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

Exponential (Euler's) Form

How do we write a complex number in Euler's (exponential) form?

• A complex number can be written in Euler's form as  

  • This relates to the modulus-argument (polar) form as 

  • This shows a clear link between exponential functions and trigonometric functions

  • This is given in the formula booklet under 'Modulus-argument (polar) form and exponential (Euler) form'

• The argument is normally given in the range 0 ≤ θ < 2π

  • However in exponential form other arguments can be used and the same convention of adding or subtracting 2π can be applied

How do we multiply and divide complex numbers in Euler's form?

• Euler's form allows for quick and easy multiplication and division of complex numbers

• If and then 

  • 

    • Multiply the moduli and add the arguments

  • 

    • Divide the moduli and subtract the arguments

• Using these rules makes multiplying and dividing more than two complex numbers much easier than in Cartesian form

• When a complex number is written in Euler's form it is easy to raise that complex number to a power

  • If ,    and  

What are some common numbers in exponential form?

• As  and  you can write:

  • 

• Using the same idea you can write:

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  • where k is any integer

• As  and  you can write:

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  • Or more commonly written as 

    • This is known as Euler's identity and is considered by some mathematicians as the most beautiful equation

• As and  you can write:

  •