# 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 $$\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R}$$.