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Multiplying complex numbers together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

( a + b i ) ( c + d i ) = a c + a d i + b c i + b d i 2

Because i 2 = 1 , we have

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

To simplify, we combine the real parts, and we combine the imaginary parts.

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

Given two complex numbers, multiply to find the product.

  1. Use the distributive property or the FOIL method.
  2. Simplify.

Multiplying a complex number by a complex number

Multiply ( 4 + 3 i ) ( 2 5 i ) .

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

( 4 + 3 i ) ( 2 5 i ) = ( 4 2 3 ( 5 ) ) + ( 4 ( 5 ) + 3 2 ) i                          = ( 8 + 15 ) + ( 20 + 6 ) i                          = 23 14 i
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Multiply ( 3 4 i ) ( 2 + 3 i ) .

18 + i

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Dividing complex numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate    of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of a + b i is a b i .

Note that complex conjugates have a reciprocal relationship: The complex conjugate of a + b i is a b i , and the complex conjugate of a b i is a + b i . Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

Suppose we want to divide c + d i by a + b i , where neither a nor b equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

c + d i a + b i  where  a 0  and  b 0

Multiply the numerator and denominator by the complex conjugate of the denominator.

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

Apply the distributive property.

= c a c b i + a d i b d i 2 a 2 a b i + a b i b 2 i 2

Simplify, remembering that i 2 = −1.

= c a c b i + a d i b d ( 1 ) a 2 a b i + a b i b 2 ( 1 ) = ( c a + b d ) + ( a d c b ) i a 2 + b 2

The complex conjugate

The complex conjugate    of a complex number a + b i is a b i . It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

  • When a complex number is multiplied by its complex conjugate, the result is a real number.
  • When a complex number is added to its complex conjugate, the result is a real number.

Finding complex conjugates

Find the complex conjugate of each number.

  1. 2 + i 5
  2. 1 2 i
  1. The number is already in the form a + b i . The complex conjugate is a b i , or 2 i 5 .
  2. We can rewrite this number in the form a + b i as 0 1 2 i . The complex conjugate is a b i , or 0 + 1 2 i . This can be written simply as 1 2 i .
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Given two complex numbers, divide one by the other.

  1. Write the division problem as a fraction.
  2. Determine the complex conjugate of the denominator.
  3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
  4. Simplify.

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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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