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2.4 Complex numbers  (Page 2/8)

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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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Multiplying complex numbers together

Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,   i 2 , it equals   −1.

( a + b i ) ( c + d i ) = a c + a d i + b c i + b d i 2 = a c + a d i + b c i − b d i 2 = −1 = ( a c − b d ) + ( a d + b c ) i Group real terms and imaginary terms .

Given two complex numbers, multiply to find the product.

  1. Use the distributive property or the FOIL method.
  2. Remember that   i 2 = −1.
  3. Group together the real terms and the imaginary terms

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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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