√933 at a glance
- Exact value
- √933
- Decimal (10 places)
- 30.5450486986
- Rounded
- 30.5 · 30.55 · 30.545
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.545049
- Prime factorization
- 3 × 311
- Cube root
- 9.771485
How to simplify √933
The prime factorization of 933 is 3 × 311. Every prime appears only once, so there is no pair to bring outside the radical — √933 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 933, 3 and 311 appear an odd number of times, so √933 is irrational and 30.5450486986 is a rounded value.
Where √933 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √933 lies between 30 and 31. 933 is 33 above 900 and 28 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.5410 (0.01% low)
- Tangent from 30, i.e. 30 + 33 ÷ 60: 30.5500 (0.02% high)
- Tangent from 31, i.e. 31 − 28 ÷ 62: 30.5484 (0.01% high)
For √933 the tangent at 31 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 933 is just 28 below 961.
Finding √933 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 933: following the tangent line down to zero simplifies to averaging x with 933 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 933 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.0967741935 | 30.5483870968 | 2 |
| 2 | 30.5483870968 | 30.5417106653 | 30.5450488810 | 6 |
| 3 | 30.5450488810 | 30.5450485162 | 30.5450486986 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √933 = 30.5450486986 to every decimal shown.
√933 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √933 the pattern is [30; 1, 1, 5, 20, 5, 1, 1, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √933 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 | 5.5 × 10⁻¹ |
| 31/1 | 31.0000000000 | 4.5 × 10⁻¹ |
| 61/2 | 30.5000000000 | 4.5 × 10⁻² |
| 336/11 | 30.5454545455 | 4.1 × 10⁻⁴ |
| 6,781/222 | 30.5450450450 | 3.7 × 10⁻⁶ |
| 34,241/1,121 | 30.5450490633 | 3.6 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 933y² = 1. Its smallest solution in positive whole numbers is x = 75,263, y = 2,464.
√933 in geometry and everyday measurements
- 933 square feet is 86.7 m². Laid out as a square — a small house footprint or a lot — it is about 30.55 ft (30 ft 7 in) on a side.
- 933 is not a sum of two whole-number squares — the prime factor 3 and 311 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √933 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 16 × 26 box, because 1² + 16² + 26² = 933.
Square roots near √933 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √930 | √930 | 30.4959 | No |
| √931 | 7√19 | 30.5123 | No |
| √932 | 2√233 | 30.5287 | No |
| √933 | √933 | 30.5450 | No |
| √934 | √934 | 30.5614 | No |
| √935 | √935 | 30.5778 | No |
| √936 | 6√26 | 30.5941 | No |
- The cube root of 933 is about 9.771485.
- Squaring undoes the root: (√933)² = 933, while 933² = 870,489 — the number whose square root is 933.
Frequently asked questions
What is the square root of 933?
The square root of 933 is √933, about 30.5450486986. The negative root, −30.545049, also squares to 933.
Is the square root of 933 rational or irrational?
Irrational. 933 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 √933 be simplified?
No. 933 = 3 × 311 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √933 rounded to two decimal places?
√933 ≈ 30.55 to two decimal places (30.5 to one, 30.545 to three). Check: 30.55² = 933.3025, close to 933.