4.4 Axioms for the reals and first approximations

Why define the reals before constructing them?

In earlier chapters, the pattern was:

1. decide what structure we want,
2. build a model that has that structure,
3. check the construction carefully.

Do the same for the real numbers. Before giving a formal construction, they first say what the reals are supposed to be from a structural point of view.

That is good mathematical practice. If you do not know the target properties first, then even a beautiful construction can feel unmotivated.

The formal target

This definition compresses the whole chapter into one sentence:

the reals are an ordered field with no missing least upper bounds or greatest lower bounds.

This theorem explains why the course can move between different constructions of $R$. Dedekind cuts and Cauchy sequences look different internally, but once both constructions satisfy the complete ordered field axioms, they represent the same real number system up to the unique structure that matters here: addition, multiplication, and order.

Why decimal expansions are not the full story

A first instinct is to say:

"A real number is just an infinite decimal expansion."

That instinct is useful, but it is not yet the cleanest definition.

There are at least two reasons.

• The same real number can have more than one decimal expansion, as in 0.9999\ldots = 1.
• Decimal expansions force a specific base, such as base 10, even though the idea of the real numbers should not depend on that arbitrary choice.

So we should not reject decimal intuition. Instead, use it as a guide toward a more structural construction.

The approximation idea

Suppose a real number is written informally as

$$r=10.234890234809234\ldots$$

Then we can approximate r from below and above by rational numbers:

$$10<r<11,$$ $$10.2<r<10.3,$$ $$10.23<r<10.24,$$

and so on.

This is the key insight of section 4.7: even if we do not yet have a final construction, a real number can already be understood through the rational approximations that trap it more and more tightly.

Dividing the rationals into two camps

From this approximation viewpoint, the real number r determines two sets of rationals:

$$A=\{q\in Q\mid q<r\}, \qquad B=\{q\in Q\mid q>r\}.$$

The important idea is not the notation itself. The important idea is that a real number can be read as a boundary that separates the rationals into a left camp and a right camp.

That is exactly the motivation behind the Dedekind-cut construction in the next part of the construction. This stage is still motivational rather than fully formal, but it already tells you what kind of object a real number should be: something that organizes the rationals by comparison.

What section 4.7 does and does not do

This distinction matters. If you treat section 4.7 as already giving the final definition, you miss the role of the next step. The argument is building motivation first, then turning that motivation into a rigorous object.

Quick checks

Exercises

Related notes

Read this after 4.3 Completeness and gaps in Q. Then continue with 4.5 Dedekind cuts and the embedding of Q, where the informal left/right split of $Q$ becomes the first fully rigorous construction of the real numbers in the public Notes sequence.