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Homework 1 (MATH 1199 Fall 2019)
1. In the following problems, write the expression in the form $a+bi$ for real numbers $a$ and $b$.
(a) $(2+i)(1-i)$
(b) $\dfrac{2}{1+i}$
(c) $\dfrac{1+i}{1-i} + \dfrac{2}{i}$
2. Plot the number in the complex plane and calculate its modulus.
(a) $2+i$
(b) $3-2i$
(c) $-2-7i$
3. Draw a picture of the set in $\mathbb{C}$ described by the inequality.
(a) $|z-1|=2$
(b) $|z-(1+i)|<1$
(c) $|z+1|\geq 2$
4. Use the properties of complex conjugates to establish the following identities.
(a) $\overline{\overline{z}-2i} = z+2i$
(b) $\overline{iz} = -i\overline{z}$
5. Find the principal argument $\mathrm{Arg}$ of the following numbers.
(a) $\dfrac{\sqrt{2}}{2} + \dfrac{\sqrt{2}}{2}i$
(b) $\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}i$
(c) $\dfrac{1}{-2-2i}$
6. Use Euler's formula, $e^{i\theta}=\cos(\theta)+i\sin(\theta)$ for $\theta \in \mathbb{R}$ to compute $e^{\frac{5\pi}{3}i}$, and compute its modulus $\left| e^{\frac{5\pi}{3}i} \right|$.
7. Write the number in polar form $z=re^{i\theta}$ by finding $r=|z|$ and $\theta=\mathrm{Arg}(z)$.
(a) $1+2i$
(b) $\dfrac{1}{i}$
(c) $(1-i)(1+i)$
8. Write the number given in polar form in standard form $z=a+bi$ by using Euler's formula appropriately.
(a) $z=2e^{\frac{\pi}{2}i}$
(b) $z=8e^{\frac{-5\pi}{6}i}$
(c) $z=22e^{14\pi i}$