Modulus & Argument (DP IB Applications & Interpretation (AI)): Revision Note

Did this video help you?

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 z is written open vertical bar z close vertical bar

  • If z equals x plus straight i y, then we can use Pythagoras to show…

    • open vertical bar z close vertical bar equals square root of x squared plus y squared end root

  • 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…

    • z z to the power of asterisk times equals z to the power of asterisk times z equals open vertical bar z close vertical bar squared

    • This is because z z to the power of asterisk times equals open parentheses x plus straight i y close parentheses open parentheses x minus straight i y close parentheses equals x squared plus y squared

  • In general, open vertical bar z subscript 1 plus z subscript 2 close vertical bar not equal to open vertical bar z subscript 1 close vertical bar plus vertical line z subscript 2 vertical line

    • e.g. both z subscript 1 equals 3 plus 4 straight i and z subscript 2 equals negative 3 plus 4 straight i have a modulus of 5, but z subscript 1 plus z subscript 2 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 straight pi

  • The argument of z is written arg space z 

  • 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 pi space less than space arg space z space less or equal than space pi   

    • Negative arguments are for complex numbers in the third and fourth quadrants

    • Occasionally you could be asked to give arguments in the range 0 space less than space arg space z space less or equal than space 2 pi

      • The question will make it clear which range to use

  • The argument of zero, arg space 0 is undefined (no angle can be drawn)

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

  • When two complex numbers, z subscript 1 and z subscript 2, are multiplied to give z subscript 1 z subscript 2, their moduli are also multiplied

    • open vertical bar z subscript 1 z subscript 2 close vertical bar equals open vertical bar z subscript 1 close vertical bar vertical line z subscript 2 vertical line

  • When two complex numbers, z subscript 1 and z subscript 2, are divided to give z subscript 1 over z subscript 2, their moduli are also divided

    • open vertical bar z subscript 1 over z subscript 2 close vertical bar equals fraction numerator open vertical bar z subscript 1 close vertical bar over denominator open vertical bar z subscript 2 close vertical bar end fraction

  • When two complex numbers, z subscript 1 and z subscript 2, are multiplied to give z subscript 1 z subscript 2, their arguments are added

    • arg space open parentheses z subscript 1 z subscript 2 close parentheses equals arg space z subscript 1 plus arg space z subscript 2

  • When two complex numbers, z subscript 1and z subscript 2, are divided to give z subscript 1 over z subscript 2, their arguments are subtracted

Examiner Tips and Tricks

  • Always draw a quick sketch to help you see what quadrant the complex number lies in when working out an argument

  • Look for the range of values within which you should give your argument

    • If it is  negative pi space less than space arg space z space less or equal than space pi then you may need to measure it in the negative direction

    • If it is  0 space less than space arg space z space less or equal than space 2 pi  then you will always measure in the positive direction (counter - clockwise)

Worked Example

a) Find the modulus and argument of z equals 2 plus 3 straight i

1-8-2-ib-hl-aa-mod-and-arg-we-a

b) Find the modulus and argument of w equals negative 1 minus square root of 3 straight i blank

1-8-2-ib-hl-aa-mod-and-arg-we-b
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Download notes on Modulus & Argument