Modulus & Argument (Edexcel A Level Further Maths: Core Pure)

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Maths

Argand Diagrams - Basics

What is an Argand diagram?

An Argand diagram is a geometrical way to represent complex numbers as either a point or a vector in two-dimensional space
- We can represent the complex number $x + y i$ by the point with cartesian coordinate $(x, y)$
The real component is represented by points on the x-axis, called the real axis, Re
The imaginary component is represented by points on the y-axis, called the imaginary axis, Im

8-2-1-argand-diagrams---basics-diagram-1

8-2-1-argand-diagrams---basics-diagram-2

You may be asked to show roots of an equation in an Argand diagram
- First solve the equation
- Draw a quick sketch, only adding essential information to the axes
- Plot the points and label clearly

How can I use an Argand diagram to visualise |z₁+ z₂| and |z₁- z₂|?

Plot two complex numbers z₁and z₂
Draw a line from the origin to each complex number
Form a parallelogram using the two lines as two adjacent sides
The modulus of their sum |z₁+ z₂| will be the length of the diagonal of the parallelogram starting at the origin
The modulus of their difference |z₁- z₂| will be the length of the diagonal between the two complex numbers

8-2-1-argand-diagram

Worked example

Plot the complex numbers z₁= 2 + 2i and z₂ = 3 – 4i as points on an Argand diagram.

1-8-3-ib-hl-aa-argand-diagrams-we-a

Write down the complex numbers represented by the points A and B on the Argand diagram below.

1-8-3-ib-hl-aa-argand-diagrams-we-b

Exam Tip

When setting up an Argand diagram you do not need to draw a fully scaled axes, you only need the essential information for the points you want to show, this will save a lot of time.

Modulus & Argument

How do I find the modulus of a complex number?

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 always positive
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 $8 i$ which has a modulus of 8

8-2-3_notes_fig1

How do I find the argument of a complex number?

The argument of a complex number is the anti-clockwise angle that it makes when starting at the positive real axis on an Argand diagram
Arguments are measured in radians
- Sometimes these 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 $- π < \arg z \leq π$
- This is called the principal argument
- 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 argument of zero, $\arg 0$ is undefined (no angle can be drawn)

8-2-3_notes_fig2

Exam Tip

Always draw a sketch to see which quadrant the complex number is in

Worked example

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

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