Modulus & the Argand diagram
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Question
What is an Argand diagram?
Answer
The plane for complex numbers: real part across (horizontal), imaginary part up (vertical).
Question
Where does z = a + bi sit on an Argand diagram?
Answer
At the point (a, b).
Question
What is the modulus |z|?
Answer
The distance from the origin to z; |z| = √(a² + b²).
Question
Why is |z| = √(a² + b²)?
Answer
It's Pythagoras — the point (a, b) is at horizontal distance a and vertical distance b from the origin.
Question
|3 + 4i| = ?
Answer
√(9 + 16) = √25 = 5.
Question
|5 − 12i| = ?
Answer
√(25 + 144) = √169 = 13.
Question
Can the modulus be negative?
Answer
No — it's a distance, so |z| ≥ 0.
Question
How does the sign of b affect the plot and the modulus?
Answer
It decides up (b > 0) or down (b < 0) on the diagram; the modulus is unaffected because b is squared.
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Full study notes for Modulus & the Argand diagram
Topic 1.12 hub
Complex numbers: Cartesian (HL only)
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