√903 at a glance
- Exact value
- √903
- Decimal (10 places)
- 30.0499584026
- Rounded
- 30.0 · 30.05 · 30.050
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.049958
- Prime factorization
- 3 × 7 × 43
- Cube root
- 9.665610
How to simplify √903
The prime factorization of 903 is 3 × 7 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √903 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 903, 3, 7 and 43 appear an odd number of times, so √903 is irrational and 30.0499584026 is a rounded value.
Where √903 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √903 lies between 30 and 31. 903 is 3 above 900 and 58 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.0492 (0% low)
- Tangent from 30, i.e. 30 + 3 ÷ 60: 30.0500 (0% high)
- Tangent from 31, i.e. 31 − 58 ÷ 62: 30.0645 (0.05% high)
For √903 the tangent at 30 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 903 is just 3 above 900.
Finding √903 with the Babylonian method
If a guess is too big, 903 ÷ 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√903) in one step.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 903 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.1000000000 | 30.0500000000 | 4 |
| 2 | 30.0500000000 | 30.0499168053 | 30.0499584027 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √903 = 30.0499584026 to every decimal shown.
√903 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √903 the pattern is [30; 20, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √903 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 | 5.0 × 10⁻² |
| 601/20 | 30.0500000000 | 4.2 × 10⁻⁵ |
| 36,090/1,201 | 30.0499583680 | 3.5 × 10⁻⁸ |
| 722,401/24,040 | 30.0499584027 | < 10⁻¹⁰ |
| 43,380,150/1,443,601 | 30.0499584026 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 903y² = 1. Its smallest solution in positive whole numbers is x = 601, y = 20.
√903 in geometry and everyday measurements
- 903 square feet is 83.9 m². Laid out as a square — a small house footprint or a lot — it is about 30.05 ft (30 ft 1 in) on a side.
- 903 is not a sum of two whole-number squares — the prime factor 3, 7 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √903 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 √903 as its space diagonal.
Square roots near √903 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √906 | √906 | 30.0998 | No |
- The cube root of 903 is about 9.665610.
- Squaring undoes the root: (√903)² = 903, while 903² = 815,409 — the number whose square root is 903.
Frequently asked questions
What is the square root of 903?
The square root of 903 is √903, about 30.0499584026. The negative root, −30.049958, also squares to 903.
Is the square root of 903 rational or irrational?
Irrational. 903 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 √903 be simplified?
No. 903 = 3 × 7 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √903 rounded to two decimal places?
√903 ≈ 30.05 to two decimal places (30.0 to one, 30.050 to three). Check: 30.05² = 903.0025, close to 903.