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3.1 Complex numbers  (Page 2/8)

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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   ( a , b ) ,   where   a   represents the coordinate for the horizontal axis and   b   represents the coordinate for the vertical axis.

Let’s consider the number −2 + 3 i .   The real part of the complex number is −2   and the imaginary part is   3 i .   We plot the ordered pair   ( −2 , 3 )   to represent the complex number −2 + 3 i   as shown in [link] .

Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.

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] .

The complex plane showing that the horizontal axis (in the real plane, the x-axis) is known as the real axis and the vertical axis (in the real plane, the y-axis) is known as the imaginary axis.

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   3 − 4 i   on the complex plane.

The real part of the complex number is   3 ,   and the imaginary part is   −4 i .   We plot the ordered pair   ( 3 , −4 )   as shown in [link] .

Plot of a complex number, 3 - 4i. Note that the real part (3) is plotted on the x-axis and the imaginary part (-4i) is plotted on the y-axis.
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Plot the complex number   −4 − i   on the complex plane.

Graph of the plotted point, -4-i.
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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

Adding complex numbers:

( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i

Subtracting complex numbers:

( a + b i ) − ( c + d i ) = ( a − c ) + ( b − d ) 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.

Adding complex numbers

Add   3 − 4 i   and   2 + 5 i .  

We add the real parts and add the imaginary parts.

( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i ( 3 − 4 i ) + ( 2 + 5 i ) = ( 3 + 2 ) + ( − 4 + 5 ) i                               = 5 + i
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Subtract   2 + 5 i   from   3 – 4 i .  

( 3 − 4 i ) − ( 2 + 5 i ) = 1 − 9 i

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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,

Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i.

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   4 ( 2 + 5 i ) .

Distribute the 4.

4 ( 2 + 5 i ) = ( 4 ⋅ 2 ) + ( 4 ⋅ 5 i )                                             = 8 + 20 i
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Find the product   − 4 ( 2 + 6 i ) .

− 8 − 24 i

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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