01 · Retrieval · 0–7 minutes

Review from last lesson

Work in your notebook.

Sketch $z = - 1 + \sqrt{3} i$ on an Argand diagram. Find its modulus and an argument in radians.
Write the same number in cis form.
For $w = 2 cis (\frac{π}{6})$ , find $z w$ in cis form. Describe the scaling and turn from $z$ to $z w$ .

We have already met Argand diagrams, modulus, argument, conversion, multiplication and rotation. Today we will use them to understand conjugates and division.

02 · Re and Im · 7–10 minutes

The imaginary part is a real number

In $z = a + b i$ , the two real numbers $a$ and $b$ tell us where the point lies.

Re (z) = a, Im (z) = b

$Re$ names the real part. $Im$ names the coefficient of $i$ , not the whole term $b i$ .

Worked example

Re (3 + 2 i) = 3

Im (3 + 2 i) = 2

The imaginary part is $2$ , not $2 i$ .

Individual check

For $z = - 4 - 5 i$ , write $Re (z)$ and $Im (z)$ . What changes for $z = 7 i$ ?

03 · Conjugate notation · 10–12 minutes

Conjugates change the sign of the imaginary part

The complex conjugate of $z = a + b i$ is written $z^{*}$ . It keeps the real part and reverses the imaginary part.

z = a + b i \Rightarrow z^{*} = a - b i

Worked example

z = 3 + 2 i \Rightarrow z^{*} = 3 - 2 i

Some sources use a bar over $z$ for the same conjugate.

Individual check

Write the conjugates of $1 - 4 i$ , $2 i$ and $5$ . Which of these stays unchanged?

04 · Argand reflection · 12–17 minutes

Reflection and a half-turn are different

Conjugation reflects a point across the real axis. Negation reverses both coordinates.

z = 3 + 2 i

z^{*} = 3 - 2 i but - z = - 3 - 2 i

$z^{*}$ stays on the same vertical line as $z$ . The point $- z$ is reached by a half-turn about the origin.

For

z = - 2 + 3 i

, sketch

z^{*}

and

- z

. Label each point.

The real axis is the mirror for conjugation.

05 · Numerical investigation · 17–25 minutes

Investigate a number times its conjugate

Expand the products using $i^{2} = - 1$ . What happens to the imaginary terms?

Multiply each number by its conjugate
$z$	$z^{*}$	$z z^{*}$	$\| z \|^{2}$
$3 + 2 i$	Row 1: conjugate	Row 1: conjugate product	Row 1: modulus squared
$1 - 4 i$	Row 2: conjugate	Row 2: conjugate product	Row 2: modulus squared
$2 i$	Row 3: conjugate	Row 3: conjugate product	Row 3: modulus squared

Write in your notebook, or type entries here. Work out the product before checking the last column.

What do all three products have in common?

Compare $z z^{*}$ with $| z |^{2}$ . Suggest a rule that could work for every complex number.

$z$	$z^{*}$	$z z^{*}$	$\| z \|^{2}$
$3 + 2 i$	$3 - 2 i$	$13$	$13$
$1 - 4 i$	$1 + 4 i$	$17$	$17$
$2 i$	$- 2 i$	$4$	$4$

06 · General identity · 25–30 minutes

Why the pattern always works

Let $z = a + b i$ , where $a$ and $b$ are real. Its conjugate is $a - b i$ .

(a + b i) (a - b i)

Expand all four terms. The imaginary terms cancel, and $i^{2} = - 1$ .

Connect with modulus

| z | = \sqrt{a^{2} + b^{2}}

Squaring this length gives $a^{2} + b^{2}$ .

Individual check

Without expanding four terms, find $z z^{*}$ for $z = - 5 + 2 i$ .

07 · Cartesian division · explanation · 30–33 minutes

Make a real denominator without changing the value

Division becomes easier when the denominator is real. Multiply by its conjugate to make that happen.

