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

Modulus & Argument

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…

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• 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…

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  • 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

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• When two complex numbers, and , are divided to give , their moduli are also divided

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• When two complex numbers, and , are multiplied to give , their arguments are added

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• When two complex numbers, and , are divided to give , their arguments are subtracted