Teacher page: Complex numbers in Cartesian form

The one thing to do with the animation

Multiply by i four times and ask them to watch the modulus.

It reads 5 at every step while the number walks round a square and returns to where it began, because i⁴ = 1. i rotates and never stretches.

Once that has landed, 1.13 is a short lesson rather than a new subject, and conjugates and division stop being tricks.

The imaginary part is a real number

For 3 + 4i the imaginary part is 4, not 4i. It is the coefficient, and questions ask for it directly.

"Imaginary" is a bad historical name and it does real damage: students treat these as not-quite-numbers. They describe alternating current and every two-dimensional rotation, which the figure makes concrete.

The answers

1. \|3 + 4i\|	√25 = 5.
2. Real part of (3+4i)(1+2i)	3 − 8 = −5.
3. Imaginary part of i(3+4i)	3; it is −4 + 3i.

Where the marks go

1 markUsing i² = −1 and collecting parts separately.

1 markThe imaginary part as a real number.

1 markMultiplying by the conjugate when dividing, and finishing in a + bi form.

What each wrong answer tells you

They give	What it means
7 (Q1)	Added 3 and 4. The modulus is a distance.
25 (Q1)	Gave the modulus squared, which is z times its conjugate.
11 (Q2)	Left i² as +1.
10 (Q2)	Gave the imaginary part.
−4 (Q3)	Gave the real part.

Other things they will say

"Are these real numbers?" They are numbers, and "real" is a technical label for a subset of them. The naming is historical and unhelpful.

"What are they for?" Rotation, which the figure shows, and therefore alternating current, signal processing and graphics. Engineers use them daily.

"Why does the conjugate trick work?" Because z times z* is always real. It is the same move as rationalising a surd denominator, for the same reason.

A possible order

	What is happening
1	Multiply by i four times. Watch the modulus.
2	Vocabulary: real part, imaginary part, conjugate, modulus, argument.
3	Adding, subtracting, multiplying.
4	Division by the conjugate.
5	Set up 1.13: if multiplying rotates, there should be a form that says so.

Two things not to say

Do not introduce i as "the square root of minus one" and move on. That definition is where the mystification starts.

Do not let "imaginary part" include the i. It is examined as a real number.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

ModulusFind |5 − 12i|.
13
MultiplyFind (2 + i)(3 − 2i).
8 − i

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

DivideWrite (3 + 4i)/(1 − 2i) in the form a + bi.
−1 + 2i
Use the conjugateGiven z = 4 + 3i, find z + z* and z z*, and say why both are real.
8 and 25. The imaginary parts cancel in the sum and in the product.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Interpret geometricallyWhat does multiplying a complex number by 2i do to it on the Argand diagram?
Doubles the modulus and rotates it a quarter turn anticlockwise.
Solve for an unknownFind real a and b with (a + bi)² = 3 + 4i.
a = 2, b = 1, so z = 2 + i. Check: 4 + 4i + i² = 3 + 4i.

Practicalities

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