# Complex Numbers: Problems with Solutions

#### Theory

Rule Equivalent Exponent
$i^1 = i$ $i^{4n + 1}$ Multiple of 4 + 1
${4n + 1, \ n \in \mathbb{Z}} = {1; 5; 9...}$
$i^2 = -1$ $i^{4n + 2}$ Multiple of 4 + 2
${4n + 2, \ n \in \mathbb{Z}} = {2; 6; 10...}$
$i^3 = -i$ $i^{4n + 3}$ Multiple of 4 + 3
${4n + 3, \ n \in \mathbb{Z}} = {3; 7; 11...}$
$i^4 = 1$ $i^{4n}$ Multiple of 4
${4n, \ n \in \mathbb{Z}} = {4; 8; 12...}$
##### Addition and subtraction of complex numbers:

Let (a + bi) and (c + di) be two complex numbers, then:
(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

Reals are added with reals and imaginary with imaginary.

##### Complex numbers division:
$\frac{a + bi}{c + di}=\frac{(ac + bd)+(bc - ad)i)}{c^2+d^2}$

#### Problems with Solutions

Problem 1
Given two complex numbers $z=\left( -1,2\right)$ and $w=\left(3,2\right)$
Calculate $5z-3w=$
Plot a graph.
Problem 2
Given two complex numbers $z=\left( 2,-1\right)$ and $w=\left( 3,2\right)$
Find $x=z\cdot \overline{w}$
Problem 3
$z=\left( 3,-2\right)$ and $w=\left( -3,1\right)$ are complex numbers. Find $x=z\div w$
Problem 4
If $z=(-2,3)$ and $w=(4,-1).$
Find $x=z^{2}-w^{2}$ ?
Problem 5
Calculate $\overline{\left( z-w\right) }-\overline{\left( z+w\right)}$, where:
$z=\left( 2,4\right)$, $w=\left( 4,-1\right)$, $x=(0,1)$ and $y=\left( 3,-1\right)$
Problem 6
If $\rho =\mid z\mid$ is the module of $z$ and $\ z=(3,-4)$
Calculate:
$x=\rho \overline{z}+\frac{10}{\rho }z$
Problem 7
If $z=(3,4)$
Find $\sqrt{z}=$
Problem 8
Let $z=(4,-3)$
$\text{Re}\left( z\cdot \overline{z}\right) =\mid z\mid ^{2}$
Problem 9
Let $z=(4,-3)$
$z+\overline{z}=2\text{Re}(z)$
Problem 10
What is the result of $i^{761}$ ?

Problem 11
Calculate
$\frac{(2-3i)-(3+2i)}{(3+2i)-(2+i)}$
Problem 12
$\frac{i^{326}-1}{i^{545}+1}=$
Problem 13
Write in the polar and binomial forms the conjugate of:
$z=4+4i$
Problem 14
Multiply the complex numbers:
$(5+2i)(2-3i)$
Problem 15
Divide the complex numbers:

$\frac{3-2i}{5+2i} =$
Problem 16
If $z=2+5i$ and $w=3+2i$, perform the following operation $(z\cdot \overline{w})^{2}$ and find its conjugate.
Problem 17
If $z=3+i$  $t=1+i$   $w=2-3i$   $s=-1+2i$, perform the following calculation $\left\vert z\cdot 3t-2w\cdot s\right\vert$
Problem 18
What is the polar form of the complex number $z=-1+\sqrt{3}i$?
Problem 19
Perform the following operation
$(5+2i)+(-8+3i)-(4-2i)$ and write the result in a polar form.
Problem 20
Find the distance between the complex numbers $z=2-3i$ and $w=-3+2i$.
Problem 21
What is the midpoint of the segment formed by $z=6-3i$ and $w=2+5i$ ?
Problem 22
Let $s$ be the sum of the complex numbers
$z=2+3i$ and $w=1-4i$ and let $r$ be the subtraction of the same numbers.

Find the midpoint of $s,r$.
Problem 23
If $z=2-i$, $w=-3+2i$, what is the midpoint between $2z$ and $\overline{w}$ ?
Problem 24
Plot a graph of the complex numbers $z=-1+i$, $w=2+3i$.
Find the distance.
Problem 25
If $z=2-i$, $w=5+i$, $t=-3+2i$, which of the following options is the result of $\frac{2z}{w-t}$ ?
Problem 26
If $z=2-i$, $w=5+i$, $t=-3+2i$, which of the following options is the result of $\overline{(w\cdot t)-3z}$?

Correct:
Wrong:
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