Modulus & Argument (DP IB Maths: AI HL)

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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 Tip

  • 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

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Amber

Author: Amber

Expertise: Maths

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.