√902 at a glance
- Exact value
- √902
- Decimal (10 places)
- 30.0333148354
- Rounded
- 30.0 · 30.03 · 30.033
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.033315
- Prime factorization
- 2 × 11 × 41
- Cube root
- 9.662040
How to simplify √902
The prime factorization of 902 is 2 × 11 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √902 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 902, 2, 11 and 41 appear an odd number of times, so √902 is irrational and 30.0333148354 is a rounded value.
Where √902 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √902 lies between 30 and 31. 902 is 2 above 900 and 59 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.0328 (0% low)
- Tangent from 30, i.e. 30 + 2 ÷ 60: 30.0333 (0% high)
- Tangent from 31, i.e. 31 − 59 ÷ 62: 30.0484 (0.05% high)
For √902 the tangent at 30 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 902 is just 2 above 900.
Finding √902 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 902 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.0666666667 | 30.0333333333 | 4 |
| 2 | 30.0333333333 | 30.0332963374 | 30.0333148354 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √902 = 30.0333148354 to every decimal shown.
√902 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √902 the pattern is [30; 30, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √902 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 | 3.3 × 10⁻² |
| 901/30 | 30.0333333333 | 1.8 × 10⁻⁵ |
| 54,090/1,801 | 30.0333148251 | 1.0 × 10⁻⁸ |
| 1,623,601/54,060 | 30.0333148354 | < 10⁻¹⁰ |
| 97,470,150/3,245,401 | 30.0333148354 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 902y² = 1. Its smallest solution in positive whole numbers is x = 901, y = 30.
√902 in geometry and everyday measurements
- 902 square feet is 83.8 m². Laid out as a square — a small house footprint or a lot — it is about 30.03 ft (30 ft) on a side.
- 902 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √902 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 30 box, because 1² + 1² + 30² = 902.
Square roots near √902 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √899 | √899 | 29.9833 | No |
| √900 | 30 | 30.0000 | Yes |
| √901 | √901 | 30.0167 | No |
| √902 | √902 | 30.0333 | No |
| √903 | √903 | 30.0500 | No |
| √904 | 2√226 | 30.0666 | No |
| √905 | √905 | 30.0832 | No |
- The cube root of 902 is about 9.662040.
- Squaring undoes the root: (√902)² = 902, while 902² = 813,604 — the number whose square root is 902.
Frequently asked questions
What is the square root of 902?
The square root of 902 is √902, about 30.0333148354. The negative root, −30.033315, also squares to 902.
Is the square root of 902 rational or irrational?
Irrational. 902 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 √902 be simplified?
No. 902 = 2 × 11 × 41 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √902 rounded to two decimal places?
√902 ≈ 30.03 to two decimal places (30.0 to one, 30.033 to three). Check: 30.03² = 901.8009, close to 902.