Mathematics for Chemistry/Complex Numbers

Introduction to complex numbers


The equation:


Does not factorise


  without complex numbers does not exist. However the number   behaves exactly like any other number in algebra without any anomalies, allowing us to solve this problem.

The solutions are  .

  is an imaginary number.   is a complex number.

Two complex numbers   are added by    .

Subtraction is obvious:  .


Division can be worked out as an exercise. It requires  as a common denominator. This is  , (difference of two squares), and is  .

This means


In practice complex numbers allow one to simplify the mathematics of magnetism and angular momentum as well as completing the number system.

There is an apparent one to one correspondence between the Cartesian   plane and the complex numbers,  . This is called an Argand diagram. The correspondence is illusory however, because say for example you raise the square root of   to a series of ascending powers. Rather than getting larger it goes round and round in circles around the origin. This is not a property of ordinary numbers and is one of the fundamental features of behaviour in the complex plane.

Plot on the same Argand diagram  





(Answers -2 plus or minus 5i, 3/2 plus or minus 2i, i(-1 plus or minus root 2)

2 important equations to be familiar with, Euler's equation:


and de Moivre's theorem:


Euler's equation is obvious from looking at the Maclaurin expansion of  .

To find the square root of   we use de Moivre's theorem.


so de Moivre's theorem gives  


Check this by squaring up to give  .

The other root comes from:


de Moivre's theorem can be used to find the three cube roots of unity by


where   can be  .

Put  ,   and  



This is the difference of two squares so


Similarly any collection of  th roots of 1 can be obtained this way.

Another example is to get the expressions for   and   without expanding  .


Remember Pascal's Triangle

                      1   2   1
                    1   3   3   1
                  1   4   6   4   1
                1   5   10  10  5   1
              1   6   15  20  15  6   1



Separating the real and imaginary parts gives the two expressions. This is a lot easier than


Use the same procedure to get

  and  .