# Lesson 13

Multiplying Complex Numbers

### Lesson Narrative

This lesson continues the idea from the previous lesson that when complex numbers are combined, the result is also a complex number and can be written in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. In this lesson, students use the fact that $$i^2=\text-1$$ to multiply imaginary numbers, and use the strategies they develop to multiply complex numbers by writing the $$i^2$$ terms as real numbers.

Students also make use of the familiar structure of distributing terms to find the result of multiplying two complex numbers (MP7). To organize their thinking, they can use the same kind of diagrams they used in a previous unit to multiply polynomials.

### Learning Goals

Teacher Facing

• Calculate the result of multiplying two complex numbers.
• Justify that two equivalent expressions involving complex numbers are equivalent, using the fact that $i^2=\text-1$.

### Student Facing

• Let's multiply complex numbers.

### Student Facing

• I can multiply complex numbers.

Building On

Building Towards

### Glossary Entries

• complex number

A number in the complex plane. It can be written as $$a + bi$$, where $$a$$ and $$b$$ are real numbers and $$i^2 = \text-1$$.

• imaginary number

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

• real number

A number on the number line.