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 ¹¹ 1 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 \sayplug the gaps between the rationals.

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