Sometimes people call the number line the real number line.

All real numbers are either positive, negative, or 0, and they can be plotted on this line. All the numbers we have used until this lesson have been real numbers. For real numbers, we know that:

• A positive number times a positive number is always positive. So when we square a positive number, the result will always be positive.
• A negative number times a negative number is always positive. So when we square a negative number, the result will always be positive.
• 0 squared is 0.

So squaring a real number never results in a negative number. We can conclude that the equation does not have any real number solutions. In other words, none of the numbers on the real number line satisfy this equation.

Mathematicians invented a new number that is not on the real number line. This new number was invented as a solution to the equation . For now, let’s write it as and draw a point to represent this number. Although you cannot have of anything, it is still a useful number in the same way that you cannot have -3 of something, but it is useful to think about.

We place our new point  one unit above 0.

This new number is a solution to the equation , so . If we draw a line that passes through 0 on the real number line and , we get the imaginary number line. The numbers on the imaginary number line are called the imaginary numbers.