Square Root Basics

A number a is a square root of a number x if a² = x (or equivalently, a ⋅ a = x).

Example: 3 is a square root of 9 because 3² = 3 ⋅ 3 = 9.

–3 is also a square root of 9 because (–3)² = (–3)(–3) = 9.

We see that 9 has two square roots: 3 and –3. We sometimes combine these two and say that the two square roots of 9 are ±3.

Square Root Symbol

Students sometimes get confused when using the square root symbol:

The symbol above represents the positive square root. So, for example, we have

So even though 9 has the two square roots ±3, when we put 9 under the symbol, the result is just 3 (and not –3). You may want to compare this with the Square Root Property (students often get the square root property confused with taking the positive square root of a number).

If we want the negative square root of a number, we need to place a minus sign before the square root symbol.

And if we want both square roots of a number, we should place the symbol ± before the square root symbol.

“Types” of Square Roots

All numbers, with the exception of 0, have two square roots. The number 0 has just one square root because both the positive and negative square root of 0 are both 0.

When taking positive and negative square roots of numbers, it’s worth trying to determine the “type” of number that you get. For example, whenever we take the square root of a nonzero integer (the set of integers is {…–3, –2, –1, 0, 1, 2, 3,…}), the result can be an integer, an irrational number, or a pure imaginary number (see this: Complex Numbers). When we identify the type of roots of a number we usually refer to this as determining the nature of the roots.

Examples: Determine the nature of the roots of 16, 11, –2, and 0.

Solution: Since 16 is a positive perfect square, the two square roots of 16 are integers (in fact, they are 4 and –4).

Since 11 is positive, but not a perfect square, the two square roots of 11 are irrational numbers.

Since –2 is negative, the two square roots of –2 are pure imaginary numbers.

Finally, 0 has just one square root…itself. In particular, 0 has one square root, which is an integer.

Note: If a positive integer is not a perfect square, then it’s square root will always be irrational. The proof of this result is beyond the scope of this article. I will discuss this further in a future post. 

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