Find the real part of a complex number

Question

I am interested in finding the real part of this

$$\frac{z+2i}{z-2i} + 2i$$

Answer 1

To express $\frac 1z$ in $a + b i$ format multiply the numerator and denominater by $\overline z$. $z\overline z$ is always a real number so $\frac 1z = \frac 1z \frac {\overline z}{\overline z} = \frac {\overline z}{z\overline z}$ will be easy to put into $a + bi$ form.

Example:

$\frac 1{a+bi} = \frac 1{a+bi}\frac {a-bi}{a-bi}=\frac {a -bi}{a^2 + b^2} = \frac a{a^2 + b^2} - \frac b{a^2 + b^2} i$.

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So in this one if we assume $z = a + bi$ then

$\frac {z + 2i}{z - 2i} +2i= \frac {a + (b+2)i}{a + (b-2)i}=$

$\frac {(a+(b+2)i)(a - (b-2)i)}{(a + (b-2)i)(a-(b-2)i)} + 2i=$

......