modulus and argument

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modulus and argument

Given a complex number, $z = a + bi$, the modulus of $z$, denoted $|z|$ is given by $$ |z| = a^2 + b^2 $$

The value of the modulus corresponds to the length of the line segment from $(0,0)$ to the point $(a,b)$, which on an argand diagram repesents the complex number $z=a + ib$.

The modulus is always non-negative and is only zero for the complex number $z=0$.

The argument, denoted $\arg(z)$, is a more difficult concept. It it the angle the line segment from (0,0) to the point $(a,b)$ makes, when measured from the positive x-axis, with a clockwise rotation being a positive angle and an anticlockwise rotation being a negative angle. 

argand diagram 1

modulus and argument 1

Determine the modulus and argument of the following complex numbers
  1. $z_1 = 4 + 4i$
  2. $z_2 = -1 + \sqrt{3}i$
  3. $z_3 = -8 - 8i$
solution - press button to display
  1. $|z_1| = \sqrt{4^2+4^2} = 4\sqrt{2}$, $\arg(z_1) =\tan^{-1}\left(\frac{4}{4}\right) = \frac{\pi}{4}$
  2. $|z_2| = \sqrt{(-1)^2 + (\sqrt{3})^2} = 2$, $\arg(z_2) = \pi - \tan\left(\frac{\sqrt{3}}{1}\right) = \frac{2\pi}{3}$
  3. $|z_3| = \sqrt{(-8)^2 + (-8)^2} = 8\sqrt{2}$, $\arg(z_3) = -\pi + \tan\left(\frac{8}{8}\right) = -\frac{3\pi}{4}$

modulus and argument 2

Show that $|zw| = |z|\times|w|$

solution - press button to display

Let $z = a + bi$ and $w = u+vi$, then $zw = (au - bv) + (bu + av)i$

It follows that $$ \begin{align}|zw| &= \sqrt{(au - bv)^2 + (bu + av)^2} \\ &= \sqrt{(au)^2 + (bv)^2 + (bu)^2 + (av)^2} \\ &= \sqrt{(a^2+b^2)(u^2 + v^2)} \\ &= |z|\times |w|\end{align} $$

modulus and argument 3

Show that $ \arg(zw) = \arg(z) + \arg(w) \pm 2n\pi $
solution - press button to display

Let $z = a + bi,\; w = u + vi$ then zw = (au - bv) + (av + bu)i

We must proceed carefully with this question and consider a number of cases that arise

Let us consider the simplest case first, $0\leq \arg(z),\arg(w) \leq \frac{\pi}{2}$

$$\tan(\arg(zw)) = \frac{av + bu}{au - bv}$$

Now let us consider $\tan(\arg(z) + \arg(w))$.

$$ \begin{align} \tan(\arg(z) + \arg(w)) &= \frac{\tan(\arg(z)) + \tan(\arg(w))}{1 - \tan(\arg(z))\tan(\arg(w))} \\ &= \frac{\frac{b}{a} + \frac{v}{u}}{1 - \frac{b}{a}\frac{v}{u}} \\ &= \frac{bu +av}{au - bv} \\ &= \tan(\arg(zw)) \end{align}$$

Care must now be given to ensure the cases of negative argument and large arguments are correctly considered.