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What is i?
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All Flashcards in Topic 1.12
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1.12.18 cards
What is i?
The imaginary unit, defined by i² = −1 (so i = √(−1)).
What is the Cartesian form of a complex number?
z = a + bi, where a is the real part and b is the imaginary part.
How do you add or subtract complex numbers?
Combine the real parts together and the imaginary parts together — they don't mix.
How do you multiply complex numbers?
Expand like two brackets (FOIL), then replace every i² with −1 and collect terms.
(3 + 2i) + (1 − 5i) = ?
4 − 3i.
(3 + 2i)(1 − 4i) = ?
3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i.
Powers of i: i, i², i³, i⁴?
i, −1, −i, 1 — then the cycle repeats.
What does i² equal, and why does it matter?
i² = −1; it's the step that turns a multiplication of complex numbers back into a + bi form.
1.12.28 cards
What is the conjugate of z = a + bi?
z* = a − bi — flip the sign of the imaginary part (real part unchanged).
What is z × z*?
a² + b², which is always a real number (= |z|²).
How do you divide complex numbers?
Multiply top and bottom by the conjugate of the bottom, making the denominator real, then write as a + bi.
Conjugate of 4 − 7i?
4 + 7i.
Why multiply by the conjugate when dividing?
Because z × z* = a² + b² is real, so it clears the i from the denominator.
Express (5 + i)/(2 − 3i) as a + bi.
Multiply by (2 + 3i)/(2 + 3i): (7 + 17i)/13 = 7/13 + (17/13)i.
On an Argand diagram, where is z*?
The mirror image of z in the real axis (same real part, opposite imaginary part).
z × z* for z = 2 + 5i?
4 + 25 = 29.
1.12.38 cards
What is an Argand diagram?
The plane for complex numbers: real part across (horizontal), imaginary part up (vertical).
Where does z = a + bi sit on an Argand diagram?
At the point (a, b).
What is the modulus |z|?
The distance from the origin to z; |z| = √(a² + b²).
Why is |z| = √(a² + b²)?
It's Pythagoras — the point (a, b) is at horizontal distance a and vertical distance b from the origin.
|3 + 4i| = ?
√(9 + 16) = √25 = 5.
|5 − 12i| = ?
√(25 + 144) = √169 = 13.
Can the modulus be negative?
No — it's a distance, so |z| ≥ 0.
How does the sign of b affect the plot and the modulus?
It decides up (b > 0) or down (b < 0) on the diagram; the modulus is unaffected because b is squared.
Topic 1.12 study notes
Full notes & explanations for Complex numbers: Cartesian (HL only)
Math AA exam skills
Paper structures, command terms & tips
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