# 3.1 Complex numbers  (Page 2/8)

 Page 2 / 8

## Plotting a complex number on the complex plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane    , which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs $\text{\hspace{0.17em}}\left(a,b\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ represents the coordinate for the horizontal axis and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ represents the coordinate for the vertical axis.

Let’s consider the number $-2+3i.\text{\hspace{0.17em}}$ The real part of the complex number is $-2\text{\hspace{0.17em}}$ and the imaginary part is $\text{\hspace{0.17em}}3i.\text{\hspace{0.17em}}$ We plot the ordered pair $\text{\hspace{0.17em}}\left(-2,3\right)\text{\hspace{0.17em}}$ to represent the complex number $-2+3i\text{\hspace{0.17em}}$ as shown in [link] .

## Complex plane

In the complex plane , the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in [link] .

Given a complex number, represent its components on the complex plane.

1. Determine the real part and the imaginary part of the complex number.
2. Move along the horizontal axis to show the real part of the number.
3. Move parallel to the vertical axis to show the imaginary part of the number.
4. Plot the point.

## Plotting a complex number on the complex plane

Plot the complex number $\text{\hspace{0.17em}}3-4i\text{\hspace{0.17em}}$ on the complex plane.

The real part of the complex number is $\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}$ and the imaginary part is $\text{\hspace{0.17em}}-4i.\text{\hspace{0.17em}}$ We plot the ordered pair $\text{\hspace{0.17em}}\left(3,-4\right)\text{\hspace{0.17em}}$ as shown in [link] .

Plot the complex number $\text{\hspace{0.17em}}-4-i\text{\hspace{0.17em}}$ on the complex plane.

## Adding and subtracting complex numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

## Complex numbers: addition and subtraction

$\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i$

Subtracting complex numbers:

$\left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i$

Given two complex numbers, find the sum or difference.

1. Identify the real and imaginary parts of each number.
2. Add or subtract the real parts.
3. Add or subtract the imaginary parts.

Add $\text{\hspace{0.17em}}3-4i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2+5i.\text{\hspace{0.17em}}$

Subtract $\text{\hspace{0.17em}}2+5i\text{\hspace{0.17em}}$ from $\text{\hspace{0.17em}}3–4i.\text{\hspace{0.17em}}$

$\left(3-4i\right)-\left(2+5i\right)=1-9i$

## Multiplying complex numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

## Multiplying a complex numbers by a real number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,

Given a complex number and a real number, multiply to find the product.

1. Use the distributive property.
2. Simplify.

## Multiplying a complex number by a real number

Find the product $\text{\hspace{0.17em}}4\left(2+5i\right).$

Distribute the 4.

$\begin{array}{l}4\left(2+5i\right)=\left(4\cdot 2\right)+\left(4\cdot 5i\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=8+20i\hfill \end{array}$

Find the product $\text{\hspace{0.17em}}-4\left(2+6i\right).$

$-8-24i$

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim