Square Root of 83
The square root of 83 is approximately 9.110434, or exactly √83 in simplified radical form. No, 83 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √83 sits
9² = 81 and 10² = 100, and 83 falls between them, so √83 is between 9 and 10. It's closer to 9 (2 above 81, 17 below 100), which matches the decimal 9.110434.
| k | k² | Compared to 83 |
|---|---|---|
| 7 | 49 | 34 below |
| 8 | 64 | 19 below |
| 9 | 81 | 2 below |
| 10 | 100 | 17 above |
| 11 | 121 | 38 above |
| 12 | 144 | 61 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 83 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 83 ÷ x | New guess (x + 83÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.222222 | 9.111111 |
| 2 | 9.111111 | 9.109756 | 9.110434 |
| 3 | 9.110434 | 9.110434 | 9.110434 |
| 4 | 9.110434 | 9.110434 | 9.110434 |
After 4 passes the guess is 9.110434, already matching √83 = 9.110434 to six decimal places.
Exact form from the prime factors
83 = 83. No prime appears twice, so nothing can be pulled out of the root -- √83 is already in simplest form.
Frequently Asked Questions
What is the square root of 83?
The square root of 83 is approximately 9.110434, or exactly √83 in simplified radical form.
Is 83 a perfect square?
No, 83 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √83 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 83?
√82 is about 9.0554, and √84 is about 9.1652. 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 √83 squared?
Exactly 83 -- squaring undoes the square root. Squaring the rounded decimal 9.110434 gives 83, which is why the radical form √83 is the exact answer and the decimal is an approximation.
What is the negative square root of 83?
−9.110434. Every positive number has two square roots, one positive and one negative, because (−9.110434)² is also 83. The radical sign √ by convention means the positive one.
More Number Theory
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