Modulus & Argument (CIE A Level Maths: Pure 3)

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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 always positive
  • 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 8 straight i which has a modulus of 8

8-2-3_notes_fig1

How do I find the argument of a complex number?

  • The argument of a complex number is the anti-clockwise angle that it makes when starting at the positive real axis on an Argand diagram
  • Arguments are measured in radians
    • Sometimes these 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
  • 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 or equal than arg space z less than 2 pi
  • The argument of zero, arg space 0 is undefined (no angle can be drawn)

8-2-3_notes_fig2

Worked example

8-2-3_example_fig1-part-1

8-2-3_example_fig1-part-2

Examiner Tip

  • Give non-exact arguments in radians to 3 significant figures.

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Modulus-Argument (Polar) Form

The complex number z equals x plus straight i y is said to be in Cartesian form. There are, however, other ways to write a complex number, such as in modulus-argument (polar) form.

How do I write a complex number in modulus-argument (polar) form?

  • The Cartesian form of a complex number, z equals x plus straight i y, is written in terms of its real part, x, and its imaginary part, y
  • If we let r equals vertical line z vertical line and theta equals arg space z, then it is possible to write a complex number in terms of its modulus, r, and its argument, theta, called the modulus-argument (polar) form, given by...
    • z equals r open parentheses cos space theta plus isin space theta close parentheses
  • It is usual to give arguments in the range negative pi space less than space theta space less or equal than space pi
    • Negative arguments should be shown clearly, e.g. z equals 2 open parentheses cos space open parentheses negative pi over 3 close parentheses plus isin space open parentheses negative pi over 3 close parentheses close parentheseswithout simplifying cos invisible function application left parenthesis negative pi over 3 right parenthesis  to either cos invisible function application open parentheses pi over 3 close parentheses or 1 half
    • Occasionally you could be asked to give arguments in the range 0 space less or equal than space theta space less than space 2 pi
  • If a complex number is given in the form z equals r open parentheses cos space theta minus isin space theta close parentheses, then it is not currently in modulus-argument (polar) form due to the minus sign, but can be converted as follows…
    • By considering transformations of trigonometric functions, we see that negative sin invisible function application theta identical to sin invisible function application left parenthesis negative theta right parenthesis and cos invisible function application theta identical to cos invisible function application left parenthesis negative theta right parenthesis
    • Therefore z equals r open parentheses cos invisible function application theta minus isin invisible function application theta close parentheses can be written as z equals r open parentheses cos invisible function application open parentheses negative theta close parentheses plus isin invisible function application open parentheses negative theta close parentheses close parentheses, now in the correct form and indicating an argument of negative theta
  • To convert from modulus-argument (polar) form back to Cartesian form, evaluate the real and imaginary parts
    • E.g. z equals 2 open parentheses cos invisible function application open parentheses negative pi over 3 close parentheses plus isin invisible function application open parentheses negative pi over 3 close parentheses close parentheses becomes z equals 2 open parentheses 1 half plus straight i open parentheses negative fraction numerator square root of 3 over denominator 2 end fraction close parentheses close parentheses equals 1 minus square root of 3 blank straight i

8-2-3_notes_fig3

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
    • arg space open parentheses z subscript 1 over z subscript 2 close parentheses equals arg space z subscript 1 minus arg space z subscript 2

How do I multiply complex numbers in modulus-argument (polar) form?

  • The main benefit of writing complex numbers in modulus-argument (polar) form is that they multiply and divide very easily (often quicker than when in Cartesian form)
  • To multiply two complex numbers, z subscript 1 and z subscript 2, in modulus-argument (polar) form we use the rules from above to multiply their moduli and add their arguments
    • open vertical bar z subscript 1 z subscript 2 close vertical bar equals open vertical bar z subscript 1 close vertical bar open vertical bar z subscript 2 close vertical bar
    • arg space left parenthesis z subscript 1 z subscript 2 right parenthesis equals arg space z subscript 1 plus arg space z subscript 2
  • So if z subscript 1 equals r subscript 1 left parenthesis cos space theta subscript 1 plus isin space theta subscript 1 right parenthesis and z subscript 2 equals r subscript 2 left parenthesis cos space theta subscript 2 plus isin space theta subscript 2 right parenthesis then the rules above give…
    • z subscript 1 z subscript 2 equals r subscript 1 r subscript 2 open parentheses cos space open parentheses theta subscript 1 plus theta subscript 2 close parentheses plus isin space open parentheses theta subscript 1 plus theta subscript 2 close parentheses close parentheses blank
  • Sometimes the new argument, theta subscript 1 plus theta subscript 2, does not lie in the range negative pi space less than space theta space less or equal than space pi (or  0 space less or equal than space theta space less than space 2 pi  if this is being used)
    • An out-of-range argument can be adjusted by either adding or subtracting 2 straight pi
    • E.g. If theta subscript 1 equals fraction numerator 2 pi over denominator 3 end fraction and theta subscript 2 equals pi over 2  then  theta subscript 1 plus theta subscript 2 space equals space fraction numerator 7 straight pi over denominator 6 end fraction 
      • This is currently not in the range , but by subtracting 2 straight pi from fraction numerator 7 straight pi over denominator 6 end fraction to give negative fraction numerator 5 straight pi over denominator 6 end fraction, a new argument is formed that lies in the correct range and represents the same angle on an Argand diagram
  • The rules of multiplying the moduli and adding the arguments can also be applied when…
    • …multiplying three complex numbers together, z subscript 1 z subscript 2 z subscript 3, or more
    • …finding powers of a complex number (e.g. z squared can be written as z z)
  • Whilst not examinable, the rules for multiplication can be proved algebraically by multiplying z subscript 1 equals r subscript 1 left parenthesis cos space theta subscript 1 plus isin space theta subscript 1 right parenthesis by z subscript 2 equals r subscript 2 left parenthesis cos space theta subscript 2 plus isin space theta subscript 2 right parenthesis, expanding the brackets and using compound angle formulae

How do I divide complex numbers in modulus-argument (polar) form?

  • To divide two complex numbers, z subscript 1 and z subscript 2 in modulus-argument (polar) form, we use the rules from above to divide their moduli and subtract their arguments
    • open vertical bar z subscript 1 over z subscript 2 close vertical bar blank equals fraction numerator open vertical bar z subscript 1 close vertical bar over denominator vertical line z subscript 2 vertical line end fraction
    • arg space open parentheses z subscript 1 over z subscript 2 close parentheses equals arg space z subscript 1 minus arg space z subscript 2
  • So if z subscript 1 equals r subscript 1 left parenthesis cos space theta subscript 1 plus isin space theta subscript 1 right parenthesis and z subscript 2 equals r subscript 2 left parenthesis cos space theta subscript 2 plus isin space theta subscript 2 right parenthesis then the rules above give…
    • z subscript 1 over z subscript 2 equals r subscript 1 over r subscript 2 open parentheses cos space open parentheses theta subscript 1 minus theta subscript 2 close parentheses plus isin space open parentheses theta subscript 1 minus theta subscript 2 close parentheses close parentheses blank
  • As with multiplication, sometimes the new argument, theta subscript 1 minus theta subscript 2, can lie out of the range negative pi space less than space theta space less or equal than space pi (or the range 0 space less or equal than space theta space less than space 2 pi if this is being used)
    • You can add or subtract 2 straight pi to bring out-of-range arguments back in range
  • Whilst not examinable, the rules for division can be proved algebraically by dividing z subscript 1 equals r subscript 1 left parenthesis cos space theta subscript 1 plus isin space theta subscript 1 right parenthesis by z subscript 2 equals r subscript 2 left parenthesis cos space theta subscript 2 plus isin space theta subscript 2 right parenthesis, using complex division and compound angle formulae

Worked example

8-2-3_example_fig2-part-1

8-2-3_example_fig2-part-2

Examiner Tip

  • The rules for multiplying and dividing in modulus-argument (polar) form must be learnt (they are not given in the formula booklet).
  • Remember to add or subtract 2 straight pi to any out-of-range arguments to bring them back in range.
  • If a question does not give a clear range for arguments, then both negative pi space less than space theta space less or equal than space pi  or  0 space less than space theta space less or equal than space 2 pi  would be accepted.

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.