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10.5 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       a + b i   is similar to plotting a real number, except that the horizontal axis represents the real part of the number,   a ,   and the vertical axis represents the imaginary part of the number,   b i .

Given a complex number   a + b i ,   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   a   units in the horizontal direction and   b   units in the vertical direction.

Plotting a complex number in the complex plane

Plot the complex number   2 − 3 i   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 of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).
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Plot the point   1 + 5 i   in the complex plane.

Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).
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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   | z | .   It measures the distance from the origin to a point in the plane. For example, the graph of   z = 2 + 4 i ,   in [link] , shows   | z | .

Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.

Absolute value of a complex number

Given   z = x + y i ,   a complex number, the absolute value of   z   is defined as

| z | = x 2 + y 2

It is the distance from the origin to the point   ( x , y ) .

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,   ( 0 ,   0 ) .

Finding the absolute value of a complex number with a radical

Find the absolute value of   z = 5 − i .

Using the formula, we have

| z | = x 2 + y 2 | z | = ( 5 ) 2 + ( − 1 ) 2 | z | = 5 + 1 | z | = 6

See [link] .

Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.
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Find the absolute value of the complex number   z = 12 − 5 i .

13

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Finding the absolute value of a complex number

Given   z = 3 − 4 i ,   find   | z | .

Using the formula, we have

| z | = x 2 + y 2 | z | = ( 3 ) 2 + ( − 4 ) 2 | z | = 9 + 16 | z | = 25 | z | = 5

The absolute value   z   is 5. See [link] .

Plot of (3-4i) in the complex plane and its magnitude |z| =5.
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
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Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
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A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
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Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
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I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
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Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
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field is a region of space under the influence of some physical properties
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Another formula for Acceleration
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a=v/t. a=f/m a
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
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What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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