8.3.1 Exponential Form of Complex Numbers

You now know how to do lots of operations with complex numbers: add, subtract, multiply, divide, raise to a power and even square root. The last operation to learn is raising the number e to the power of an imaginary number.

How do we calculate e to the power of an imaginary number?

• Given an imaginary number (iθ) we can define exponentiation as
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  • is the complex number with modulus 1 and argument θ
• This works with our current rules of exponents
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    • This shows e to the power 0 would still give the answer of 1
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    • This is because when you multiply complex numbers you can add the arguments
    • This shows that when you multiply two powers you can still add the indices
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    • This is because when you divide complex numbers you can subtract the arguments
    • This shows that when you divide two powers you can still subtract the indices

What is the exponential form of a complex number?

• Any complex number  can be written in polar form 
  • r is the modulus
  • θ is the argument
• Using the definition of  we can now also write  in exponential form
  • 

Why do I need to use the exponential form of a complex number?

• It's just a shorter and quicker way of expressing complex numbers
• It makes a link between the exponential function and trigonometric functions
• It makes it easier to remember what happens with the moduli and arguments when multiplying and dividing
• If and  then
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    • You can clearly see that the moduli have been multiplied and the arguments have been added
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    • You can clearly see that the moduli have been divided and the arguments have been subtracted

What are some common numbers in exponential form?

• As  and  you can write:
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• Using the same idea you can write:
  • where k is any integer
• As  and  you can write:
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  • Or more commonly written as 
• As and  you can write:
  •