√23 at a glance
- Exact value
- √23
- Decimal (10 places)
- 4.7958315233
- Rounded
- 4.8 · 4.80 · 4.796
- Perfect square?
- No — between 4² and 5²
- Rational?
- Irrational
- Both square roots
- ±4.795832
- Prime factorization
- 23
- Cube root
- 2.843867
How to simplify √23
23 is a prime number, so its only factors are 1 and 23. There is no perfect-square factor to pull out, which means √23 is already in its simplest radical form.
The square root of any prime is irrational. If √23 were a fraction a/b in lowest terms, then a² = 23b², so 23 would divide a — and then 23 would divide b too, contradicting “lowest terms.” That is why the decimal 4.7958315233 is only a rounded value.
Where √23 sits between perfect squares
16 = 4² and 25 = 5² are the nearest perfect squares, so √23 lies between 4 and 5. 23 is 7 above 16 and 2 below 25, so the root is closer to 5.
- Straight line between 16 and 25: 4.7778 (0.38% low)
- Tangent from 4, i.e. 4 + 7 ÷ 8: 4.8750 (1.65% high)
- Tangent from 5, i.e. 5 − 2 ÷ 10: 4.8000 (0.09% high)
For √23 the tangent at 5 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 23 is just 2 below 25.
Finding √23 with the Babylonian method
If a guess is too big, 23 ÷ 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√23) in one step.
Start from the nearest whole number, 5 (5² = 25):
| Step | Guess x | 23 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 4.6000000000 | 4.8000000000 | 2 |
| 2 | 4.8000000000 | 4.7916666667 | 4.7958333333 | 5 |
| 3 | 4.7958333333 | 4.7958297133 | 4.7958315233 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √23 = 4.7958315233 to every decimal shown.
√23 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √23 the pattern is [4; 1, 3, 1, 8] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √23 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 |
|---|---|---|
| 4/1 | 4.0000000000 | 8.0 × 10⁻¹ |
| 5/1 | 5.0000000000 | 2.0 × 10⁻¹ |
| 19/4 | 4.7500000000 | 4.6 × 10⁻² |
| 24/5 | 4.8000000000 | 4.2 × 10⁻³ |
| 211/44 | 4.7954545455 | 3.8 × 10⁻⁴ |
| 235/49 | 4.7959183673 | 8.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 23y² = 1. Its smallest solution in positive whole numbers is x = 24, y = 5.
√23 in geometry and everyday measurements
- A square tile with an area of 23 square inches has sides about 4.796 in long.
- 23 is not a sum of two whole-number squares — 23 is itself a prime that is one less than a multiple of 4, which rules that out — so √23 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 √23 as its space diagonal.
Square roots near √23 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √20 | 2√5 | 4.4721 | No |
| √21 | √21 | 4.5826 | No |
| √22 | √22 | 4.6904 | No |
| √23 | √23 | 4.7958 | No |
| √24 | 2√6 | 4.8990 | No |
| √25 | 5 | 5.0000 | Yes |
| √26 | √26 | 5.0990 | No |
- The cube root of 23 is about 2.843867.
- Four times the radicand doubles the root: √92 = 2 × √23 ≈ 9.591663.
Frequently asked questions
What is the square root of 23?
The square root of 23 is √23, about 4.7958315233. The negative root, −4.795832, also squares to 23.
Is the square root of 23 rational or irrational?
Irrational. 23 is not a perfect square — it falls between 16 and 25 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √23 be simplified?
No. 23 is prime, so there is no perfect square to take out of the radical.
What is √23 rounded to two decimal places?
√23 ≈ 4.80 to two decimal places (4.8 to one, 4.796 to three). Check: 4.80² = 23.04, close to 23.