R - the Reals and Completeness

We note that $\mathbb{R}$ really is just an extension of $\mathbb{Q}$, filling in the irrational holes like $\sqrt{2},\sqrt{3},...$ This definition of $\mathbb{R}$ really was chosen for this property of completeness, so even though the justification as to why this property is important will have to wait, you'll see why as we use this axiom a ton. Just know that by the 1870s, we had a pretty rigorous way to construct $\mathbb{R}$ from $\mathbb{Q}$, which we'll get to see in 8.6.

An Initial Definition of $\mathbb{R}$

First $\mathbb{R}$ contains $\mathbb{Q}$ with the same operations of addition, multiplication:

• All elements $x\in \mathbb{R}$ has their additive inverse and if $x\ne 0$ then the multiplicative inverse $x^{-1}$ exists too.
• $\mathbb{R}$ is a Field, giving the properties of:
  • Commutativity
  • Associativity
  • Distributive Property
• $\mathbb{R}$ also gets the Order from $\mathbb{Q}$ to all of $\mathbb{R}$. So things like "If $a<b$ and $c>0$ implies $ac<bc$" are silently derived from this ordering.
• Thus, $\mathbb{R}$ is an Ordered Field, containing $\mathbb{Q}$ as a subfield

This brings us to the idea of completeness. We need to say that $\mathbb{R}$ doesn't have "any gaps" where the irrationals should go:

See Existence of the Reals for a proof/construction of these.

References

1. [[Abbott Real Analysis.pdf#page=27]]