IB Mathematics Analysis and Approaches Topic AHL…

The complex number $z$ is defined by $z = \frac{4-4i}{1+i}$ .

(a) Find an expression for $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

(b) Find the two square roots of $z$ , giving your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

The complex number $z$ is given in Euler form as $z = 4e^{i\frac{\pi}{3}}$ .

Express $z$ in the form $x+iy$ , where $x$ and $y$ are real.

The complex number $z$ is given by $z = 1+i$ .

(b) Find the argument of $z$ , giving your answer in radians.

Let the complex number $z$ be defined by $z = \sqrt{3} - i$ . Express $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

Given the complex numbers $z_1 = 4\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ and $z_2 = \frac{1+i\sqrt{3}}{1-i}$ .

(a) Write $z_1$ in exponential form.

(b) Write $z_2$ in exponential form.

(a) the modulus of $z$ .

IB Mathematics Analysis and Approaches AHL 1.13—polar and Euler Form Questionbank