 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.6393 (0.01% low)
- Tangent from 30, i.e. 30 + 39 ÷ 60: 30.6500 (0.02% high)
- Tangent from 31, i.e. 31 − 22 ÷ 62: 30.6452 (0.01% high)
For √939 the tangent at 31 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 939 is just 22 below 961.
Finding √939 with the Babylonian method
If a guess is too big, 939 ÷ 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√939) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 939 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.2903225806 | 30.6451612903 | 2 |
| 2 | 30.6451612903 | 30.6410526316 | 30.6431069610 | 7 |
| 3 | 30.6431069610 | 30.6431068232 | 30.6431068921 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √939 = 30.6431068921 to every decimal shown.
√939 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √939 the pattern is [30; 1, 1, 1, 4, 20, 4, 1, 1, 1, 60] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √939 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 6.4 × 10⁻¹ |
| 31/1 | 31.0000000000 | 3.6 × 10⁻¹ |
| 61/2 | 30.5000000000 | 1.4 × 10⁻¹ |
| 92/3 | 30.6666666667 | 2.4 × 10⁻² |
| 429/14 | 30.6428571429 | 2.5 × 10⁻⁴ |
| 8,672/283 | 30.6431095406 | 2.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 939y² = 1. Its smallest solution in positive whole numbers is x = 122,695, y = 4,004.
√939 in geometry and everyday measurements
- 939 square feet is 87.2 m². Laid out as a square — a small house footprint or a lot — it is about 30.64 ft (30 ft 8 in) on a side.
- 939 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √939 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 17 × 25 box, because 5² + 17² + 25² = 939.
Square roots near √939 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √936 | 6√26 | 30.5941 | No |
| √937 | √937 | 30.6105 | No |
| √938 | √938 | 30.6268 | No |
| √939 | √939 | 30.6431 | No |
| √940 | 2√235 | 30.6594 | No |
| √941 | √941 | 30.6757 | No |
| √942 | √942 | 30.6920 | No |
- The cube root of 939 is about 9.792386.
- Squaring undoes the root: (√939)² = 939, while 939² = 881,721 — the number whose square root is 939.
Frequently asked questions
What is the square root of 939?
The square root of 939 is √939, about 30.6431068921. The negative root, −30.643107, also squares to 939.
Is the square root of 939 rational or irrational?
Irrational. 939 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √939 be simplified?
No. 939 = 3 × 313 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √939 rounded to two decimal places?
√939 ≈ 30.64 to two decimal places (30.6 to one, 30.643 to three). Check: 30.64² = 938.8096, close to 939.