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Solve both parts and justify the conditions used. Part A: Simplify the sum and explain the cancellation. Part B: Write the conjugate.

Read the idea, work independently, then explain what changed.

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TOPIC 01

Complex numbers: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

Solve both parts and justify the conditions used. Part A: Simplify the sum and explain the cancellation. Part B: Write the conjugate.

\begin{gathered}\text{A: }1+i+i^2+i^3\\\text{B: }z=8-3i\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

B: Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Worked solution

Part A reasoning
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$1+i-1-i=0$
The four points cancel in opposite pairs on the unit circle.
Part B reasoning
Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$\overline z=8+3i$
Only the sign of the imaginary term changes.

A: The requested value is 0. B: 8+3i.

Checks and common pitfalls: The four points cancel in opposite pairs on the unit circle. Only the sign of the imaginary term changes.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.