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

Cartesian form

What is an imaginary number?

• Up until now, equations like have “no real solutions”

  • The solutions are

    • but you cannot square root a negative number

• To extend this idea, mathematicians have defined one of the square roots of negative one as

  • This is called an imaginary number

    • 

    • 

• The square roots of other negative numbers can be found by rewriting them as a multiple of  

  • using

    • e.g.

What is a complex number?

• Complex numbers have both a real part and an imaginary part

  • For example: 

    • The real part is 3

    • The imaginary part is 4

    • The imaginary part does not include the ''

• Complex numbers are often denoted by 

  • The real part of is

  • The imaginary part of is

• The set of all complex numbers is given the symbol 

• Complex numbers can have one part

  • e.g. is purely real

    • and real numbers are a subset of complex numbers

  • e.g. is purely imaginary

    • and imaginary numbers are a subset of complex numbers

• Two complex numbers are equal if, and only if, both the real and imaginary parts are identical.

  • For example,  and  are not equal

What is Cartesian form?

• The form is known as Cartesian form

  • 

• In general, for

  • 

  •