A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Geometry and algebra
CSCA Mathematics · Lesson 13 of 16Complex numbers
Nombres complexes · الأعداد العقدية
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Complex numbers add i to the real numbers, where i² = −1. Calculate with i like a letter, then replace i² by −1.
Key ideas
- z = a + bi: real part a, imaginary part b.
- i² = −1, i³ = −i, i⁴ = 1, and the cycle repeats.
- Conjugate z̄ = a − bi, and z·z̄ = a² + b².
- Modulus |z| = √(a² + b²): the distance from z to 0 in the complex plane.
Formulas to know
- (a + bi)(c + di) = (ac − bd) + (ad + bc)i
- To divide, multiply top and bottom by the conjugate of the bottom
- z = a + bi ↔ point (a, b)
Worked example
Compute (2 + 3i)(1 − i).
- Expand: 2 − 2i + 3i − 3i²
- Use i² = −1: 2 + i + 3
- Result: 5 + i
Answer: 5 + i
Practice questions
1. |3 − 4i| = ?
1 · 5 ✓ · 7 · 25
√(9 + 16) = 5.
2. i¹⁰ = ?
1 · i · −1 ✓ · −i
i¹⁰ = i⁸ · i² = 1 · (−1) = −1.
3. (1 + i)/(1 − i) = ?
i ✓ · −i · 1 · 2i
Multiply by (1 + i)/(1 + i): (1 + i)²/2 = 2i/2 = i.