Multiply complex numbers by scaling and rotating

Try

Represent z=1+i as the point (1,1). Its modulus is √2 and one argument is π/4, so z=√2(cos(π/4)+i sin(π/4)). Now multiply it by itself using i²=−1: (1+i)²=2i. The modulus has squared to 2 and the argument has doubled to π/2.

Predict

Predict (1+i)⁴ from that intermediate result. Squaring 2i gives −4. In polar terms, four copies multiply moduli to (√2)⁴=4 and add arguments to π. Both calculations locate the same negative real point.

Experiment

Draw a complex plane with real and imaginary axes on the same scale. Plot the following powers and connect each multiplication to its rotation and scaling.

Expression	Rectangular form	Modulus	One argument
1+i	1+i	√2	π/4
(1+i)²	2i	2	π/2
(1+i)⁴	−4	4	π

Arguments are not unique: adding 2kπ leaves the complex number unchanged. Zero has modulus zero but no distinguished argument, so formulas requiring an argument must handle it separately.

Understand

For nonzero z=a+bi, write z=r(cos θ+i sin θ) with r=√(a²+b²)>0. Multiplying this by s(cos φ+i sin φ) and expanding gives real part rs(cos θ cos φ−sin θ sin φ) and imaginary part rs(sin θ cos φ+cos θ sin φ). The addition formulas identify the product as rs(cos(θ+φ)+i sin(θ+φ)).

Repeated multiplication gives De Moivre’s formula for integer powers: zⁿ=rⁿ(cos nθ+i sin nθ). Nonnegative powers follow directly; negative powers require z nonzero and use reciprocals. A zero base with exponent zero is not assigned a universal value by this formula.

For the nth roots of a nonzero number R(cos φ+i sin φ), solve the modulus and angle conditions. Each root has modulus R^(1/n), and its angle is (φ+2kπ)/n for k=0,…,n−1. These n values are distinct; additional integer k repeat them. The extra 2kπ terms are essential before dividing the angle.

For cube roots of 8, the modulus is 2 and arguments are 0, 2π/3 and 4π/3. The rectangular roots are 2, −1+i√3 and −1−i√3. A real cube-root calculation returns only 2, while the complex equation has all three.

Apply

The practice checks modulus and powers exactly, then asks you to construct a complete root list with a completeness argument. Verify by cubing or by checking both modulus and multiplied angle. Do not count coterminal versions as extra roots.

Continue

Paired coordinate functions describe a curve as a parameter changes. The next lesson separates the set of points from their order and rate of traversal, preparing for later calculus of motion.