Field of Real Numbers
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Theorem
The set of real numbers $\R$ forms a totally ordered field under addition and multiplication: $\left({\R, +, \times, \le}\right)$.
Proof
- From Additive Group of Real Numbers, we have that $\left({\R, +}\right)$ forms an abelian group.
- From Multiplicative Group of Real Numbers, we have that $\left({\R^*, \times}\right)$ forms an abelian group.
- Next we have that Real Multiplication Distributes over Addition.
- Finally we have that Real Numbers are Totally Ordered.
Thus all the criteria are fulfilled, and $\left({\R, +, \times, \le}\right)$ is a totally ordered field.
$\blacksquare$
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$: Example $2$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.15$: Example $15$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): $\S 1.1$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19$
