Exponential Form of Complex Numbers (Cambridge International A-Level Mathematics)

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.

What is e to the power of an imaginary number?

• Given an imaginary number (iθ) we can define exponentiation as

  • $e^{iθ}=\mathrm{cos}θ+\mathrm{isin}θ$
  • $e^{iθ}$ is the complex number with modulus 1 and argument θ

• This works with our current rules of exponents

  • $\mathrm{e}^{0 = \mathrm{e}^{0} \mathrm{i} } = \cos 0 + \text{isin} 0 = 1$

    • This shows e to the power 0 would still give the answer of 1

  • $e^{iθ_{1}}\times e^{iθ_{2}}=e^{i(θ_{1}+θ_{2})}$

    • 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

  • $\frac{e^{\mathrm{iθ}_{1}}}{e^{\mathrm{iθ}_{2}}}=e^{i(θ_{1}-θ_{2})}$

    • 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 $z=a+bi$ can be written in polar form$z = r ( \cos \\theta + \text{isin} \\theta )$

  • r is the modulus
  • θ* *is the argument

• Using the definition of $e^{iθ}$ we can now also write $z$ in exponential form

  • $z=re^{iθ}$

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 $z_{1}=r_{1}e^{iθ_{1}}$ and $z_{2}=r_{2}e^{iθ_{2}}$ then

  • $z_{1} \times z_{2} = r_{1} r_{2} \mathrm{e}^{ \mathrm{i} ( \theta_{1} + \theta_{2} )}$

    • You can clearly see that the moduli have been multiplied and the arguments have been added

  • z _ 1 / z _ 2 = r _ 1 / r _ 2 e ^ i ( theta _ 1 - theta _ 2 )

    • 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 $\mathrm{cos}(2π)=1$ and $\mathrm{sin}(2π)=0$ you can write:

  • $1=e^{2πi}$

• Using the same idea you can write:

  • $1=e^{0}=e^{2πi}=e^{4πi}=e^{6πi}=e^{2kπi}$ where k is any integer

• As $\cos ( \pi ) = - 1$ and $\sin ( \pi ) = 0$ you can write:

  • $e^{πi}=-1$
  • Or more commonly written as $e^{\mathrm{iπ}}+1=0$

• As $\cos ( \frac{\pi }{2} ) = 0$ and $\sin ( \frac{\pi }{2} ) = 1$ you can write:

  • $i=e^{\frac{π}{2}i}$

Worked Example

Exam Hint

• The powers can be long and contain fractions so make sure you write the expression clearly.
• You don’t want to lose marks because the examiner can’t read your answer