The conjugate & dividing
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Flip to reveal answersWhat is the conjugate of z = a + bi?
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All 8 Flashcards — The conjugate & dividing
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Question
What is the conjugate of z = a + bi?
Answer
z* = a − bi — flip the sign of the imaginary part (real part unchanged).
Question
What is z × z*?
Answer
a² + b², which is always a real number (= |z|²).
Question
How do you divide complex numbers?
Answer
Multiply top and bottom by the conjugate of the bottom, making the denominator real, then write as a + bi.
Question
Conjugate of 4 − 7i?
Answer
4 + 7i.
Question
Why multiply by the conjugate when dividing?
Answer
Because z × z* = a² + b² is real, so it clears the i from the denominator.
Question
Express (5 + i)/(2 − 3i) as a + bi.
Answer
Multiply by (2 + 3i)/(2 + 3i): (7 + 17i)/13 = 7/13 + (17/13)i.
Question
On an Argand diagram, where is z*?
Answer
The mirror image of z in the real axis (same real part, opposite imaginary part).
Question
z × z* for z = 2 + 5i?
Answer
4 + 25 = 29.
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Full study notes for The conjugate & dividing
Topic 1.12 hub
Complex numbers: Cartesian (HL only)
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