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2.4 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. We plot the ordered pair   ( −2 , 3 )   to represent the complex number   −2 + 3 i , as shown in [link] .

Coordinate plane with the x and y axes ranging from negative 5 to 5. The point negative 2 plus 3i is plotted on the graph. An arrow extends leftward from the origin two units and then an arrow extends upward three units from the end of the previous arrow.

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] .

A blank coordinate plane with the x-axis labeled: real and the y-axis labeled: imaginary.

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. We plot the ordered pair   ( 3 , −4 )   as shown in [link] .

Coordinate plane with the x and y axes ranging from -5 to 5. The point 3 – 4i is plotted, with an arrow extending rightward from the origin 3 units and an arrow extending downward 4 units from the end of the previous arrow.
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Plot the complex number   −4 − i   on the complex plane.

Coordinate plane with the x and y axes ranging from negative 5 to 5. The point -4 i is plotted.
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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 then 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 and subtracting complex numbers

Add or subtract as indicated.

  1. ( 3 − 4 i ) + ( 2 + 5 i )
  2. ( −5 + 7 i ) − ( −11 + 2 i )

We add the real parts and add the imaginary parts.


  1. ( 3 − 4 i ) + ( 2 + 5 i ) = 3 − 4 i + 2 + 5 i = 3 + 2 + ( −4 i ) + 5 i = ( 3 + 2 ) + ( −4 + 5 ) i = 5 + i

  2. ( −5 + 7 i ) − ( −11 + 2 i ) = −5 + 7 i + 11 − 2 i = −5 + 11 + 7 i − 2 i = ( −5 + 11 ) + ( 7 − 2 ) i = 6 + 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 number by a real number

Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example,   3 ( 6 + 2 i ) :

Multiplication of a real number and a complex number. The 3 outside of the parentheses has arrows extending from it to both the 6 and the 2i inside of the parentheses. This expression is set equal to the quantity three times six plus the quantity three times two times i; this is the distributive property. The next line equals eighteen plus six times i; the simplification.

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:   1 2 ( 5 − 2 i ) .

5 2 − i

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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