Modulus & Argument (Cambridge (CIE) A Level Maths): 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…

  • 

• A modulus is always positive

• the modulus is related to the complex conjugate by…

  • 

  • 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)

Modulus-argument (polar) form

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...

  • 

• 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

  • 

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

  • 

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

  • 

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

  • 

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

  • 

  • 

• So if and  then the rules above give…

  • 

• 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

  • 

  • 

• So if  and  then the rules above give…

  • 

• 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