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A LevelMaths: Pure 3CIERevision Notes8. Complex Numbers8.2 Argand Diagrams8.2.1 Argand Diagrams - Basics

Argand Diagrams - Basics (CIE A Level Maths: Pure 3)

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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

8-2-1-argand-diagrams---basics-example-solution

Examiner 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.

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    Amber

    Author: Amber

    Expertise: Maths

    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.