√863 at a glance
- Exact value
- √863
- Decimal (10 places)
- 29.3768616431
- Rounded
- 29.4 · 29.38 · 29.377
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.376862
- Prime factorization
- 863
- Cube root
- 9.520730
How to simplify √863
863 is a prime number, so its only factors are 1 and 863. There is no perfect-square factor to pull out, which means √863 is already in its simplest radical form.
The square root of any prime is irrational. If √863 were a fraction a/b in lowest terms, then a² = 863b², so 863 would divide a — and then 863 would divide b too, contradicting “lowest terms.” That is why the decimal 29.3768616431 is only a rounded value.
Where √863 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √863 lies between 29 and 30. 863 is 22 above 841 and 37 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.3729 (0.01% low)
- Tangent from 29, i.e. 29 + 22 ÷ 58: 29.3793 (0.01% high)
- Tangent from 30, i.e. 30 − 37 ÷ 60: 29.3833 (0.02% high)
For √863 the tangent at 29 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 863 is just 22 above 841.
Finding √863 with the Babylonian method
If a guess is too big, 863 ÷ 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√863) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 863 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.7586206897 | 29.3793103448 | 2 |
| 2 | 29.3793103448 | 29.3744131455 | 29.3768617452 | 6 |
| 3 | 29.3768617452 | 29.3768615411 | 29.3768616431 | 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 √863 = 29.3768616431 to every decimal shown.
√863 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √863 the pattern is [29; 2, 1, 1, 1, 7, 1, 3, 3, 5, 29, 5, 3, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √863 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 3.8 × 10⁻¹ |
| 59/2 | 29.5000000000 | 1.2 × 10⁻¹ |
| 88/3 | 29.3333333333 | 4.4 × 10⁻² |
| 147/5 | 29.4000000000 | 2.3 × 10⁻² |
| 235/8 | 29.3750000000 | 1.9 × 10⁻³ |
| 1,792/61 | 29.3770491803 | 1.9 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 863y² = 1. Its smallest solution in positive whole numbers is x = 18,524,026,608, y = 630,565,199.
√863 in geometry and everyday measurements
- 863 square feet is 80.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.38 ft (29 ft 5 in) on a side.
- 863 is not a sum of two whole-number squares — 863 is itself a prime that is one less than a multiple of 4, which rules that out — so √863 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 √863 as its space diagonal.
Square roots near √863 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √860 | 2√215 | 29.3258 | No |
| √861 | √861 | 29.3428 | No |
| √862 | √862 | 29.3598 | No |
| √863 | √863 | 29.3769 | No |
| √864 | 12√6 | 29.3939 | No |
| √865 | √865 | 29.4109 | No |
| √866 | √866 | 29.4279 | No |
- The cube root of 863 is about 9.520730.
- Squaring undoes the root: (√863)² = 863, while 863² = 744,769 — the number whose square root is 863.
Frequently asked questions
What is the square root of 863?
The square root of 863 is √863, about 29.3768616431. The negative root, −29.376862, also squares to 863.
Is the square root of 863 rational or irrational?
Irrational. 863 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √863 be simplified?
No. 863 is prime, so there is no perfect square to take out of the radical.
What is √863 rounded to two decimal places?
√863 ≈ 29.38 to two decimal places (29.4 to one, 29.377 to three). Check: 29.38² = 863.1844, close to 863.