4+i divided by 5+i

Question

I really need help with this question.

Answer 1

Note:

     4 +i    .   5 - i   =

     5 +i        5 - i

 

     20 +5i-4i +1  = 

      25  + 1

 

     21  + i  

     26    26

 

     To divide 2 complex numbers .

      Multiply the denominator by its conjugate,  i.e.  5+i , by 5- i 

      the denominator becomes real number, i.e. 26

       and the result is a complex number  with real , and imaginary part.

Answer 2

4 + i

------

5 + i

 

Multiply top and bottom by the conjugate, 5-i.

 

(4+i)(5-i)

-----------

(5+i)(5-i)

 

Using FOIL (first outside inside last)

 

20 - 4i + 5i - i²

-----------------

25 -5i + 5i - i²

 

20 + i - (-1)

--------------

25 - (-1)

 

20+1 + i

------------

25+1

 

Answer:

 

21 + i

-------

   26

Answer 3

Division of Complex Numbers:

 

1) Find the conjugate of the denominator.  The conjugate of a complex number is simply the same expression with the opposite sign.  In this case the conjugate of 5+i is 5-i.

 

2) Multiply the given fraction by the conjugate fraction.

 

(4+i)    (5-i)          20-4i+5i-i²

------ x ------- =    --------------   ,  you can see that this new fraction simplifies quite a bit

(5+i)    (5-i)          25+5i-5i-i²

 

 

3) Simplify the fraction, i² =-1, and add the common terms  (5i-4i=i, 5i-5i=0)

 

20+i+1       21+i

----------  =  ---------

25+1              26

 

 

That is the final answer.