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# 2.E: Exercises for Chapter 2

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## Calculational Exercises

1. Express the following complex numbers in the form $$x + yi$$ for $$x, y \in \mathbb{R}:$$

(a) $$(2 + 3i) + (4 + i)$$

(b) $$(2 + 3i)^2 (4 + i)$$

(c) $$\frac{2+3i}{4+i}$$

(d) $$\frac{1}{i}+\frac{3}{1+i}$$

(e) $$(−i)^{−1}$$

(f) $$(−1 + i \sqrt{3})^3$$

2. Compute the real and imaginary parts of the following expressions, where $$z$$ is the
complex number $$x + yi$$ and $$x, y \in \mathbb{R}:$$

(a) $$\frac{1}{z^2}$$

(b) $$\frac{1}{3z+2}$$

(c) $$\frac{z+1}{2z-5}$$

(d) $$z^3$$

3. Find $$r > 0$$ and $$\theta \in [0, 2\pi)$$ such that $$(1 − i)/ 2 = re^{i \theta}.$$

4. Solve the following equations for $$z$$ a complex number:
(a) $$z^5 − 2 = 0$$
(b) $$z^4 + i = 0$$
(c) $$z^6 + 8 = 0$$
(d) $$z^3 − 4i = 0$$

5. Calculate the
(a) complex conjugate of the fraction $$(3 + 8i)^4 /(1 + i)^10 .$$
(b) complex conjugate of the fraction $$(8 − 2i)^10 /(4 + 6i)^5 .$$
(c) complex modulus of the fraction $$i(2 + 3i)(5 − 2i)/(−2 − i).$$
(d) complex modulus of the fraction $$(2 − 3i)^2 /(8 + 6i)^2 .$$

6. Compute the real and imaginary parts:
(a) $$e^{2+i}$$
(b) $$sin(1 + i)$$
(c) $$e^{3−i}$$
(d) $$cos(2 + 3i)$$

7. Compute the real and imaginary part of $$e^{e^{z}}$$ for $$z \in \mathbb{C}.$$

## Proof-Writing Exercises

1. Let $$a \in \mathbb{R}$$ and $$z, w \in \mathbb{C}.$$ Prove that
(a) $$Re(az) = aRe(z)$$ and $$Im(az) = aIm(z).$$
(b) $$Re(z + w) = Re(z) + Re(w)$$ and $$Im(z + w) = Im(z) + Im(w).$$

2. Let $$z \in \mathbb{C}.$$ Prove that $$Im(z) = 0$$ if and only if $$Re(z) = z.$$

3. Let $$z, w \in \mathbb{C}.$$ Prove the parallelogram law $$|z − w|^2 + |z + w|^2 = 2(|z|^2 + |w|^2).$$

4. Let $$z, w \in \mathbb{C}$$ with $$\bar{z}w \neq 1$$ such that either $$|z| = 1$$ or $$|w| = 1.$$ Prove that $$\left| \frac{z−w}{1 − \bar{z}w} \right| =1.$$

5. For an angle $$\theta \in [0, 2\pi),$$ ﬁnd the linear map $$f_\theta : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$, which describes the rotation by the angle $$\theta$$ in the counterclockwise direction.

Hint: For a given angle $$\theta$$, ﬁnd $$a, b, c, d \in \mathbb{R}$$ such that $$f_\theta (x_1 , x_2 ) = (ax_1 +bx_2 , cx_1 +dx_2 ).$$

### Contributors

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