How would you simplify 2/3-i?

Question

I tried to multiply the 3-i on the top and bottom but then I get lost on what belongs on the bottom and on the top. 

 

I thought I would have something like 6-2i/9 but that doesn't seem right.

Answer 1

 This is division of 2 complex numbers.

 

 

      2__  =

     3 - i

   

 Multiply both numerator and denominator by conjugate number of the denominator.

 

    2     *   3+ i    =                   

 3 - i       3 +i                                                        

                                                                                 separate the rational and irrational numbers from

   each other                  

 

 

  6 + 2i    =  3  + 1  i            

    10           5     5

  

  This is also division of 2 complex numbers into each other.

Answer 2

 

Multiply top and bottom by the conjugate, 3+i:

 

2           3 + i

---   *    ------

3-i         3 + i

 

 

2(3+i)

--------

(3-i)(3+i)

 

6 + 2i

----------

9 + 3i - 3i - i^2 

 

6 + 2i

-------

9 - (-1)

 

6+2i

-------    or    (3/5) + (1/5) i

  10

 

(3/5) + (i/5) works as well.