√823 at a glance
- Exact value
- √823
- Decimal (10 places)
- 28.6879765756
- Rounded
- 28.7 · 28.69 · 28.688
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.687977
- Prime factorization
- 823
- Cube root
- 9.371302
How to simplify √823
823 is a prime number, so its only factors are 1 and 823. There is no perfect-square factor to pull out, which means √823 is already in its simplest radical form.
The square root of any prime is irrational. If √823 were a fraction a/b in lowest terms, then a² = 823b², so 823 would divide a — and then 823 would divide b too, contradicting “lowest terms.” That is why the decimal 28.6879765756 is only a rounded value.
Where √823 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √823 lies between 28 and 29. 823 is 39 above 784 and 18 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.6842 (0.01% low)
- Tangent from 28, i.e. 28 + 39 ÷ 56: 28.6964 (0.03% high)
- Tangent from 29, i.e. 29 − 18 ÷ 58: 28.6897 (0.01% high)
For √823 the tangent at 29 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 823 is just 18 below 841.
Finding √823 with the Babylonian method
If a guess is too big, 823 ÷ 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√823) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 823 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.3793103448 | 28.6896551724 | 2 |
| 2 | 28.6896551724 | 28.6862980769 | 28.6879766247 | 7 |
| 3 | 28.6879766247 | 28.6879765265 | 28.6879765756 | 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 √823 = 28.6879765756 to every decimal shown.
√823 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √823 the pattern is [28; 1, 2, 4, 1, 7, 2, 1, 1, 1, 1, 8, 1, …] with the block of 44 terms after the semicolon repeating forever (only the first 12 of the 44 are shown). A pattern that never ends is one more proof that √823 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 | 6.9 × 10⁻¹ |
| 29/1 | 29.0000000000 | 3.1 × 10⁻¹ |
| 86/3 | 28.6666666667 | 2.1 × 10⁻² |
| 373/13 | 28.6923076923 | 4.3 × 10⁻³ |
| 459/16 | 28.6875000000 | 4.8 × 10⁻⁴ |
| 3,586/125 | 28.6880000000 | 2.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 823y² = 1. Its smallest solution in positive whole numbers is x = 235,170,474,903,644,006,168, y = 8,197,527,430,497,636,651 — 21 digits for x, even though 823 is small, which is what makes Pell’s equation famous.
√823 in geometry and everyday measurements
- 823 square feet is 76.5 m². Laid out as a square — a small house footprint or a lot — it is about 28.69 ft (28 ft 8 in) on a side.
- 823 is not a sum of two whole-number squares — 823 is itself a prime that is one less than a multiple of 4, which rules that out — so √823 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 √823 as its space diagonal.
Square roots near √823 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √820 | 2√205 | 28.6356 | No |
| √821 | √821 | 28.6531 | No |
| √822 | √822 | 28.6705 | No |
| √823 | √823 | 28.6880 | No |
| √824 | 2√206 | 28.7054 | No |
| √825 | 5√33 | 28.7228 | No |
| √826 | √826 | 28.7402 | No |
- The cube root of 823 is about 9.371302.
- Squaring undoes the root: (√823)² = 823, while 823² = 677,329 — the number whose square root is 823.
Frequently asked questions
What is the square root of 823?
The square root of 823 is √823, about 28.6879765756. The negative root, −28.687977, also squares to 823.
Is the square root of 823 rational or irrational?
Irrational. 823 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 √823 be simplified?
No. 823 is prime, so there is no perfect square to take out of the radical.
What is √823 rounded to two decimal places?
√823 ≈ 28.69 to two decimal places (28.7 to one, 28.688 to three). Check: 28.69² = 823.1161, close to 823.