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A square root of a number a is a number that, when multiplied by itself, equals a. For example, 4 and - 4 are the square roots of 16. 4* 4 &= 16 [0.5em]
-4 * (-4)&=16
All positive numbers have two square roots — one positive and one negative. To avoid ambiguity, when talking about the square root of a number, only the positive root, also known as its principal root, is considered. Furthermore, to denote the square root, the symbol sqrt()
is used. For example, the square root of 16 is denoted as sqrt(16).
sqrt(16) = 4
In the example above, the principal root of 16 is 4 because 4 multiplied by itself equals 16 and 4 is positive. When a number is a perfect square, its square roots are integers. The square roots of positive integers that are non-perfect squares are irrational numbers.
| Principal Root of Perfect Squares | Principal Root of Non-Perfect Squares | ||
|---|---|---|---|
| Perfect Square | Principal Root (Integer Number) |
Non-Perfect Square | Principal Root (Irrational Number) |
| 1 | sqrt(1)=1 | 2 | sqrt(2)≈ 1.414213... |
| 4 | sqrt(4)=2 | 3 | sqrt(3)≈ 1.732050... |
| 9 | sqrt(9)=3 | 5 | sqrt(5)≈ 2.236067... |
| 16 | sqrt(16)=4 | 10 | sqrt(10)≈ 3.162277... |
| 25 | sqrt(25)=5 | 20 | sqrt(20)≈ 4.472135... |
The square root of a negative number is not a real number. This is because there is no real number that, when multiplied by itself, results in a negative number. a * a ≥ 0, for any real numbera Instead, the square roots of negative numbers are imaginary numbers.