Edexcel A Level Further Maths: Core Pure:复习笔记1.1.4 Modulus-Argument Form

Category: A-level课程, 国际课程, 教材笔记, 福利干货 Date: 2022年10月17日上午11:15

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
- 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

Worked Example

Write in the form whereand are exact.

al-fm-1-1-4-mod-and-arg-form-we-solution-1

Operations using Modulus-Argument Form

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