# Lesson 12

Arithmetic with Complex Numbers

### Lesson Narrative

The purpose of this lesson is to lay the foundation for establishing that complex numbers are closed under addition, subtraction, and multiplication. In other words, the sum, difference, or product of two complex numbers is itself a complex number. Students see this by using usual arithmetic and the fact that $$i^2=\text-1$$ to write sums and products in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. In this lesson, students will add and subtract complex numbers, and raise imaginary numbers to powers. In the next lesson, students will multiply complex numbers. It is beyond the scope of this course to consider quotients of complex numbers.

In this lesson, students focus on the structure of complex numbers as a real term plus an imaginary term in order to calculate sums, differences, and powers. Students do this by visualizing the numbers on the complex plane and by strategically regrouping terms (MP7). Visualizing numbers on the complex plane also helps students understand that the complex plane is a way of representing individual complex numbers and relationships between them, not a way of representing functions or pairs of numbers like the coordinate plane.

### Learning Goals

Teacher Facing

• Determine powers of imaginary numbers and represent them on the complex plane.
• Determine the result of adding complex numbers, and represent it in the form $a+bi$, where $a$ and $b$ are real numbers.
• Use the complex plane to visualize why the sum of two complex numbers is another complex number.

### Student Facing

• Let’s work with complex numbers.

### Student Facing

• I can add complex numbers and calculate powers of imaginary numbers.

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.