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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:




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





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


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)






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-i 

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



           =   1-i

               -----                  Write in standard form



           =  1     1

              --- _ ---(i)

               2      2



For Additional Help:









Comments (1)

Reid Knuth said

at 5:45 pm on Dec 16, 2008

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