<< Chapter < Page Chapter >> Page >

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

18 + i

Got questions? Get instant answers now!

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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask