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Intro to Complex Numbers (DP IB Maths: AI HL)
Revision Note
Cartesian Form
What is an imaginary number?
- Up until now, when we have encountered an equation such as we would have stated that there are “no real solutions”
- The solutions are which are not real numbers
- To solve this issue, mathematicians have defined one of the square roots of negative one as ; an imaginary number
- The square roots of other negative numbers can be found by rewriting them as a multiple of
- using
What is a complex number?
- Complex numbers have both a real part and an imaginary part
- For example:
- The real part is 3 and the imaginary part is 4
- Note that the imaginary part does not include the ''
- Complex numbers are often denoted by
- We refer to the real and imaginary parts respectively using and
- Two complex numbers are equal if, and only if, both the real and imaginary parts are identical.
- For example, and are not equal
- The set of all complex numbers is given the symbol
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
- a, b ∈
- 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
Examiner Tip
- 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)
Worked example
a)
Solve the equation
b)
Solve the equation , giving your answers in Cartesian form.
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Complex Addition, Subtraction & Multiplication
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
- Just like you would when collecting like terms in algebra and surds, or dealing with different components in vectors
How do I multiply complex numbers in Cartesian Form?
- Complex numbers can be multiplied by a constant in the same way as algebraic expressions:
- Multiplying two complex numbers in Cartesian form is done in the same way as multiplying two linear expressions:
- This is a complex number with real part and imaginary part
- The most important thing when multiplying complex numbers is that
- 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
- Sometimes when a question describes multiple complex numbers, the notation is used to represent each complex number
How do I deal with higher powers of i?
- Because this can lead to some interesting results for higher powers of i
- We can use this same approach of using i2 to deal with much higher powers
- Just remember that -1 raised to an even power is 1 and raised to an odd power is -1
Examiner Tip
- 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
Worked example
a)
Simplify the expression .
b)
Given two complex numbers and , find .
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Complex Conjugation & Division
When dividing complex numbers, the complex conjugate is used to change the denominator to a real number.
What is a complex conjugate?
- For a given complex number , the complex conjugate of is denoted as , where
- If then
- You will find that:
- is always real because
- For example:
- is always imaginary because
- For example:
- is always real because (as )
- For example:
- is always real because
How do I divide complex numbers?
- To divide two complex numbers:
- STEP 1: Express the calculation in the form of a fraction
- STEP 2: Multiply the top and bottom by the conjugate of the denominator:
- 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
Examiner Tip
- We can speed up the process for finding by using the basic pattern of
- We can apply this to complex numbers:
(using the fact that )- So multiplied by its conjugate would be
Worked example
Find the value of .
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