Extra (optional) exercises – Long Day

These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

Extra exercise 1: Integer powers of complex numbers

This exercise continues Exercise 9 from the day’s main exercises.

Let \(n\) be a natural number. Show the following:

\[(-1+\sqrt{3}i)^{3n}=2^{3n},\]

\[(-1+\sqrt{3}i)^{3n+1}=2^{3n}(-1+\sqrt{3}i)\]

and

\[(-1+\sqrt{3}i)^{3n+2}=2^{3n+1}(-1-\sqrt{3}i).\]

Hint

First, show that \((-1+\sqrt{3}i)^{3}=2^3\). What can we then say about \((-1+\sqrt{3}i)^{3n}\)?

Extra exercise 2: Complex numbers and Pythagorean triples

A Pythagorean triple \((a,b,c)\) consists of three natural numbers that satisfy \(a>b\) and \(a^2+b^2=c^2\). An example is \((4,3,5)\).

Question a

Show that if a triple \((a,b,c)\) of natural numbers is a Pythagorean triple, then the complex number \(z=\frac{a}{c}+\frac{b}{c}i\) satisfies \(|z|=1\) and \(\mathrm{Re}(z)>\mathrm{Im}(z)\).

Question b

Now assume that a complex number \(z\) satisfies \(|z|=1\) and \(\mathrm{Re}(z)>\mathrm{Im}(z)\) and that \(z\) can be written in the form \(z=\frac{a}{c}+\frac{b}{c}i\), where \(a\), \(b\) and \(c\) are natural numbers. Show that \((a,b,c)\) is a Pythagorean triple.

Question c

Use the above results and complex numbers to construct other Pythagorean triples based on the triple \((4,3,5)\).

Hint

If the number \(z=\frac{a}{c}+\frac{b}{c}i\) gives rise to the Pythagorean triple \((a,b,c)\), then what about the number \(z^2\) (or maybe \(i \cdot \overline{z}^2\))?