Chapter 3

Show that the three cube roots of unity are \(1,(-1+i \sqrt{3}) / 2\) and \((-1-i \sqrt{3}) / 2\)

Show that if \(a\) is real and non-zero then (a) \(z \mapsto \bar{z}+a\) is a glide reflection along the real axis, and (b) \(z \mapsto-\bar{z}+i a\) is a glide reflection along the imaginary axis.

Find the radius of the circle whose equation is \(z \bar{z}+5 z+5 \bar{z}+9=0\).

Show that \(1+i=\sqrt{2} e^{i \pi / 4}\) and \(\sqrt{3}-i=2 e^{-i \pi / 6}\).

Show that \(i^{-1}=-i,(1+i)^{-1}=\frac{1}{2}(1-i)\), and \((1+i)^{2}=2 i\).

Show that all roots of \(a+b z+c z^{2}+z^{3}=0\) lie inside the circle \(|z|=\max \\{1,|a|+|b|+|c|\\}\)

Solve the equation \(z^{3}+6 z=20\) (this was considered by Cardan in \(A r s\) magna).

Suppose that the vertices of a regular pentagon lie on the circle \(|z|=1\). Show that the distance between any two distinct vertices is \(2 \sin (\pi / 5)\) or \(2 \sin (2 \pi / 5)\)

Find the equation of the line \(y=x\) in the form \(\bar{a} z+a \bar{z}=b\).

Show that if \(w=r e^{i \theta}\) and \(w \neq 0\), then \(z^{2}=w\) if and only if \(|z|=\sqrt{r}\) and \(\arg z\) is \(\theta / 2\) or \(\theta / 2+\pi\)

Show that \(z^{2}=2 i\) if and only if \(z=\pm(1+i)\).

Suppose that \(n \geq 2 .\) Show that (i) all roots of \(1+z+z^{n}=0\) lie inside the circle \(|z|=1+1 /(n-1)\); (ii) all roots of \(1+n z+z^{n}=0\) lie inside the circle \(|z|=1+2 /(n-1)\).

If we place a unit mass at each vertex of a regular \(n\)-gon whose vertices are on the circle \(|z|=1\), the centre of gravity of the masses should be at the origin. Prove (algebraically) that this is so. Let \(\omega=e^{2 \pi i / n}\), and let \(k\) be a positive integer. Show that $$ 1+\omega^{k}+\cdots+\omega^{k(n-1)}= \begin{cases}n & \text { if } n \text { divides } k \\ 0 & \text { otherwise }\end{cases} $$

If \(g\) is a reflection then there are infinitely many lines \(L\) satisfying \(g(L)=L\). Show that if \(f\) is a glide reflection then there is only one line \(L\) such that \(f(L)=L\); we call this line the axis of \(f\). Show that if \(f\) is a glide reflection with axis \(L\), then \(\frac{1}{2}(z+f(z))\) lies on \(L\) for every \(z\). This shows how to find \(L\) (choose two different values of \(z\) ).

Show that \(z^{4}=1\) if and only if \(z \in\\{1, i,-1,-i\\}\).

Show that \(\overline{z w}=\bar{z} \bar{w}\).

Suppose that \(\zeta\) is a solution of the \(3-2 z+z^{4}+z^{5}=0\). Use the inequalities $$ \begin{aligned} 3 &=\left|2 \zeta-\zeta^{4}-\zeta^{5}\right| \leq 2|\zeta|+|\zeta|^{4}+|\zeta|^{5} \\ |\zeta|^{5} &=\left|-\zeta^{4}+2 \zeta-3\right| \leq 3+2|\zeta|+|\zeta|^{4} \end{aligned} $$ to show that \(0.89426<|\zeta|<1.7265\).

Let \(f(z)=a z+b\) and \(g(z)=\alpha z+\beta\), where neither is the identity. Show that \(f g f^{-1} g^{-1}\) is a translation. Show also that \(f\) commutes with \(g\) if and only if either \(f\) and \(g\) are translations, or \(f\) and \(g\) have a common fixed point.

