4.6 Decimal expansions and irrational numbers

Once Dedekind cuts are available, return to the familiar language of decimal expansions and ask a natural question:

How does a decimal such as 10.4352902543... determine a real number?

The answer is that a decimal expansion produces a nested family of rational approximations, and those approximations can be turned into a cut.

Decimal expansions give lower and upper fences

Consider the decimal

$$x=10.4352902543\ldots$$

Form the sequence of lower bounds

$$S=\{10,\ 10.4,\ 10.43,\ 10.435,\ \ldots\}$$

and the sequence of upper bounds

$$T=\{11,\ 10.5,\ 10.44,\ 10.436,\ \ldots\}.$$

The idea is simple:

• the lower list truncates the decimal and stays below the target,
• the upper list moves one step above the truncation and stays above the target.

So the real number is trapped inside narrower and narrower rational intervals.

Figure. A decimal expansion does not merely list digits. It creates a chain of smaller and smaller rational intervals containing the target number.

Turning a decimal expansion into a cut

Define

$$A=\{q\in Q\mid \exists s\in S\text{ such that }q<s\}$$

and

$$B=\{q\in Q\mid \exists t\in T\text{ such that }q>t\}.$$

Intuitively:

• $A$ contains every rational that is definitely to the left of some lower approximation,
• $B$ contains every rational that is definitely to the right of some upper approximation.

The claim is that (A,B) forms a Dedekind cut, and that this cut is the real number represented by the decimal expansion.

So decimal notation is not being discarded. It is being absorbed into the cut-based model.

Why decimal strings are useful but not the primary definition

The previous note already hinted at the main problem with taking decimal strings as the primary definition of real numbers:

• the same real number can have more than one decimal expansion,
• for example 0.999...=1.

So decimal notation is excellent for intuition and approximation, but it needs a deeper structural interpretation. Dedekind cuts supply that interpretation.

Build the interval chain interactively

Use the builder below to reveal one more decimal digit at a time and watch the lower bound, upper bound, and interval width update together.

Irrational numbers inside $R$

Once the real numbers are constructed, define:

This definition is short, but its meaning is deep. An irrational number is not “mysterious” or “unfinished”. It is a perfectly legitimate real number that is simply not represented by any rational cut.

The real number $\sqrt{2}$

Now consider the classical example

$$A=\{r\in Q\mid r\le 0 \text{ or } r^2<2\},$$ $$B=Q\setminus A=\{r\in Q\mid r>0 \text{ and } r^2>2\}.$$

This cut defines the real number called $\sqrt{2}$.

The key idea is that the boundary between “rationals whose square is still below 2” and “rationals whose square is already above 2” is exactly the positive number whose square should be 2.

This is one of the cleanest illustrations of why the real numbers are needed at all: the rational numbers describe the two sides of the boundary, but they do not contain the boundary point itself.

What decimals and irrationals are teaching you together

There are two compatible claims here.

1. A real number can be approximated arbitrarily well by rationals.
2. Some real numbers are nevertheless not rational.

Those statements do not conflict. In fact, they are exactly what makes the real number system powerful. Irrational numbers can be reached by rational approximations without ever turning into rational numbers themselves.

Quick checks

Exercises

Related notes

Read this after 4.5 Dedekind cuts and the embedding of Q and 3.5 Gaps in Q and why sqrt(2) is not rational. Then continue to 5.1 Sequences and epsilon-N limits.