A LevelFurther MathsEdexcelCore PureRevision NotesComplex NumbersComplex Numbers & Argand DiagramsModulus & Argument

Modulus & Argument (Edexcel A Level Further Maths): Revision Note

Exam code: 9FM0

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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 plus y straight i by the point with cartesian coordinate left parenthesis x comma space y right parenthesis

  • 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 z1 and 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

a) 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

b) 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

Examiner Tips and Tricks

  • 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 open vertical bar z close vertical bar

  • If z equals x plus straight i y, then we can use Pythagoras to show…

    • open vertical bar z close vertical bar equals square root of x squared plus y squared end root

  • A modulus is always positive

  • the modulus is related to the complex conjugate by…

    • z z to the power of asterisk times equals z to the power of asterisk times z equals open vertical bar z close vertical bar squared

    • This is because z z to the power of asterisk times equals open parentheses x plus straight i y close parentheses open parentheses x minus straight i y close parentheses equals x squared plus y squared

  • In general, open vertical bar z subscript 1 plus z subscript 2 close vertical bar not equal to open vertical bar z subscript 1 close vertical bar plus vertical line z subscript 2 vertical line

    • e.g. both z subscript 1 equals 3 plus 4 straight i and z subscript 2 equals negative 3 plus 4 straight i have a modulus of 5, but z subscript 1 plus z subscript 2 simplifies to 8 straight 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 straight pi

  • The argument of z is written arg space 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 negative pi space less than space arg space z space less or equal than space pi

    • 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 space less than space arg space z space less or equal than space 2 pi

  • The argument of zero, arg space 0 is undefined (no angle can be drawn)

8-2-3_notes_fig2

Examiner Tips and Tricks

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

Worked Example

a) Find the modulus and argument of z equals 2 plus 3 straight i

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

b) Find the modulus and argument of w equals negative 1 minus square root of 3 straight i blank

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

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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