CIE A Level Maths: Pure 3复习笔记8.2.1 Argand Diagrams - Basics

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  by the point with cartesian coordinate
  • 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



  • You may be asked to show roots of an equation in an Argan 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


Worked Example


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.