# 8.5 Polar form of complex numbers

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In this section, you will:
• Plot complex numbers in the complex plane.
• Find the absolute value of a complex number.
• Write complex numbers in polar form.
• Convert a complex number from polar to rectangular form.
• Find products of complex numbers in polar form.
• Find quotients of complex numbers in polar form.
• Find powers of complex numbers in polar form.
• Find roots of complex numbers in polar form.

“God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras , Descartes , De Moivre, Euler , Gauss , and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.

We first encountered complex numbers in Complex Numbers . In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.

## Plotting complex numbers in the complex plane

Plotting a complex number     $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ is similar to plotting a real number, except that the horizontal axis represents the real part of the number, $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ and the vertical axis represents the imaginary part of the number, $\text{\hspace{0.17em}}bi.$

Given a complex number $\text{\hspace{0.17em}}a+bi,\text{\hspace{0.17em}}$ plot it in the complex plane.

1. Label the horizontal axis as the real axis and the vertical axis as the imaginary axis.
2. Plot the point in the complex plane by moving $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units in the horizontal direction and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ units in the vertical direction.

## Plotting a complex number in the complex plane

Plot the complex number $\text{\hspace{0.17em}}2-3i\text{\hspace{0.17em}}$ in the complex plane    .

From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See [link] .

Plot the point $\text{\hspace{0.17em}}1+5i\text{\hspace{0.17em}}$ in the complex plane.

## Finding the absolute value of a complex number

The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude , or $\text{\hspace{0.17em}}|z|.\text{\hspace{0.17em}}$ It measures the distance from the origin to a point in the plane. For example, the graph of $\text{\hspace{0.17em}}z=2+4i,\text{\hspace{0.17em}}$ in [link] , shows $\text{\hspace{0.17em}}|z|.$

## Absolute value of a complex number

Given $\text{\hspace{0.17em}}z=x+yi,\text{\hspace{0.17em}}$ a complex number, the absolute value of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ is defined as

$|z|=\sqrt{{x}^{2}+{y}^{2}}$

It is the distance from the origin to the point $\text{\hspace{0.17em}}\left(x,y\right).$

Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin,

## Finding the absolute value of a complex number with a radical

Find the absolute value of $\text{\hspace{0.17em}}z=\sqrt{5}-i.$

Using the formula, we have

$\begin{array}{l}|z|=\sqrt{{x}^{2}+{y}^{2}}\hfill \\ |z|=\sqrt{{\left(\sqrt{5}\right)}^{2}+{\left(-1\right)}^{2}}\hfill \\ |z|=\sqrt{5+1}\hfill \\ |z|=\sqrt{6}\hfill \end{array}$

Find the absolute value of the complex number $\text{\hspace{0.17em}}z=12-5i.$

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## Finding the absolute value of a complex number

Given $\text{\hspace{0.17em}}z=3-4i,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}|z|.$

Using the formula, we have

$\begin{array}{l}|z|=\sqrt{{x}^{2}+{y}^{2}}\hfill \\ |z|=\sqrt{{\left(3\right)}^{2}+{\left(-4\right)}^{2}}\hfill \\ |z|=\sqrt{9+16}\hfill \\ \begin{array}{l}|z|=\sqrt{25}\\ |z|=5\end{array}\hfill \end{array}$

The absolute value $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ is 5. See [link] .

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
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?