A Brief Introduction to Complex Numbers

Why do this problem?

This problem can be used with students who have never met complex numbers. The challenges posed offer practice in manipulating complex numbers while at the same time offering insight into their structure.

Possible approach

To set the scene for the lesson, you may wish to remind students of the different sets of numbers they have met before (Naturals, Integers, Rationals, Reals).

Show students the video below, or recreate it on the board.

Then pose the following questions:
"Choose some pairs of complex numbers and add them together.
Can you find some pairs of complex numbers whose sum is a real number?
Can you find some pairs of complex numbers whose sum is an imaginary number?"
"In general, what would you need to add to $a+ib$ to get a real number? Or an imaginary number?"

Next, invite students to multiply some pairs of complex numbers:
"Can you find some pairs of complex numbers whose product is a real number?
Can you find some pairs of complex numbers whose product is an imaginary number?"

Give students some time to work on their own or in pairs before bringing the class together to discuss the question:
"In general, what would you need to multiply by $a+ib$ to get a real number? Or an imaginary number?"

Students could tackle A Brief Introduction to the Argand Diagram next.

Key questions

When you add two complex numbers, what contributes to the real part of the answer? What contributes to the imaginary part?
When you multiply two complex numbers, what contributes to the real part of the answer? What contributes to the imaginary part.

Possible extension

After working on this problem and A Brief Introduction to the Argand Diagram, students could try Complex Squares.

Possible support

Finding Factors offers practise in expanding and factorising brackets, and might offer useful preparation for multiplying pairs of complex numbers.