Edexcel A Level Further Maths: Core Pure:復習筆記1.2.1 Exponential Form

Category: A-level課程, 國際課程, 教材筆記, 福利干貨 Date: 2022年10月17日下午4:47

Exponential Form

You now know how to do lots of operations with complex numbers: add, subtract, multiply, divide, raise to a power and even square root. The last operation to learn is raising the number e to the power of an imaginary number.

How do we calculate e to the power of an imaginary number?

Given an imaginary number (iθ) we can?define exponentiation?as
- is the complex number with modulus 1 and argument θ
This works with our current rules of exponents
- - This shows e to the power 0 would still give the answer of 1
- - This is because when you?multiply complex numbers?you can?add?the?arguments
  - This shows that when you multiply two powers you can still add the indices
- - This is because when you?divide complex numbers?you can?subtract?the?arguments
  - This shows that when you divide two powers you can still subtract the indices

What is the exponential form of a complex number?

Any complex number ?can be written in?polar form
- r?is the modulus
- θ?is the argument
Using the definition of we can now also write ?in?exponential form

Why do I need to use the exponential form of a complex number?

It's just a?shorter?and?quicker?way of expressing complex numbers
It makes a link between the?exponential function?and?trigonometric functions
It makes it easier to remember what happens with the moduli and arguments when multiplying and dividing

What are some common numbers in exponential form?

As ?and ?you can write:
Using the same idea you can write:
- where k?is any integer
As and ?you can write:
- Or more commonly written as
As ?and ?you can write:

Exam Tip

The powers can be long and contain fractions so make sure you write the expression clearly.
You don’t want to lose marks because the examiner can’t read your answer

Worked Example

Two complex numbers are given by ?and .

a) Write

?in the form

1-9-2-ib-aa-hl-forms-of-cn-we-solution-3-a

b) Write

?in?the form

1-9-2-ib-aa-hl-forms-of-cn-we-solution-3-b

Operations using Exponential Form

How do I multiply and divide exponential forms of complex numbers?

If ?and ?then
- - You can clearly see that the?moduli have been multiplied?and the?arguments have been added
- - You can clearly see that the?moduli have been divided?and the?arguments have been subtracted