Edexcel A Level Further Maths: Core Pure:复习笔记1.2.1 Exponential Form

Exponential Form

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
    • is the complex number with modulus 1 and argument θ
  • This works with our current rules of exponents
      • This shows e to the power 0 would still give the answer of 1
      • 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
      • 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

What are some common numbers in exponential form?

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

Exam Tip

  • 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

Worked Example

Two complex numbers are given by  and .

a) Write  in the form


b) Write  in the form .




Operations using Exponential Form

How do I multiply and divide exponential forms of complex numbers?

  • If  and  then
      • You can clearly see that the moduli have been multiplied and the arguments have been added
      • You can clearly see that the moduli have been divided and the arguments have been subtracted

How do I find the complex conjugate of a complex number in exponential form?

  • Simply change the sign of the argument(s)
    • If  then
    • then

Worked Example

Consider the complex number . Calculate  giving your answer in the form