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

What is an imaginary number?

What is a complex number?

What is Cartesian Form?

• 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

How do I add and subtract complex numbers in Cartesian Form?

• 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

How do I multiply complex numbers in Cartesian Form?

• 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

How do I deal with higher powers of i?

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

What is a complex conjugate?

How do I divide complex numbers?

• 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