Up until this point, the only definition that students have of a real number is that it is a number on the real number line. In this lesson, students expand their concept of numbers again to include non-real numbers — that is, numbers that are not on the real number line. These include imaginary numbers, numbers that can be written as , where is a real number and .

Students first note that the equation has no solutions on the real number line, and then define a new number that is a solution to this equation, and therefore not a real number. In order to make sense of what it means for a number to be not real, we place this new number off of the real number line, eventually resulting in the imaginary number line (MP1).

It is important to note that the words “real” and “imaginary” are just names for different kinds of numbers, and do not mean that one type of number is “actual” while the other type is “fantasy.” Imaginary numbers are numbers whose squares aren’t positive, and real numbers are numbers whose squares aren’t negative. For now, imaginary numbers are described as real multiples of  to remind students of the origin of the numbers. Students will be introduced to the symbol later.