Unit 9 · Advanced TopicsN4 · Complex Numbers · Propulsion

Complex Numbers

Use i squared equals negative one, multiply cleanly, and divide with conjugates.

28 min
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7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

imaginary unit: The number i defined by i squared equals negative one.
complex number: A number a+bi.
conjugate: For a+bi, the conjugate is a-bi.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Powers of i repeat every four. To divide complex numbers, multiply numerator and denominator by the denominator conjugate so the denominator becomes real.

Key idea

The decision that matters

Treat i like an algebraic symbol during expansion, then replace every i squared with -1.

Imaginary unit

i^2=-1

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

$(3+2i)(4-i)=12-3i+8i-2i^2=14+5i$ .

Cue	Best move
Power of i	reduce exponent modulo 4
Complex denominator	multiply by its conjugate

Trap

Replacing i squared with 1

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Square a complex binomial

Problem

Simplify

(3-2i)^2

Expand all three terms:

9-12i+4i^2

Replace $i^2$ with $-1$ : $9-12i-4$ .

Which expansion correctly squares the complex binomial?

Combine real parts: $5-12i$ .

Why must the cross term and i squared both be handled?

Worked example

Example 2: Divide using a conjugate

Problem

Write

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

in the form

a+bi

Multiply numerator and denominator by the conjugate

2+i

Numerator: $(4+i)(2+i)=7+6i$ . Denominator: $(2-i)(2+i)=5$ .

Which factor makes the complex denominator real?

The quotient is $7/5+(6/5)i$ .

Why does multiplying by the conjugate preserve the quotient?

Faded example · you finish it

Example 3: You finish it

Problem

Complete

i^{27}=i^{24}i^3=\_\_

Reduce the exponent modulo 4 and use the repeating cycle

1,i,-1,-i

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.