1.5.1 Intro to Complex Numbers

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
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• 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 4i
• 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

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
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How do I multiply complex numbers in Cartesian Form?

• Complex numbers can be multiplied by a constant in the same way as algebraic expressions:
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• Multiplying two complex numbers in Cartesian form is done in the same way as multiplying two linear expressions:
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  • This is a complex number with real part and imaginary part 
  • The most important thing when multiplying complex numbers is that
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• 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
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• We can use this same approach of using i² to deal with much higher powers
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  • Just remember that -1 raised to an even power is 1 and raised to an odd power is -1

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:

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:
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• 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