Square Root of 13
The square root of 13 is approximately 3.605551, or exactly √13 in simplified radical form. No, 13 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √13 sits
3² = 9 and 4² = 16, and 13 falls between them, so √13 is between 3 and 4. It's closer to 4 (4 above 9, 3 below 16), which matches the decimal 3.605551.
| k | k² | Compared to 13 |
|---|---|---|
| 1 | 1 | 12 below |
| 2 | 4 | 9 below |
| 3 | 9 | 4 below |
| 4 | 16 | 3 above |
| 5 | 25 | 12 above |
| 6 | 36 | 23 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 3), divide 13 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 13 ÷ x | New guess (x + 13÷x) ÷ 2 |
|---|---|---|---|
| 1 | 3 | 4.333333 | 3.666667 |
| 2 | 3.666667 | 3.545455 | 3.606061 |
| 3 | 3.606061 | 3.605042 | 3.605551 |
| 4 | 3.605551 | 3.605551 | 3.605551 |
After 4 passes the guess is 3.605551, already matching √13 = 3.605551 to six decimal places.
Exact form from the prime factors
13 = 13. No prime appears twice, so nothing can be pulled out of the root -- √13 is already in simplest form.
Frequently Asked Questions
What is the square root of 13?
The square root of 13 is approximately 3.605551, or exactly √13 in simplified radical form.
Is 13 a perfect square?
No, 13 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √13 between?
Between 3 and 4, because 3² = 9 and 4² = 16.
What are the square roots of the numbers right next to 13?
√12 is about 3.4641, and √14 is about 3.7417. 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 √13 squared?
Exactly 13 -- squaring undoes the square root. Squaring the rounded decimal 3.605551 gives 13, which is why the radical form √13 is the exact answer and the decimal is an approximation.
What is the negative square root of 13?
−3.605551. Every positive number has two square roots, one positive and one negative, because (−3.605551)² is also 13. The radical sign √ by convention means the positive one.
More Number Theory
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