## Complex numbers in Mathematica

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

```   Sqrt[-1]
```
```   I
```
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
Conjugate[a]
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: