Questions

1. Question 1

   Find the multiplicative inverse of each complex number.

   (i)

   $(-4, 7)$

   (ii)

   $(\sqrt{2}, -\sqrt{5})$

   (iii)

   $(1, 0)$

   SolutionTheory
2. Question 2

   Separate the real and imaginary parts (write as a + bi).

   (i)

   $\dfrac{2-7i}{4+5i}$

   (ii)

   $\dfrac{(-2+3i)^2}{1+i}$

   (iii)

   $\dfrac{i}{1+i}$

   (iv)

   $\dfrac{(4+3i)^2}{4-3i}$

   SolutionTheory
3. Question 3

   Prove that $z=\bar{z}$ if and only if $z$ is real.

   SolutionTheory
4. Question 4

   For $z \in \mathbb{C}$, show the following identities.

   (i)

   $\dfrac{z+\bar{z}}{2}=\operatorname{Re}(z)$

   (ii)

   $\dfrac{z-\bar{z}}{2i}=\operatorname{Im}(z)$

   (iii)

   $|z|^2=z\,\bar{z}$

   SolutionTheory
5. Question 5

   If $z_1=2+i$, $z_2=3-2i$, $z_3=1+3i$, express $\dfrac{z_1z_3}{z_2}$ in the form $a+ib$.

   SolutionTheory
6. Question 6

   If $z_1=2+7i$ and $z_2=-5+3i$, evaluate the following.

   (i)

   $|2z_1-4z_2|$

   (ii)

   $|3z_1+2z_2|$

   (iii)

   $|-7z_2+2\bar{z_2}|$

   (iv)

   $|(z_1+z_2)^3|$

   SolutionTheory
7. Question 7

   Show that $i^{n+1}+i^{n+2}+i^{n+3}+i^{n+4}=0$, for all $n\in\mathbb{N}$.

   SolutionTheory
8. Question 8

   Find the least positive value of $n$, if $\left(\dfrac{1+i}{1-i}\right)^{2n}=1$.

   SolutionTheory
9. Question 9

   Show that for $n\in\mathbb{N}$, $n>4$, we have $i^n=i^r$, where $r$ is the remainder when $n$ is divided by $4$.

   SolutionTheory