Chapter 5 A brief digression on numbers

The "natural numbers" are what would happen if someone were to spontaneously break out counting: 0,1,2,3,4,5,0,1,2,3,4,5,... (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 xx\in\mathbb{N} (where xx 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,...,-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/qn=p/q. So 11 or 5555 or 0.50.5 or 0.750.75 or 55155\frac{55}{155} or 0.55532342423523535225342343420.5553234242352353522534234342 are all rational numbers. This set can be denoted using the symbol \mathbb{Q}.

Some numbers aren’t rational, however. 2\sqrt{2} or π\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}.