1. I proved that z^-1=conj z, converting into mod-arg form and by doing de moivres theorem so this must therefore be true
2. yet when i substituted z for an actual complex number eg. z=2+3i
and then did the realisation the answer is different
1/(2+3i) X (2-3i)/(2-3i)=(2-3i)/13
3. and i learned of a complex number rule where
z^-1=conj z/(mod z)^2
and this rule is consistent with what got ... which was (2-3i)/13
so now the question finally is why are there 2 answers that seem right?
What were you asked to do originally?
You have all these work based on something wrong.
Are you asked to prove or disprove if If so, an example of why it doesn't suffices which you have. If not, what are you trying to do?
there was a question regarding graphing 1/z and conj z and they were the same line
then i realised that through demoivre's theorem that the conj and 1/z could be proved to be equal
however when i realised the denominator in 1/z where eg. z=2+3i the result is (2+3i)/13 which is consistent to the rule z^-1=conj z/(mod z)^2
so why are there 2 answers by using 2 methods (demoivre's theorem and realising the denominator), or am i misunderstanding something?