1.8 The real numbers

Simplify expressions with square roots

Remember that when a number n is multiplied by itself, we write $n^2$ and read it “n squared.” The result is called the square of n . For example,

Similarly, 121 is the square of 11, because ${11}^2$ is 121.

Complete the following table to show the squares of the counting numbers 1 through 15.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because ${10}^2=100,$ we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is $m$ is called a square root of m .

Notice ${\left(\mathrm{-10}\right)}^2=100$ also, so $\mathrm{-10}$ is also a square root of 100. Therefore, both 10 and $\mathrm{-10}$ are square roots of 100.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign    , $\sqrt{m},$ denotes the positive square root. The positive square root is called the principal square root . When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because $0^2=0,$ $\sqrt{0}=0.$ Notice that zero has only one square root.

Square root notation

$\sqrt{m}$ is read “the square root of m ”

If $m=n^2,$ then $\sqrt{m}=n,$ for $n\ge 0.$

The square root of m , $\sqrt{m},$ is the positive number whose square is m .

Since 10 is the principal square root of 100, we write $\sqrt{100}=10.$ You may want to complete the following table to help you recognize square roots.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{100}=10.$ If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $\text{−}\sqrt{100}=\mathrm{-10}.$ We read $\text{−}\sqrt{100}$ as “the opposite of the square root of 10.”