Exponential Form of Complex Numbers (Cambridge (CIE) A Level Maths): Revision Note

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

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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:

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