8.2.3 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 always positive
• 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  which has a modulus of 8

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 
• 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
• 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 argument of zero,  is undefined (no angle can be drawn)

The complex number  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, , is written in terms of its real part, , and its imaginary part, 
• If we let and , then it is possible to write a complex number in terms of its modulus, , and its argument, , called the modulus-argument (polar) form, given by...
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• It is usual to give arguments in the range 
  • Negative arguments should be shown clearly, e.g. without simplifying   to either  or 
  • Occasionally you could be asked to give arguments in the range 
• If a complex number is given in the form , 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  and 
  • Therefore  can be written as , now in the correct form and indicating an argument of 
• To convert from modulus-argument (polar) form back to Cartesian form, evaluate the real and imaginary parts
  • E.g.  becomes 

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
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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,  and , in modulus-argument (polar) form we use the rules from above to multiply their moduli and add their arguments
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  • 
• So if and  then the rules above give…
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• Sometimes the new argument, , does not lie in the range (or   if this is being used)
  • An out-of-range argument can be adjusted by either adding or subtracting 
  • E.g. If and   then   
    • This is currently not in the range , but by subtracting  from  to give , 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, , or more
  • …finding powers of a complex number (e.g.  can be written as )
• Whilst not examinable, the rules for multiplication can be proved algebraically by multiplying by , expanding the brackets and using compound angle formulae

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

• To divide two complex numbers,  and in modulus-argument (polar) form, we use the rules from above to divide their moduli and subtract their arguments
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  • 
• So if  and  then the rules above give…
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• As with multiplication, sometimes the new argument, , can lie out of the range  (or the range  if this is being used)
  • You can add or subtract  to bring out-of-range arguments back in range
• Whilst not examinable, the rules for division can be proved algebraically by dividing by , using complex division and compound angle formulae