Complex numbers in Mathematica

Mathematica uses the capital letter I to represent the square root of -1. Type

and you'll get the answer
You can use I in expressions: the complex number 2 + 3i is represented as
   2 + 3 I
in Mathematica. The generic complex number (x + y i) is written as
   x + y I
or, equivalently,
   x + I y
Mathematica has a tendency to "alphabetize" things, so it will usually print out (x + y i) in the second form.

Mathematica uses the function Conjugate to take the complex conjugate of a number. Try it:

   a = 2 + 3 I
   2 - 3 I
We know that the complex conjugate of (x + y i) is (x - y i). But Mathematica gives us
   Conjugate[x + y I]
   Conjugate[x + I y]
which, needless to say, is not very informative.

The problem is that we have not specified whether x and y are real or complex numbers, and Mathematica won't make any assumptions about x and y without our help. If x and y are themselves complex numbers, then the conjugate of (x + y i) is not simply (x - i y). To tell Mathematica that x and y are real numbers, use the ComplexExpand command:

   ComplexExpand[ Conjugate[x + y I] ]
   x - I y
Now we get the expected result. Remember to use ComplexExpand when working with complex functions like wavefunctions.

Other parts of the Mathematica tutorial: