Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Associate a + bi with the point (a, b).
- Use i² = −1 in multiplication.
- Interpret a counterclockwise quarter-turn and check distance.
Before you start
Read Cartesian coordinates, distribute multiplication, and work with negative numbers.
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
On a mathematical coordinate plane, a game marker is at (3, 2). Represent it as z = 3 + 2i and multiply by i. Where is the marker after this transformation?
Why this math matters
Complex numbers combine two coordinates into one algebraic object. Their multiplication can describe rotations and scaling, providing a connection between symbolic arithmetic and plane motion. Multiplication by i gives a particularly simple quarter-turn example.
Set up the model
A useful answer starts with clear assumptions:
- The real axis points right and the imaginary axis points upward, unlike some screen-pixel coordinates.
- The rotation is about the origin with equal coordinate scales.
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.
Rotate a point using the imaginary unit
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On a mathematical coordinate plane, a game marker is at (3, 2). Represent it as z = 3 + 2i and multiply by i. Where is the marker after this transformation?
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.
Encode the coordinates
(3, 2) ↔ z = 3 + 2i
The real part gives the horizontal coordinate and the coefficient of i gives the vertical coordinate. They remain separate components.
Multiply and simplify
iz = i(3 + 2i) = 3i + 2i² = −2 + 3i
Distribute i to both terms, then replace i² by negative one. The transformed point is (−2, 3).
Check the geometry
3² + 2² = 13; (−2)² + 3² = 13
The distance from the origin remains √13. The rule (a, b) → (−b, a) describes a ninety-degree counterclockwise rotation.
The result
The marker moves to (−2, 3), a quarter-turn counterclockwise about the origin.
Applying the transformation twice multiplies by i² = −1, giving a half-turn. Four applications multiply by i⁴ = 1 and return the marker to its starting point.
Common mistakes to catch
- Treating i² as positive one reverses the essential rule.
- A screen coordinate system with downward-positive vertical coordinates changes the visual direction unless coordinates are converted.
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
Multiply −1 + 4i by i and identify the new point.
Show a hint
Distribute and replace i² by −1.
Reveal answer and explanation
−4 − i, corresponding to (−4, −1)
i(−1 + 4i) = −i + 4i² = −4 − i.
Practice 2
Where does the original marker (3, 2) go after two multiplications by i?
Show a hint
Combine the multipliers before applying them.
Reveal answer and explanation
(−3, −2)
i²(3 + 2i) = −(3 + 2i) = −3 − 2i. Both coordinates change sign in a half-turn.
Take the idea with you
Compare an algebraic transformation with its geometric effect. Checking distance and direction can catch different kinds of errors.
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: Use a clock to meet your first algebraic group
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