Exponential (Euler's) Form (DP IB Analysis & Approaches (AA)): Revision Note

Exponential (Euler's) form

What is exponential (Euler) form?

• The exponential (Euler) form of a complex number is

  • 

  • where

    • 

    • 

• e.g. has a modulus of 2 and an argument of

  • so

• To convert back to Cartesian form

  • go via the modulus-argument (polar) form

  • e.g. is

How do I multiply and divide complex numbers in exponential (Euler) form?

• If and then 

  • 

    • Multiply the moduli and add the arguments

  • 

    • Divide the moduli and subtract the arguments

• These rules makes multiplying and dividing easier in exponential (Euler) form than in Cartesian form!

• Powers of complex numbers are also easier in exponential (Euler) form

  • You can use index laws

    • e.g. ,    and  

What numbers do I need to know in exponential (Euler) form?

• You should know the following numbers in exponential (Euler) form

• 

  • as  and

  • similarly

    • for all even multiples of

• 

  • as  and  

  • can be rearranged to 

    • known as Euler's identity

    • often considered the most elegant relationship in Mathematics!

• 

  • as and