# 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 #### 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) #### Worked Example  #### 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  #### 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 divide complex numbers in modulus-argument (polar) form? #### Worked Example   