Key Idea: A complex number can be written by its size and direction instead of a + bi. This makes multiplying, dividing and powers quick, and it lets you add waves and work with AC circuits.
| Task | Rule |
|---|---|
| Size | r = √(a²+b²) |
| Direction | θ = arg z; sketch first to fix the quadrant |
| Same number | z = r cis θ = r eⁱθ |
| Multiply | Moduli ×, arguments + |
| Divide | Moduli ÷, arguments − |
De Moivre and applications
- De Moivre: zⁿ = rⁿ cis(nθ). Power the modulus, multiply the argument by n.
- Impedance Z = R + iX: |Z| is the total opposition, arg Z is the phase angle. Series impedances add.
- Adding waves: write each wave as a phasor A eⁱφ, add them, then read off the modulus (amplitude) and argument (phase).
- Example: 5 cos t + 12 sin t gives 13 cos(t − 67.4°), because |5 − 12i| = 13.
Multiply the sizes and add the angles. That one idea gives you De Moivre too.
Why sketch the point before finding θ? A calculator's arctan only gives −π/2 to π/2, so it can pick the wrong quadrant.
z = 2 cis(π/4). What is z²? 4 cis(π/2), which is 4i.
An argument comes out as −5π/3. How do you bring it into range? Add 2π to get π/3, inside −π < θ ≤ π.
Two impedances are in series. How do you find the total? Add them as complex numbers.