Complex Numbers in Rectangular and Polar Forms

The rectangular form of a complex number is
                     where     

          
                              Complex Plane

The point can be associated with the complex number

     See Example 1, page 628, of the textbook


               Polar Form

Using the conversion equations
     
     
the complex number
     
can be written in polar form
     
         

                         This conversion is shown in the graph

          

                                                  FIGURE 1

In the calculus the famous relation
     
is proved

Then the polar form of a complex number also can be written as



Note that if we let     , we get

     
or
                         

an amazing equation relating these five very important numbers!

Since the sine and the cosine functions are periodic functions with period ,
we can write

     
     
                                                                                for any integer

Then we can write the

          General Polar Form Of A Complex Number

               

                   

                 
                                                            for any integer

The number      is called the modulus, or absolute value, of      and is denoted
by   mod or  
The polar angle      that the line joining    to the origin makes with the polar axis is called
the argument of     and is denoted by   arg


From Figure 1 we see that

               Modulus and Argument for
                This quantity is never negative


                      for any integer 

                    where     and  

The argument      is conventionally chosen so that
                                                                                                            (radians)
or
                                                                                           (degrees)



Conversion Rectangular to Polar Form –
                                                                           See Example 2, pages 630 - 631, of the textbook

                       Polar to Rectangular Form –
                                                                           See Example 3, pages 631 - 632, of the textbook



               Multiplication and Division in Polar Form

               If        and       , then

     1]
                         

     2]
                             


     See Example 4, page 633, of the textbook



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