What is a Real Number? A 5 Minute Introduction

[Greg Bernhardt]

Definition/Summary
The real numbers are the most commonly encountered number system, familiar to the layman via the number line, and as the number system lying behind decimal notation.
Because real numbers have much nice arithmetic and geometric properties, they feature prominently in many fields of mathematics.
Extended explanation
This is a preliminary article. More information will be added in the future.
A Brief History
The real numbers originate from a need to quantify the geometric notion of ‘length’. It was known in ancient times that rational numbers are not adequate, culminating with Eudoxus’s ‘method of exhaustion.
Formal Definition
With the development of modern calculus, it became increasingly clear that a rigorous definition of real numbers was required. Cantor provided the first definition, by identifying a real number with the set of Cauchy sequences of rational that ought to...

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[fresh_42]

There are two main ways to introduce real numbers. Cauchy sequences are one possibility, Dedekind cuts are another. It is surprising how such a for us common thing like real numbers need some elaborated mathematics in order to properly define them. Even the approach by Cauchy sequences needs the concept of equivalence classes.