Question 120862
I'm pretty sure you meant to say 8 + 10i is the <i>denominator</i> of both <i>fractions</i>, and your problem actually looks like:


{{{(1/(8+10i))-(1/(8+10i))}}}


Just like working with any other fractions, if the denominators are equal, then you simply add the numerators.


{{{(1-1)/(8+10i)=0}}}


We could have made this a little more complicated by using a process called rationalizing the denominators, but the result will still be the same.  Watch closely.


First thing is to remember that {{{(a+b)(a-b)=a^2+b^2}}}.   Having said that, we are now going to multiply each of the fractions by 1, in the form of {{{(8-10i)/(8-10i)}}}:


{{{((8-10i)/(8-10i))(1/(8+10i))-((8-10i)/(8-10i))(1/(8+10i))}}}


{{{((8-10i)(8+10i))=64-(-100)}}} (remember that {{{i^2=-1}}}), so our denominator becomes 164, giving us:


{{{((8-10i)-(8-10i))/164}}}


Now, to add complex numbers, you just add the real parts to the real parts and the imaginary parts to the imaginary parts:  {{{(a+bi)+(c+di)=(a+b)+(c+d)i}}}


In our problem, we have {{{((8-8)-(10-10)i)/164}}} giving us:


{{{0/164=0}}}


If you need to express 0 in complex number ({{{(a+bi)}}}) form, then you would write {{{(0+0i)}}}.


Hope that helps.