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Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Make a complex denominator real without changing the quotient.
- Justify the conclusion "The quotient is −1/2+(7/2)i" using the stated assumptions.
Before you start
Complex multiplication and i²=−1.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Simplify (3+4i)/(1−i).
Why this math matters
Make a complex denominator real without changing the quotient. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The denominator is nonzero.
- Complex arithmetic uses i²=−1.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Divide complex numbers using a conjugate
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Simplify (3+4i)/(1−i).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Multiply numerator and denominator by 1+i
The conjugate creates a real denominator.
Work through the mathematics
(3+4i)(1+i)=−1+7i; (1−i)(1+i)=2
The i² term changes the real part's sign.
Check the conclusion
The quotient is −1/2+(7/2)i
Multiplying this result by 1−i recovers 3+4i.
The result
The quotient is −1/2+(7/2)i
Multiplying this result by 1−i recovers 3+4i.
Common mistakes to catch
- Keep both cross terms in the numerator product.
- The conjugate changes only the imaginary sign.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Simplify 1/i.
Show a hint
Multiply by −i or use i²=−1.
Reveal answer and explanation
−i
i(−i)=1.
Practice 2
Why is multiplying only the denominator invalid?
Show a hint
A quotient must be multiplied by one.
Reveal answer and explanation
It changes the value
Both numerator and denominator require the same nonzero factor.
Take the idea with you
Compare conjugate multiplication with rationalizing a real radical denominator.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Describe a complex number by magnitude and angle
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