Let \(z=3+5i\). Draw on the complex plane the point \(3z, \frac{1}2{z}\), and \(-z\). What do you notice?
Show that if \(x\) and \(y\) are complex numbers, then
\[\overline{\left(\frac{x}{y}\right)}=\frac{\bar x}{\bar y}\]
Describe the set of complex numbers \(z\) for which \(z=\bar z\).
Let points \(A,B,C,D\) on the plane be represented by complex numbers \(a,b,c,d\). Show that the segments \(AB\) and \(CD\) are parallel if and only if \[\frac{a-b}{\bar a-\bar b}=\frac{c-d}{\bar c - \bar d}.\]
Let \(A,B,C,D\) be points on the plane represented by complex numbers \(a,b,c,d\). Show that the segments \(AB\) and \(CD\) are perpendicular if and only if \[\frac{d-c}{b-a}=-\frac{\bar d-\bar c}{\bar b- \bar a}\]
Draw some points \(a,b,c\) in the complex plane (whichever you like), and then draw the points \(ia, ib, ic\). Do you notice what geometric action corresponds to multiplying by \(i\)? Can you prove that this is the case?