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## 4+i divided by 5+i

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Note:
4 +i    .   5 - i   =
5 +i        5 - i

20 +5i-4i +1  =
25  + 1

21  +
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.
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

21 + i
-------
26
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