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, 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?