For a nonzero denominator $w$ :

\frac{w^{*}}{w^{*}} = 1

\frac{z}{w} = \frac{z}{w} \cdot \frac{w^{*}}{w^{*}} = \frac{z w^{*}}{| w |^{2}}

We multiply both numerator and denominator by the same nonzero number, so the quotient keeps its value.

A familiar idea

This is like rationalising a surd denominator: use a conjugate pair to replace the denominator with something simpler.

Here the product $w w^{*} = | w |^{2}$ is a positive real number because $w \neq 0$ .

For a denominator

1 - i

, which factor would you multiply both top and bottom by?

08 · Cartesian division · worked example · 33–36 minutes

Work through a Cartesian quotient

Find the quotient

\frac{3 + i}{1 - i}

The denominator’s conjugate is $1 + i$ . Multiply top and bottom by it.

\frac{(3 + i) (1 + i)}{(1 - i) (1 + i)}

A short individual check

Use the same denominator to find

\frac{1 + i}{1 - i}

Explain why the denominator becomes $2$ .

09 · Same question · method 1 · 36–38 minutes

One quotient, first in Cartesian form

\frac{1 + \sqrt{3} i}{\sqrt{3} - i}

Use the denominator’s conjugate, $\sqrt{3} + i$ .

\frac{(1 + \sqrt{3} i) (\sqrt{3} + i)}{(\sqrt{3} - i) (\sqrt{3} + i)}

Before we check another method: where should $i$ lie on the Argand diagram? What are its modulus and argument?

10 · Individual attempt · 38–41 minutes

Your turn: divide, then explain

Work independently first. Name the conjugate you use and show how it makes the denominator real.

\frac{4 + 2 i}{1 + i}

Write the answer in Cartesian form. Then check it by multiplying your answer by $1 + i$ .

My Cartesian answer

Your GDC can check the calculation. Your written steps explain why it works.

11 · GDC · radians and rectangular output · 41–45 minutes

Use your GDC as a check

TI-84 Plus / CE family

Press MODE. Choose RADIAN and a+bi.
Enter $i$ with 2nd then the decimal-point key.
Press MATH, then choose CPX / CMPLX:
1 conj · 2 real · 3 imag · 4 angle · 5 abs
Put parentheses around each part of a quotient: (3+i)/(1-i).

Menu labels vary across the family. Decimal output is normal on many TI-84 models.

Check these functions on your model

`conj(3+2i)`	$3 - 2 i$
`real(3+2i)`	$3$
`imag(3+2i)`	$2$
`abs(-1+√(3)i)`	$2$
`angle(-1+√(3)i)`	$\frac{2 π}{3} \approx 2.094$

The last point is in quadrant II. A negative acute angle would not match its position.

Check (3+i)/(1-i): expect

1 + 2 i

. To write cis form, use abs for the modulus and angle for the argument.

12 · Cis division · explanation · 45–48 minutes

Division undoes scaling and rotation

Multiplying moduli and adding angles was our rule for multiplication. Division reverses those actions.

For $r_{2} > 0$ :

\frac{r_{1} cis θ_{1}}{r_{2} cis θ_{2}} = \frac{r_{1}}{r_{2}} cis (θ_{1} - θ_{2})

Divide the lengths. Subtract the denominator’s angle from the numerator’s angle.

Think of undoing a transformation

Dividing by $r_{2} cis θ_{2}$ scales by $1 / r_{2}$ and rotates by $- θ_{2}$ .

A negative denominator angle therefore produces an anticlockwise undoing turn.

When dividing by

2 cis (- \frac{π}{6})

, what happens to the length and direction?

14 · Short individual check · 51–53 minutes

Check the order of the subtraction

Try this before looking at the answer. Describe both the length and the turn.

\frac{2 cis (\frac{π}{2})}{4 cis (\frac{π}{6})}

Write your result in cis form.

My cis answer

If you reverse the angle subtraction, you reverse the direction. Keep numerator minus denominator.

