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

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 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

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  is written 
• If , then we can use Pythagoras to show…
  • 
• A modulus is always positive
• the modulus is related to the complex conjugate by…
  • 
  • This is because 
• In general, 
  • e.g. both  and  have a modulus of 5, but  simplifies to  which has a modulus of 8

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 
• The argument of is written  
• 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 
  • 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 
• The argument of zero,  is undefined (no angle can be drawn)