(i) Show that \(\left|z_{1}+\cdots+z_{n}\right| \leq\left|z_{1}\right|+\cdots+\left|z_{n}\right|\). (ii) Show that if \(|\arg z| \leq \pi / 4\), then \(x \geq 0\) and \(|z| \leq \sqrt{2} x\), where \(z=x+i y\). Deduce that if \(\left|\arg z_{j}\right| \leq \pi / 4\) for \(j=1, \ldots, n\), then $$ \frac{\left|z_{1}\right|+\cdots+\left|z_{n}\right|}{\sqrt{2}} \leq\left|z_{1}+\cdots+z_{n}\right| \leq\left|z_{1}\right|+\cdots+\left|z_{n}\right| $$

Verify directly that \(z w=0\) if and only if \(z=0\) or \(w=0\).

Show that the set of roots of unity for all \(n\) (that is, the set \(z\) for which \(z^{n}=1\) for some \(n\) ) is a group with respect to multiplication.

Suppose that \(f\) is a reflection in the line \(L\), and that \(f(z)\) that \(f(z)=a(\bar{z}-\bar{b})\). As \(|a|=1\) we can write \(a=e^{i \theta} ;\) let \(c=e^{i \theta / 2}\) By considering the fixed points of \(f\), show that \(L\) is given by \(c \bar{z}-\bar{c} z=c \bar{b}\)

Suppose that \(a \neq 0\). Show that the equation of the line that passes through \(z_{0}\), and is in the direction \(a\), is \(z \bar{a}-\bar{z} a=z_{0} \bar{a}-\overline{z_{0}} a\).

Show that \(\cos (\pi / 5)=\lambda / 2\), where \(\lambda=(1+\sqrt{5}) / 2\) (the Golden Ratio). [Hint: As \(\cos 5 \theta=1\), where \(\theta=2 \pi / 5\), we see from De Moivre's theorem that \(P(\cos \theta)=0\) for some polynomial \(P\) of degree five. Now observe that \(P(z)=(1-z) Q(z)^{2}\) for some quadratic polynomial \(\left.Q .\right]\)

Suppose that \(z w \neq 0\). Show that the segment joining 0 to \(z\) is perpendicular to the segment joining 0 to \(w\) if and only if \(\operatorname{Re}[z \bar{w}]=0\).

An \(n\)-th root of unity \(z\) is said to be primitive if \(z^{m} \neq 1\) for \(m=1,2, \ldots\), \(n-1\). Show that the primitive fourth roots of unity are \(i\) and \(-i\). Show that there are only two primitive 6-th roots of unity and find them. Show that, for a general \(n, e^{2 \pi i k / n}\) is a primitive \(n\)-th root of unity if and only if \(k\) and \(n\) have no common divisor other than \(1 .\)

Use De Moivre's theorem and the binomial theorem to show that \(\cos n \theta\) is a polynomial in \(\cos \theta\). This means that there are polynomials \(T_{0}, T_{1}, \ldots\) such that \(\cos n \theta=T_{n}(\cos \theta)\). The polynomial \(T_{n}\) is called the \(n\)-th Chebychev polynomial. By considering appropriate trigonometric identities, show that \(T_{n+1}(z)+T_{n-1}(z)=2 z T_{n}(z)\), and hence show that \(T_{3}(z)=4 z^{3}-3 z\).

Let \(T\) be a triangle in \(\mathbb{C}\) with vertices at \(0, w_{1}\) and \(w_{2}\). By applying the mapping \(z \mapsto \bar{w}_{2} z\), show that the area of \(T\) is \(\frac{1}{2}\left|\operatorname{Im}\left[w_{1} \bar{w}_{2}\right]\right|\).

Show that if \(\theta\) is real then \(\left|e^{i \theta}-1\right|=2 \sin (\theta / 2)\). Use this to derive Ptolemy's theorem: if the four vertices of a quadrilateral \(Q\) lie on a circle. then \(d_{1} d_{2}=\ell_{1} \ell_{3}+\ell_{2} \ell_{4}\), where \(d_{1}\) and \(d_{2}\) are the lengths of the diagonals of \(Q\), and \(\ell_{1}, \ell_{2}, \ell_{3}\) and \(\ell_{4}\) are the lengths of its sides taken in this order around \(Q .\)

Show that for any positive integer \(n\), $$ z^{n}-w^{n}=(z-w)\left(z^{n-1}+z^{n-2} w+\cdots+z w^{n-2}+w^{n-1}\right) $$ Deduce that \(z^{3}-w^{3}=(z-w)^{3}+3 z w(z-w)\)