15 · Individual practice · 53–60 minutes

Choose a method and check the geometry

Work independently. Use cis division, then convert your result to Cartesian form.

\frac{6 cis (\frac{5 π}{6})}{2 cis (\frac{π}{3})}

Sketch the result on an Argand diagram. Use your GDC to check its modulus and argument.

My result and a reason

Why does subtracting

\frac{π}{3}

make sense as an undoing of multiplication? Explain in a sentence.

16 · Final maths check and questions · 60–70 minutes

Finish with three mathematical checks

Use these to decide which idea you are ready to use and which one you want to revisit.

For $z = - 2 + 3 i$ , state $Re (z)$ , $Im (z)$ , $z^{*}$ and $z z^{*}$ .
Find $\frac{2 + i}{1 + i}$ in Cartesian form.
Find $\frac{4 cis (- \frac{π}{6})}{2 cis (\frac{π}{3})}$ in cis and Cartesian form.

One question I still have

Which line of working would you like us to explain once more?

17 · TOK · context, preparation and debate · 17 minutes · 70–87 minutes

Setup

Context · 4 minutes

On 6 October 2026, OpenAI released AI-generated mathematical results. Many proofs have Lean formalizations that computers can check; the collection has different verification stages and is not all verified.

Computer checking can support correctness while a conceptual explanation still takes work. The formal statement must match the intended question.

If we can verify that a mathematical result is true but don’t understand why, to what extent can we claim to know it?

Write individually · 2 minutes

Write your initial position on the statement and one reason in your notebook. You may agree, disagree or qualify it.

OpenAI announcement and repository, 6 Oct 2026. Source links are in Teacher notes.

A verified mathematical result counts as knowledge, even if we cannot explain why it is true.

Original groups · 5 minutes

Four groups of three: A and B agree; C and D disagree.

Prepare your strongest argument, an example and a reply to a challenge. Number yourselves 1–3 in each group.

Regroup and debate · 6 minutes

Join the group with your number. Each new group has one representative from A, B, C and D.

Debate 1	Debate 2	Debate 3
A1 · B1 · C1 · D1	A2 · B2 · C2 · D2	A3 · B3 · C3 · D3

Two agree and two disagree in each debate. Each person presents their argument and responds to another.

18 · TOK · individual reflection · 3 minutes · 87–90 minutes

Reflection

If we can verify that a mathematical result is true but don’t understand why, to what extent can we claim to know it?

Write your personal conclusion

Give your conclusion and reasons. Your view does not have to match your assigned group’s position.

Explain which argument influenced you, or what remains uncertain. Distinguish personal understanding from the mathematical community’s understanding where useful.

My conclusion, reasons and what influenced me or remains uncertain

Compare this with your initial view. You may keep it, revise it or add a condition.

Use the mathematical examples and the debate to support your reasoning. There is no required conclusion about AI.

Teacher notes

90-minute route · 10:10–11:40 MYT

0–7	Retrieval
7–17	Re, Im, conjugate and reflection
17–30	Conjugate product investigation and identity
30–45	Cartesian division, independent attempt, GDC
45–60	Cis division, same quotient, individual checks
60–70	Final mathematical checks and questions
70–90	TOK: Setup and Reflection

TOK · preserve all 20 minutes

4 context + 2 individual writing + 5 preparation in four groups of three + 6 debate in three groups of four + 3 personal reflection.

Original groups A/B agree and C/D disagree. Number students 1–3. Regroup by number: A1/B1/C1/D1, A2/B2/C2/D2, A3/B3/C3/D3. Each debate has two agree and two disagree. Everyone writes before and after; personal conclusions may differ from assigned roles.

Core text and diagrams work offline. Sources are optional external links. All reveals are manual.

Prepared for Anis · 13 October 2026. Prior Argand, modulus/argument, cis conversion and multiplication were confirmed taught. Crowded spine sections were split without adding content.