| 
  • If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • Stop wasting time looking for files and revisions. Connect your Gmail, DriveDropbox, and Slack accounts and in less than 2 minutes, Dokkio will automatically organize all your file attachments. Learn more and claim your free account.

View
 

Complex-Numbers

Page history last edited by Kristen Fouss 11 years, 7 months ago

 

2.4 Complex Numbers

 

Complex numbers are expressions that include real and non-real portions.  These are used to find solutions to problems where there is no real solution.  Standard form of complex numbers is written as "a+bi", where a is the real portion of the answer and bi is the imaginary portion (b is a real coefficient).

 

i is equal to the square root of -1.

For example, to symplify eq=\sqrt{-9} -5:

 

Factor out an i from eq=\sqrt{-9}, so you get i times the square root of 9, or 3i.

Now put the answer in standard form with the real portion first. -5+3i

 

Two complex numbers, a+bi and c+di are equal. Therefore, a=c and b=d.

 

To add and subtract complex numbers, group the real parts together and the imaginary parts together like so:

 

(a+bi)+(c+di)=(a+c)+(b+d)i  

 

ex. (3+2i)+(4-i)-(7+i)=3+2i+4-i-7-i

                              =3+4-7+2i-i-i

                              =0+0i

                              =0

 

Multiplying Complex Numbers: 

 

Example 1: 

 

eq=\sqrt{-4} *eq=\sqrt{-16}=(2i)+(4i)      Write the factor in standard form

                     =8i^2           Multiply out the factors

                    =8(-1)            Change the squared, imaginary number to -1

                    =-8

Example 2:

 

(2-i)(4+2i)=8+6i-4i-3i^2      Factor out

              = 8+6i-4i-3(-1)     Change the squared, imaginary nember to -1

              = 8+3+6i-4i         Group Like Terms

              =11+2i                Set In Standard From        

 

Complex Conjugates: occurs with pairs of complex numbers of the form a+bi and a-bi

          (a+bi)(a-bi)=a^2 -abi =abi-(B^2)(i^2)

                          = (a^2)-b^2(-1)

                          = (a^2)+(b^2)

 

To find the quotient of a+bi and c+di, multiply the numerator and the denominator by teh conjugate of the denominator:

   

     a+bi     a+bi    (c-di)

     ------ = ------ * ------

     c+di     c+di     (c-di)

 

     =(ac+bd)+(bc-ad)i

       ---------------------

           (c^2)+(d^2)

 

Dividing Complex Numbers:

 

     1          1       (1-i)

  ------ = ------ * ------          Multiply Numerator and denominator by conjugate of denominator

   1+i       1+i      (1-i)

 

           =     1-i              

             ------------             Expand

             (1^2)-(i^2)        

 

           =     1-i 

                 -------              i^2 = -1

                 1-(-1)

          

           =   1-i

               -----                  Write in standard form

                  2

 

           =  1     1

              --- _ ---(i)

               2      2

 

 

For Additional Help:

http://www.purplemath.com/modules/complex.htm

http://www.uncwil.edu/courses/mat111hb/Izs/complex/complex.html

 

 

 

 

 

 

Comments (1)

Reid Knuth said

at 5:45 pm on Dec 16, 2008

Garrett, if u wanna write in color just use the tool at the top of our page.

You don't have permission to comment on this page.