Modulus-Argument 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
Worked example
Write in the form where and are exact.