Calculate the difference 1/(5+2i) - 1/(5-2i) .

For the beginning, before calculating the difference
of the 2 ratios, we have to transform the denominator of each ratio into a real number,
instead of complex numbers.

We'll multiply the complex
number from denominator by it's conjugate.

If the complex
number is z=a+b*i, it's conjugate is z'=a-b*i.

So, if the
complex number is 5+2i, it's conjugate is 5-2i.

We'll
multiply the first ratio by the conjugate number (5-2i) and the second ratio by
(5+2i).

(5-2i)/(5+2i)(5-2i) - (5+2i)/(5-2i)(5+2i) =
(5-2i)/(25+4)-(5+2i)/(25+4)

(5-2i)/(25+4)-(5+2i)/(25+4) =
(5-2i-5-2i)/29

We'll reduce like
terms:

(5-2i-5-2i)/29= (-4i) /
29

So, the result of the difference between
the 2 ratios is the complex number (-4i) /
29.

1/(5+2i) - 1/(5-2i) =
(-4i) / 29