A calculus of the absurd

Chapter 5 A brief digression on numbers

The "natural numbers" are what would happen if someone were to spontaneously break out counting: \(0, 1, 2, 3, 4, 5, ...\) (and so on). These are called the "natural numbers" and denoted as \(\mathbb {N}\). To say a number is a natural number, we can say that it is in the set of natural numbers, and can write this as \(x \in \mathbb {N}\) (where \(x\) is our number).

From the natural numbers, a logical next step is the set of "integers". These are just the numbers \(..., -3, -2, -1, 0, 1, 2, 3, ...\) (going infinitely far in either direction).

After this, there are the "rational numbers". A "rational" number is a number which can be written as \(n = p / q\). So \(1\) or \(55\) or \(0.5\) or \(0.75\) or \(\frac {55}{155}\) or \(0.5553234242352353522534234342\) are all rational numbers. This set can be denoted using the symbol \(\mathbb {Q}\).

Some numbers aren’t rational, however. \(\sqrt {2}\) or \(\pi \) are "irrational" numbers 4242 There’s a proof of this in the proof section.. The "real" numbers, \(\mathbb {R}\), include the "irrational" numbers (as well as some other numbers). If we think of the rational numbers on a number line, then the real numbers are (informally) an attempt to “plug” the gaps between the rationals.

Note that \(\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R}\).