CIE A Level Maths: Pure 3复习笔记8.2.3 Modulus & Argument

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

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
  • 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)

8-2-3_notes_fig2

Worked Example

8-2-3_example_fig1-part-1

8-2-3_example_fig1-part-2

Exam Tip

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

 

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, 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

8-2-3_notes_fig3

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?

Worked Example

8-2-3_example_fig2-part-1

8-2-3_example_fig2-part-2

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