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I'm having trouble learning and understanding questions relating to complex numbers and was wondering if I can get any help. Thanks in advance for any help I can get!

Given $x$ is a complex number (not $0$), we also have a new complex number $x^{-1}$, called inverse of $x$, so we have the property that $x\cdot x^{-1} = 1$.

a) Prove that this inverse is unique.

b) $x^{-1} = \bar{x}/\lvert x\rvert^2$

c) Prove if $\lvert a\rvert = \lvert b\rvert = \lvert c\rvert = 1$, then $\bar{a} + \bar{b} + \bar{c} = 1/a + 1/b + 1/c$.

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$\overline{stuff}$ gives you $\overline{stuff}$, for future reference. – Cameron Buie Mar 26 '13 at 22:24
Thank you, I will use that in the future – Michael Mar 26 '13 at 22:24
up vote 1 down vote accepted

(1) Suppose $x\cdot a=1$ and $x\cdot b=1$. Since multiplication commutes, then $a\cdot x=1$, so $$a=a\cdot 1=a\cdot (x\cdot b)=(a\cdot x)\cdot b=1\cdot b=b.$$ Thus, inverses are unique.

(2) The first part told us that if $x\cdot a=1$, then $a=x^{-1}$. How can we use this to show that $$\frac{\overline z}{|z|^2}=z^{-1}?$$

(3) $\frac1x$ is another way to write $x^{-1}$. How can we use the second part to show that $$\overline a+\overline b+\overline c=a^{-1}+b^{-1}+c^{-1}$$ whenever $|a|=|b|=|c|=1$?

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