L3 Complex Numbers
Question Bank
The complex number $z$ has $|z| = 1$.
(a) Prove that $\dfrac{1}{z} = \bar{z}$.
(b) Hence prove that $z^2 + \dfrac{1}{z^2}$ is a real number.
Show that $\displaystyle 1 + x + x^2 + \frac{1}{x - 1} = \frac{x^3}{x - 1}$ for $x \neq 1$.
Let $z$ be a non-zero complex number. Prove that
$ \frac{z^2 + |z|^2}{z} = 2\,\text{Re}(z) $
The complex number $z$ satisfies $|z| = 1$ and $z \neq 1$. Prove that
$ \frac{(z - 1)^2}{z} $
is a negative real number.
Let $z$ and $w$ be two complex numbers.
By letting $z = a + bi$ and $w = c + di$, prove that:
$ |zw|^2 = |z|^2 |w|^2 $
Let $z$ and $w$ be two non-zero complex numbers such that $|z + w|^2 = |z|^2 + |w|^2$.
Prove that the complex number $\displaystyle \frac{z}{w}$ is purely imaginary.
Prove that for any two complex numbers $u$ and $v$:
$ |u + v|^2 + |u - v|^2 = 2(|u|^2 + |v|^2) $
Let $u = x + iy$ and $v = a + bi$.
Prove that
$ |u + v|^2 = |u|^2 + |v|^2 + 2\text{Re}(u\bar{v}) $
Show that, for any integer $n \geq 2$, the sum of the $n$ distinct roots of the equation $z^n = 1$ is zero.
Hint: You may use the formula for the sum of a geometric series.
If $|z| = 1$ and $z \neq 1$, prove that $\displaystyle \frac{z + 1}{z - 1}$ is purely imaginary.
The Arithmetic mean ($A$) and Harmonic mean ($H$) of two positive real numbers, $x$ and $y$, are defined as:
$ A = \frac{x+y}{2} \quad \text{and} \quad H = \frac{2}{\frac{1}{x} + \frac{1}{y}} $
Prove that:
$ \frac{1}{\sqrt{H}} - \frac{1}{\sqrt{A}} = \frac{(\sqrt{x} - \sqrt{y})^2}{\sqrt{2xy(x+y)}} $
If $z$ is a complex number with $|z| = 1$, and $w$ is any complex number such that $w \neq z$, prove that:
$ \left| \frac{z - w}{1 - \bar{z}w} \right| = 1 $