Real Numbers

Why

We want a set which corresponds to our notion of points on a line.¹

Rational cuts

We call a subset $R$ of $\Q $ a rational cut if (a) $R \neq \varnothing$, (b) $R \neq \Q $, (c) for all $q \in R$, $r \leq q \implies r \in R$, and (d) $R$ has no greatest element. Briefly, the intuition is that the point is the set of all rationals to less than (or, potentially, equal to) some particular rational number.²

Definition

Notation

We follow tradition and denote the set of real numbers by $\R $, likely a mnemonic for “real.”

Other terminology

Some authors call a real number a quantity or a continuous quantity. The real numbers, then, are said to be continuous. When contrasting (using this terminology) a finite set with the real numbers, one refers to the finite set as discrete.³

1. Future editions will modify and expand this justification. ↩︎
2. This brief intuition will be expanded upon in future sheets. ↩︎
3. Future editions may move this discussion later, to the discussion of the cardinality of the reals. ↩︎