# 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 did you get the value of 2000N.What calculations are needed to arrive at it
hi
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4
hil
hi
A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center