The Real Numbers

We have four infinite sets of familiar objects, in increasing order of complication:

$\mathbb {N}$ -- the Natural numbers

are defined as the set {0, 1, 2,..., n,...}. Contrast these with the positive integers; the same set without 0.

$\mathbb {Z}$ -- the Integers

are defined as the set {0,±1,±2,...,±n,...}.

$\mathbb {Q}$ -- the Rational numbers

are defined as the set {p/q : p, q $\in$ $\mathbb {Z}$, q$\ne$0}.

$\mathbb {R}$ -- the Reals

are defined in a much more complicated way. In this course you will start to see why this complication is necessary, as you use the distinction between $\mathbb {R}$ and  $\mathbb {Q}$.

Note:We have a natural inclusion $\mathbb {N}$ $\subset$ $\mathbb {Z}$ $\subset$ $\mathbb {Q}$ $\subset$ $\mathbb {R}$, and each inclusion is proper. The only inclusion in any doubt is the last one; recall that $\sqrt{2}$ $\in$ $\mathbb {R}$ $\setminus$ $\mathbb {Q}$ (i.e. it is a real number that is not rational).

One point of this course is to illustrate the difference between $\mathbb {Q}$ and $\mathbb {R}$. It is subtle: for example when computing, it can be ignored, because a computer always works with a rational approximation to any number, and as such can't distinguish between the two sets. We hope to show that the complication of introducing the ``extra'' reals such as $\sqrt{2}$ is worthwhile because it gives simpler results.