√803 at a glance
- Exact value
- √803
- Decimal (10 places)
- 28.3372546306
- Rounded
- 28.3 · 28.34 · 28.337
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.337255
- Prime factorization
- 11 × 73
- Cube root
- 9.294767
How to simplify √803
The prime factorization of 803 is 11 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √803 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 803, 11 and 73 appear an odd number of times, so √803 is irrational and 28.3372546306 is a rounded value.
Where √803 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √803 lies between 28 and 29. 803 is 19 above 784 and 38 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.3333 (0.01% low)
- Tangent from 28, i.e. 28 + 19 ÷ 56: 28.3393 (0.01% high)
- Tangent from 29, i.e. 29 − 38 ÷ 58: 28.3448 (0.03% high)
For √803 the tangent at 28 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 803 is just 19 above 784.
Finding √803 with the Babylonian method
If a guess is too big, 803 ÷ 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√803) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 803 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.6785714286 | 28.3392857143 | 2 |
| 2 | 28.3392857143 | 28.3352236925 | 28.3372547034 | 7 |
| 3 | 28.3372547034 | 28.3372545578 | 28.3372546306 | 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 √803 = 28.3372546306 to every decimal shown.
√803 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √803 the pattern is [28; 2, 1, 27, 1, 2, 56] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √803 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 3.4 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.6 × 10⁻¹ |
| 85/3 | 28.3333333333 | 3.9 × 10⁻³ |
| 2,352/83 | 28.3373493976 | 9.5 × 10⁻⁵ |
| 2,437/86 | 28.3372093023 | 4.5 × 10⁻⁵ |
| 7,226/255 | 28.3372549020 | 2.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 803y² = 1. Its smallest solution in positive whole numbers is x = 7,226, y = 255.
√803 in geometry and everyday measurements
- 803 square feet is 74.6 m². Laid out as a square — a small house footprint or a lot — it is about 28.34 ft (28 ft 4 in) on a side.
- 803 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 √803 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 19 × 21 box, because 1² + 19² + 21² = 803.
Square roots near √803 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √800 | 20√2 | 28.2843 | No |
| √801 | 3√89 | 28.3019 | No |
| √802 | √802 | 28.3196 | No |
| √803 | √803 | 28.3373 | No |
| √804 | 2√201 | 28.3549 | No |
| √805 | √805 | 28.3725 | No |
| √806 | √806 | 28.3901 | No |
- The cube root of 803 is about 9.294767.
- Squaring undoes the root: (√803)² = 803, while 803² = 644,809 — the number whose square root is 803.
Frequently asked questions
What is the square root of 803?
The square root of 803 is √803, about 28.3372546306. The negative root, −28.337255, also squares to 803.
Is the square root of 803 rational or irrational?
Irrational. 803 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √803 be simplified?
No. 803 = 11 × 73 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √803 rounded to two decimal places?
√803 ≈ 28.34 to two decimal places (28.3 to one, 28.337 to three). Check: 28.34² = 803.1556, close to 803.