Problems

Let distinct 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 distinct 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?

Let \(a,b\) be complex numbers with \(|a|=|b|=1\). Let \(z\) be some other complex number. Show that the reflection of \(z\) about the line that connects \(a\) and \(b\) is given by \[a+b-ab\bar z\]

Let \(\ell\) be a line in the complex plane through the origin. Show that multiplication by a nonzero complex number sends \(\ell\) into another line through the origin.

The Pythagorean Theorem is a very useful tool in geometry. It says that if you have a right-angled triangle with sides measuring \(a,b,c\) where \(c\) is the longest side of the three, then \(a^2+b^2=c^2\). Explain how the following diagram gives a visual proof of the Pythagoraen Theorem.

Without carrying out the multiplications, which is larger: \[(2015+2026)^2\qquad \text{or} \qquad 4\times 2015\times 2016\]

Let \(z\) be a complex number. Show that

1. For a real number \(k\), \(|kz|=|k|\cdot |z|\).

2. \(|iz|=|z|\).