Free Mathematics Tutorials
Home
Math and Precalculus
Math Problems
Algebra Questions and Problems
Graphs of Functions, Equations, and Algebra
Free Math Worksheets to Download
Analytical Tutorials
Solving Equation and Inequalities
Online Math Calculators and Solvers
graphing
Free Graph Paper
Math Software
Applied Math
The Applications of Mathematics in Physics and Engineering
Antennas
Exercises de Mathematiques Utilisant les Applets
Calculus
Calculus Tutorials and Problems
Calculus Questions With Answers
Free Calculus Worksheets to Download
Free Calculus Worksheets to Download
Geometry
Geometry Tutorials and Problems
Online Geometry Calculators and Solvers
Free Geometry Worksheets to Download
Trigonometry
Trigonometry Tutorials and Problems for Self Tests
Free Trigonometry Questions with Answers
Free Trigonometry Worksheets to Download
More
Statistics(2)
Site(3)
Primary Math
Middle School Math
High School Math
Free Practice for SAT, ACT and Compass Math tests
More
Statistics (2)
Site (3)
Primary Math
Middle School Math
High School Math
Free Practice for SAT, ACT and Compass Math tests
Elementary Statistics and...
Mathematics pages in French
About the author
Download
E-mail
Primary Math
Middle School Math
High School Math
Free Practice for SAT, ACT and Compass Math tests
Complex Numbers - Basic Operations
Definition of Complex Numbers
Complex Conjugate
Addition of Complex Numbers
Subtractiontion of Complex Numbers
Multiply Complex Numbers
Divide two Complex Numbers
Definition of Complex Numbers
A complex number z is a number of the form
z = a + b i
where a and b are real numbers and i is the imaginary unit defined by
a is called the real part of z and b is the imaginary part of z.
Note that the set
R
of all real numbers is a subset of the complex number
C
since any real number may be considered as having the imaginary part equal to zero..
Complex Conjugate
The conjugate of a complex number a + b i is a complex number equal to
a - b i
Examples:
Find the conjugate of the following complex numbers.
a) 2 - i , b) -3 + 4i , c) 5 , d) -5i
Solution to above example
a) 2 + i , b) -3 - 4i , c) 5 , d) 5i
Use calculator below to generate complex conjugate of any complex number.
Applet to generate complex conjugate : enter your complex number and press enter
Your browser is completely ignoring the <APPLET> tag!
Addition of Complex Numbers
Addition of two complex numbers a + b i and c + d i is defined as follows.
(a + b i) + (c + d i) = (a + c) + (b + d) i
This is similar to grouping like terms: real parts are added to real parts and imaginary parts are added to imaginary parts.
Example:
Express in the form of a complex number a + b i.
(2 + 3i) + (-4 + 5i)
(3i) + (-5 + 6i)
(2) + (-2 + 9i)
Solution to above example.
(2 + 3i) + (-4 + 5i) = (2 - 4) + (3 + 5) i = - 2 + 8 i
(3i) + (-5 + 6i) = (0 - 5) + (3 + 6) i = -5 + 9 i
(2) + (-2 + 9i) = (2 - 2) + (9) i = 9i
Addition can be done by grouping like terms.
(2 + 3i) + (-4 + 5i) = 2 + 3 i - 4 + 5 i = -2 + 8 i
Applet that Adds two complex numbers: enter your complex numbers and press enter
Your browser is completely ignoring the <APPLET> tag!
Subtraction of Complex Numbers
The subtraction of two complex numbers a + b i and c + d i is defined as follows.
(a + b i) - (c + d i) = (a - b) + (b - d) i
Example:
Express in the form of a complex number a + b i.
(2 - 5i) - (-4 - 5i)
(-7i) - (-5 - 6i)
(2) - (2 + 6i)
Solution to above example
(2 - 5i) - (-4 - 5i) = (2 - (-4)) + (-5 - (-5)) i = 6
(-7i) - (-5 - 6i) = (0 - (-5)) + (-7 - (-6)) i = 5 - i
(2) - (2 + 6i) = (2 - 2) - 6 i = -6 i
Note: subtraction can be done as follows:
(a + b i) - (c + d i) = (a + bi) + (- c - d i) and then group like terms
Exapmple:
(2 - 5i) - (-4 - 5i) = 2 - 5 i + 4 + 5 i = 6
Applet that subtract two complex numbers: enter your complex numbers and press enter
Your browser is completely ignoring the <APPLET> tag!
Mulyiply Complex Numbers
The multiplication of two complex numbers a + b i and c + d i is defined as follows.
(a + b i)(c + d i) = (a c - b d) + (a d + bc) i
However you do not need to memorize the above definition as the multiplication can be carried out using properties similar to those of the real numbers and the added property i
^{ 2}
= -1. (see the example below)
Example:
Express in the form of a complex number a + b i.
(3 + 2 i)(3 - 3i)
Solution to above example
(3 + 2 i)(3 - 3i)
Using the distributive law, (3 + 2 i)(3 - 3 i) can be written as
(3 + 2 i)(3 - 3 i) = (3 + 2 i)(3) + (3 + 2 i)(-3 i) = 9 + 6 i - 9 i -6 i
^{ 2}
Group like terms and use i
^{ 2}
= -1 to simplify (3 + 2 i)(3 - 3 i)
(3 + 2 i)(3 - 3 i) = 15 - 3 i
Applet that multiplies two complex numbers: enter your complex numbers and press enter
Your browser is completely ignoring the <APPLET> tag!
Divide two Complex Numbers
We use the multiplication property of complex number and its conjugate to divide two complex numbers.
Example:
Express in the form of a complex number a + b i.
(8 + 4 i) / (1 - i)
We first multiply the numerator and denominator by the complex conjugate of the denominator
[ (8 + 4 i)
(1 + i)
]
Multiply and group like terms
[ (8 + 4 i)
(1 + i)
] / [ (1 - i)
(1 + i)
] =
[ 8 + 4 i + 8 i + 4 i
^{ 2}
] / [ 1 - i + i - i
^{ 2}
]
= (4 + 12 i ) / (2)
= 2 + 6 i
Applet that divides two complex numbers: enter your complex numbers and press enter
Your browser is completely ignoring the <APPLET> tag!
Exercises
Find the complex conjugate of the following complex numbers
a) 2 + 6 i
b) -8 i
c) 12
Write the following expressions in the form a + b i
a) (2 - 8 i) + (-6 i)
b) -8 i + (3 - 9 i)
c) 6 - (3 - i)
d) (2 - 3 i)(7 - i)
e) (2 + 2 i) / (2 - 2i)
Solutions to above exercises
Find the complex conjugate.
a) 2 - 6 i
b) 8 i
c) 12
Write the following expressions in the form a + b i
a) (2 - 8 i) + (-6 i) = 2 - 14 i
b) -8 i + (3 - 9 i) = 3 - 17 i
c) 6 - (3 - i) = 3 + i
d) (2 - 3 i)(7 - i) = 11 - 23 i
e) (2 + 2 i) / (2 - 2i) = i
Popular Pages
Grade 12 Problems on Complex Numbers with Solutions and Answers
Free Calculus Tutorials and Problems
Online Step by Step Calculus Calculators and Solvers
Convert Polar to Rectangular Coordinates and Vice Versa
Math Problems, Questions and Online Self Tests
More Info
© analyzemath.com. All rights reserved.
Privacy Policy