CIE A Level Maths: Pure 3复习笔记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  z  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 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 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, arg 0  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, r, and its argument, Φ, 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
  • 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,  andare 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, andare 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?

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