Modulus & Argument (DP IB Analysis & Approaches (AA)): Revision Note

Modulus & argument

What is the modulus of a complex number?

• The modulus of a complex number (written or ) is its distance from the origin when plotted on an Argand diagram

  • If , then by Pythagoras

    • 

  • A modulus is never negative

• In general, adding two complex numbers does not mean you can add their moduli 

  • 

  • e.g. both  and  have a modulus of 5

    • but  has a modulus of 8

How does the modulus relate to the complex conjugate?

• The modulus is the result of multiplying a complex number by its conjugate

  • 

  • e.g. 

What is the argument of a complex number?

• The argument of a complex number (written  or ) is the angle that it makes to the positive real axis on an Argand diagram

  • measured

    • counter-clockwise

    • in radians

  • where

• Arguments are

  • positive and acute in the first quadrant

  • positive and obtuse in the second quadrant

  • negative and obtuse in the third quadrant

  • negative and acute in the fourth quadrant

• Arguments can be calculated using right-angled trigonometry

  • A sketch is needed to decide the quadrant

    • positive / negative, or acute / obtuse

  • then the tan ratio can be used

• The argument of zero, , is undefined

  • does not exist

    • as no angle can be drawn

What happens to the modulus and argument under multiplication or division?

• When two complex numbers,  and , are multiplied,

  • their moduli are also multiplied

    • 

  • but their arguments are added

    • 

• When two complex numbers, and , are divided,

  • their moduli are also divided

    • 

  • but their arguments are subtracted

    •