Modulus & Argument (DP IB Maths: AA HL): Revision Note

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 never negative

What features should I know about the modulus of a complex number?

• 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 8i which has a modulus of 8

How do I find the argument of a complex number?

• The argument of a complex number is the angle that it makes on an Argand diagram
  • The angle must be taken from the positive real axis
  • The angle must be in a counter-clockwise direction
• Arguments are measured in radians
  • They 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

What features should I know about the argument of a complex number?

• Arguments are usually given in the range    
  • Negative arguments are for complex numbers in the third and fourth quadrants
  • Occasionally you could be asked to give arguments in the range 
    • The question will make it clear which range to use
• The argument of zero,  is undefined (no angle can be drawn)

What are the rules for moduli and arguments under multiplication and division?

• When two complex numbers,  and , are multiplied to give , their moduli are also multiplied
  • 
• When two complex numbers, and , are divided to give , their moduli are also divided
  • 
• When two complex numbers, and , are multiplied to give , their arguments are added
  • 
• When two complex numbers, and , are divided to give , their arguments are subtracted