Square Root of 23
The square root of 23 is approximately 4.795832, or exactly √23 in simplified radical form. No, 23 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √23 sits
4² = 16 and 5² = 25, and 23 falls between them, so √23 is between 4 and 5. It's closer to 5 (7 above 16, 2 below 25), which matches the decimal 4.795832.
| k | k² | Compared to 23 |
|---|---|---|
| 2 | 4 | 19 below |
| 3 | 9 | 14 below |
| 4 | 16 | 7 below |
| 5 | 25 | 2 above |
| 6 | 36 | 13 above |
| 7 | 49 | 26 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 23 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 23 ÷ x | New guess (x + 23÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 5.75 | 4.875 |
| 2 | 4.875 | 4.717949 | 4.796474 |
| 3 | 4.796474 | 4.795189 | 4.795832 |
| 4 | 4.795832 | 4.795831 | 4.795832 |
After 4 passes the guess is 4.795832, already matching √23 = 4.795832 to six decimal places.
Exact form from the prime factors
23 = 23. No prime appears twice, so nothing can be pulled out of the root -- √23 is already in simplest form.
Frequently Asked Questions
What is the square root of 23?
The square root of 23 is approximately 4.795832, or exactly √23 in simplified radical form.
Is 23 a perfect square?
No, 23 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √23 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 23?
√22 is about 4.6904, and √24 is about 4.899. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √23 squared?
Exactly 23 -- squaring undoes the square root. Squaring the rounded decimal 4.795832 gives 23, which is why the radical form √23 is the exact answer and the decimal is an approximation.
What is the negative square root of 23?
−4.795832. Every positive number has two square roots, one positive and one negative, because (−4.795832)² is also 23. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 23
- GCF Calculator
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