1. Home
  2. Algebra and trigonometry
  3. Equations and inequalities
  4. Complex numbers

2.4 Complex numbers  (Page 2/8)

<< Chapter < Page
  Algebra and trigonometry     Page 2 / 8
Chapter >> Page >

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

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.
Got questions? Get instant answers now!

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

Subtract   2 + 5 i   from   3 – 4 i .

( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i

Got questions? Get instant answers now!

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

Find the product:   1 2 ( 5 − 2 i ) .

5 2 − i

Got questions? Get instant answers now!

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

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 4

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask