At this point you should begin to have the feeling that this process of
generating supersets of the numbers we already have figured out will
never end. You would be right, and some of the extensions (ones we will
not cover here) are actually very useful in more advanced physics.
However, we have a finite amount of time to *review* numbers here,
and complex numbers are the most we will need in *this* course (or
even ``most'' undergraduate physics courses even at a somewhat more
advanced level). They are important enough that we'll spend a whole
section discussing them below; for the moment we'll just define them.

We start with the unit imaginary number^{11} ,
.
You *might* be familiar with the *naive* definition of
as
the square root of
:

(10) |

This definition is common but slightly unfortunate. If we adopt it, we have to be careful

(11) |

Oops.

A better definition for that it is just the algebraic number such that:

(12) |

and to leave the square root bit out. Thus we have the following cycle:

(13) |

where we can use these rules to do the following sort of simplification:

(14) |

but where we never actually write .

We can make all the purely imaginary numbers by simply scaling
with
a real number. For example,
is a purely imaginary number of
magnitude
.
is a purely imaginary number of magnitude
. All the purely imaginary numbers therefore form an *imaginary line* that is basically the real line, times
.

With this definition, we can define an arbitrary complex number as the sum of an arbitrary real number plus an arbitrary imaginary number:

(15) |

where and are both real numbers. It can be shown that the roots of any polynomial function can always be written as complex numbers, making complex numbers of great importance in physics. However, their

Complex numbers (like real numbers) form a *division
algebra*^{12} - that is, they are closed under
addition, subtraction, multiplication, *and division*. Division
algebras permit the factorization of expressions, something that is
obviously very important if you wish to algebraically solve for
quantities.

Hmmmm, seems like we ought to look at this ``algebra'' thing. Just what
*is* an algebra? How does algebra work?