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IB DP Maths: AA HL复习笔记1.8.1 Intro to Complex Numbers

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月11日下午3:48

There are a number of different forms that complex numbers can be written in
The form z = a + bi is known as Cartesian Form
- This is the first form given in the formula booklet
In general, for z = a + bi
- Re(z) = a
- Im(z) = b
A complex number can be easily represented geometrically when it is in Cartesian Form
Your GDC may call this rectangular form
- When your GDC is set in rectangular settings it will give answers in Cartesian Form
- If your GDC is not set in a complex mode it will not give any output in complex number form
- Make sure you can find the settings for using complex numbers in Cartesian Form and practice inputting problems
Cartesian form is the easiest form for adding and subtracting complex numbers

Remember that complex numbers have both a real part and an imaginary part
- 1 is purely real (its imaginary part is zero)
- i is purely imaginary (its real part is zero)
- 1 + i is a complex number (both the real and imaginary parts are equal to 1)

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1-8-1-ib-hl-aa-cartesian-form-we-b

Adding and subtracting complex numbers should be done when they are in Cartesian form
When adding and subtracting complex numbers, simplify the real and imaginary parts separately

Complex numbers can be multiplied by a constant in the same way as algebraic expressions:
Your GDC will be able to multiply complex numbers in Cartesian form
- Practise doing this and use it to check your answers
It is easy to see that multiplying more than two complex numbers together in Cartesian form becomes a lengthy process prone to errors
- It is easier to multiply complex numbers when they are in different forms and usually it makes sense to convert them from Cartesian form to either Polar form or Euler’s form first

When revising for your exams, practice using your GDC to check any calculations you do with complex numbers by hand
- This will speed up using your GDC in rectangular form whilst also giving you lots of practice of carrying out calculations by hand

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1-8-1-ib-hl-aa-adding-subtracting-mulitplying-we-b

When dividing complex numbers, the complex conjugate is used to change the denominator to a real number.

To divide two complex numbers:
- STEP 1: Express the calculation in the form of a fraction
- This ensures we are multiplying by 1; so not affecting the overall value
- STEP 3: Multiply out and simplify your answer
  - This should have a real number as the denominator
- STEP 4: Write your answer in Cartesian form as two terms, simplifying each term if needed
  - OR convert into the required form if needed
Your GDC will be able to divide two complex numbers in Cartesian form
- Practise doing this and use it to check your answers if you can