Lesson 10

A New Kind of Number

Lesson Narrative

In past grades, students have expanded their concept of number several times, from whole numbers to integers to rational numbers to real numbers. 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 are imaginary numbers, numbers that can be written as $$bi$$, where $$b$$ is a real number and $$i^2=\text-1$$.

Students first note that the equation $$x^2=\text-1$$ 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 just 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 $$\sqrt{\text-1}$$. Students will be introduced to the symbol $$i$$ in the next lesson.

Learning Goals

Teacher Facing

• Justify that $x^2=\text-1$ has no real solutions by using a graph of $y=x^2$.
• Represent imaginary numbers visually with the imaginary number line.

Student Facing

• Let’s invent a new number.

Student Facing

• I can represent $\sqrt{\text-1}$ and multiples of it.

CCSS Standards

A number on the imaginary number line. It can be written as $$bi$$, where $$b$$ is a real number and $$i^2 = \text-1$$.