1.5.2 Modulus & Argument

The modulus of a complex number is its distance from the origin when plotted on an Argand diagram
The modulus of $z$ is written $|z|$
If $z = x + i y$ , then we can use Pythagoras to show…
- $|z| = \sqrt{x^{2} + y^{2}}$
A modulus is never negative

the modulus is related to the complex conjugate by…
- $z z^{*} = z^{*} z = {|z|}^{2}$
- This is because $z z^{*} = (x + i y) (x - i y) = x^{2} + y^{2}$
In general, $|z_{1} + z_{2}| \neq |z_{1}| + | z_{2} |$
- e.g. both $z_{1} = 3 + 4 i$ and $z_{2} = - 3 + 4 i$ have a modulus of 5, but $z_{1} + z_{2}$ simplifies to 8i which has a modulus of 8

The argument of a complex number is the angle that it makes on an Argand diagram
- The angle must be taken from the positive real axis
- The angle must be in a counter-clockwise direction
Arguments are measured in radians
- They can be given exact in terms of $π$
The argument of $z$ is written $\arg z$
Arguments can be calculated using right-angled trigonometry
- This involves using the tan ratio plus a sketch to decide whether it is positive/negative and acute/obtuse

Arguments are usually given in the range $- π < \arg z \leq π$
- Negative arguments are for complex numbers in the third and fourth quadrants
- Occasionally you could be asked to give arguments in the range $0 < \arg z \leq 2 π$
  - The question will make it clear which range to use
The argument of zero, $\arg 0$ is undefined (no angle can be drawn)

When two complex numbers, $z_{1}$ and $z_{2}$ , are multiplied to give $z_{1} z_{2}$ , their moduli are also multiplied
- $|z_{1} z_{2}| = |z_{1}| | z_{2} |$
When two complex numbers, $z_{1}$ and $z_{2}$ , are divided to give $\frac{z_{1}}{z_{2}}$ , their moduli are also divided
- $|\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}$
When two complex numbers, $z_{1}$ and $z_{2}$ , are multiplied to give $z_{1} z_{2}$ , their arguments are added
- $\arg (z_{1} z_{2}) = \arg z_{1} + \arg z_{2}$
When two complex numbers, $z_{1}$ and $z_{2}$ , are divided to give $\frac{z_{1}}{z_{2}}$ , their arguments are subtracted

Always draw a quick sketch to help you see what quadrant the complex number lies in when working out an argument
Look for the range of values within which you should give your argument
- If it is $- π < \arg z \leq π$ then you may need to measure it in the negative direction
- If it is $0 < \arg z \leq 2 π$ then you will always measure in the positive direction (counter - clockwise)

Find the modulus and argument of

z = 2 + 3 i

1-8-2-ib-hl-aa-mod-and-arg-we-a

Find the modulus and argument of

w = - 1 - \sqrt{3} i

1-8-2-ib-hl-aa-mod-and-arg-we-b

DP IB Maths: AI HL

1.5.2 Modulus & Argument