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Rational Numbers
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Regressors
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Third Degree Equations
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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

The set of real numbers is the set of all rational cuts. This set exists by an application of the principle of selection (see Set Selection) to the power set (see Set Powers) of $\Q $. We call an element of the set of real numbers a real number or a real. We call the set of real numbers the set of reals or reals for short.

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. ↩︎
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