Square root is a number that, when multiplied by itself, produces a given number. For example, 2 is a square root of 4 because 2 × 2 = 4. Likewise, the square root of 25 is 5 and the square root of 144 is 12.
Finding a square root is the opposite of squaring a number—that is, multiplying the number by itself. For example, 4 squared or 4 × 4 is 16. So 4 is a square root of 16, often written:
Square roots are indicated using a symbol called a radical sign:
It is also true that –4 × –4 is 16, because multiplying two negative numbers results in a positive number. In fact, all positive numbers have two square roots—a positive one and negative one. To avoid having to write a double answer for square roots, we define the radical sign to only mean the positive square root. To refer to the negative square root of 16, we write:
Negative numbers also have square roots, but their square roots are a completely different kind of number. The square root of zero is just zero.
Sometimes a tiny number accompanies the radical symbol, as in:
This symbol indicates a cube root—a number that, when multiplied by itself three times, produces a given number (see Cube root ). Similarly, there are fourth roots, fifth roots, and so on.
Square roots of positive numbers.
People have thought of a variety of ways to figure out the square root of a positive number. Some methods resemble a complex version of long division. Others methods are based on a series of approximations. Today, electronic calculators and computers can quickly find a positive number’s square root.
Some positive numbers have square roots that are relatively simple. For example:
Such numbers are all rational numbers. Rational numbers are integers (positive and negative counting numbers) or ratios of integers.
The square roots of most other numbers cannot be written so easily. They are irrational numbers. The square root of 2 is an example of an irrational number. It is approximately 1.41421356. But writing out the actual number in this way would require infinitely many decimal places.
Square roots of negative numbers.
The square root of –16 cannot be 4 or –4, because neither number gives a negative result when multiplied by itself. Thus, it would seem as though negative numbers can have no square roots. But mathematicians have found many uses for the square roots of negative numbers. Early mathematicians, however, were somewhat confused about the nature of such numbers, so they called them “imaginary.” Mathematicians still call such numbers imaginary numbers. They call ordinary, non-imaginary numbers real numbers, including rational numbers and irrational numbers. Imaginary numbers, however, are not fake or made-up. They have important, real-world uses in physics and engineering.
Imaginary numbers are written as multiples of the imaginary unit, i. The value of i is defined as:
Another way to think about this concept is that i × i = –1. So, for example, the square root of –16 is equal to the imaginary number 4i. This number is equal to 4 times i. We know that 4 × 4 = 16 and i × i = –1, so it makes sense that 4i × 4i = –16.
Numbers may have both a real part and an imaginary part, such as the number 4 + 2i or the number 3 – 3/5 i. This kind of number is called a complex number.
