Introduction to Complex Numbers (DP IB Applications & Interpretation (AI)): 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

  • 

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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 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