Construction of the Reals from Decimals

In typical real analysis classes, the presentation of real numbers that is generally given in schools (starting from rationals) is considered insufficiently rigorous, and replaced by a construction involving Dedekind cuts. However, in this post, we investigate a formalization that derives directly from the presentation given in schools — specifically, using decimals — and study its equivalence to the Dedekind cut formulation.

For my learning, I start from as fundamental a place as possible and build up from there.

Decimal Representations of Rationals

We start with some properties of decimal representations for integers, which are building blocks for the decimals of rationals and reals. We then use this to motivate decimal representations for rationals. This will be a steppingstone to the construction of the reals.

Actually, we work with positive and nonnegative integers and construct the nonnegative reals; the full real line follows afterward from constructing additive inverses.

Properties of Decimal Representations of Integers. We associate to each $n \in \mathbb{Z}_{+}$ its decimal representation (decimal for short) $d_{k}\ldots d_{0}$ , where $d_{i} \in \left\{ 0,\ldots,9 \right\} = \mathbb{Z}_{10}$ , $d_{k} \neq 0$ , and $n = d_{k}10^{k} + \ldots + d_{0}$ . Such a representation always exists and is unique. Furthermore, if $a = a_{k}\ldots a_{0},b = b_{l}\ldots b_{0} \in \mathbb{Z}_{+}$ , we have the following results:

1. Addition of decimals: The decimal of $a + b$ is given by the additive carry-over procedure, which is taught in school.

2. Multiplication of decimals: The decimal of $ab$ is given by the multiplicative carry-over procedure, which is taught in school.

These properties follow from distributivity of multiplication over addition.

Now, rationals are equivalence classes of $\mathbb{Z}_{+}^{2}$ ; for $(a,b) \in \mathbb{Z}_{+}$ we denote its equivalence class by $\left\lbrack \frac{a}{b} \right\rbrack \in \mathbb{Q}_{+}$ .

In analogy with adding more zeros at the end and “sliding the decimal point to the right” when multiplying by 10, when dividing by 10 we “slide the decimal point to the left.” This leads to our first attempt at a definition:

Provisional Definition (Decimal Representation, Rational). A decimal is a representation of a number $q \in \mathbb{Q}_{+}$ consisting of a sequence $d_{i}$ with $i \in \lbrack - l,k\rbrack$ for $l,k \geq 0$ . We denote this as $d_{k}\ldots d_{0}.d_{- 1}\ldots d_{- l}$ ; the dot in between is called a decimal point.

It turns out we can’t always do this for every rational; indeed, relatively “few” rationals admit such decimals. However, when it can be done, the decimal is unique.

The decimal representations for integers are clearly contained within this definition by letting the sequence after the decimal point be empty.

When the decimal exists, there is an algorithm to find the decimal, mirroring repeated Euclidean division to get the decimal representation of an integer. Under the provisional definition, a decimal exists if and only if the algorithm terminates.

For some numbers, for example $x = \frac{1}{3}$ , the algorithm doesn’t terminate. In that case, we simply define the decimal to be the (possibly infinite) sequence of numbers produced by the algorithm.

From this definition, every rational number has a unique decimal, and we can prove that the decimal always repeats with a sequence other than 9. Furthermore, every decimal that is finite or repeats infinitely corresponds to a unique rational number.

We have operations on addition and multiplication for rational numbers (as equivalence classes.) When decimals are finite, the properties of decimals for integers above (carry-over procedures) generalize to rationals, using the “standard” known school procedure of aligning with the decimal points.

Construction of Reals

The definition above can be used to construct a mapping from rationals to their decimals. This function is not surjective; only certain decimals map to rational numbers.

We naturally define an order on the decimals based on the order of the digits and lexicographic order, and we can prove that the order on rationals corresponds to the order on their decimals.

We can imagine “listing out” these rationals on a “number line” where larger numbers are to the right of smaller ones. There are lots of “holes” in this line, and we can imagine decimals in between whose ordering is well-defined under the order.

We define all these decimals to be real numbers:

Definition (Real Number). A real number is a sequence $\left\{ d_{i} \right\}_{i = - \infty}^{i = l}$ with $l \geq 0$ and $d_{l} \neq 0$ , such that there does not exist $N > 0$ where $d_{- i} = 9$ for $i \geq N$ . We say it terminates if there exists $N > 0$ such that $d_{- i} = 0$ for $i \geq N$ ; otherwise, we say that it has an infinite decimal.

This “completes” the number line and produces what is called the “real number line”, where we define a total order on all real numbers (based on order of digits and lexicographic order.) Some real numbers correspond to rationals. Others don’t, and we call these irrational.

As a consequence, all irrational numbers must have infinite decimals.

Equivalence to Dedekind Cuts

We now show that real numbers as defined above are equivalent to Dedekind cuts: there is a bijection between the two sets, which we take to be canonical.

Let $r = \left\{ d_{i} \right\}_{i = - \infty}^{i = l}\mathbb{\in R}$ , with $l \geq 0$ . The motivation is to bound $r$ above and below using the decimal; for example, if $r = 0.12345\ldots$ , then $0.123 < r < 0.124$ . This will help us produce our Dedekind cut. We associate to $r$ the Dedekind cut

$\displaystyle \left\{ \bigcup_{i \leq l}^{}\left\{ x\mathbb{\in Q:}\left( x_{i} < d_{i} \right) \land \left( x_{j} \leq d_{j}\ (\forall j > i) \right) \right\},\bigcup_{i \leq l}^{}\left\{ x\mathbb{\in Q:}\left( x_{i} \geq d_{i} \right) \land \left( x_{j} \geq d_{j}\ (\forall j > i) \right) \right\} \right\}.$

Or, with the standard order (already defined), this is equivalent to

$\displaystyle \left\{ \left\{ x\mathbb{\in Q:}x < r \right\},\left\{ x\mathbb{\in Q:}x \geq r \right\} \right\}.$

This cut is in fact a partition and is uniquely associated to $r$ .

Field Structure

Using the Dedekind cut formulation, we can extend the definitions of addition and multiplication to all real numbers. Then, we can show that $\mathbb{R}$ is a field. We can now show that $x^{2} = 2$ has a real solution

$\displaystyle \left\{ \left\{ x\mathbb{\in Q:}\left( x^{2} < 2 \right) \vee (x < 0) \right\},\left\{ x\mathbb{\in Q:}\left( x^{2} \geq 2 \right) \land (x > 0) \right\} \right\},$

corresponding to the (positive) square root of 2, and in a similar way we can define all square roots, cube roots, etc.

Supremum and Infimum

We prove the supremum and infimum properties for real numbers (which we can see as not being true for rational numbers by taking the Dedekind cut of an irrational number); this is traditionally done using Dedekind cuts. Something like real analysis – Proof of least upper bound property of $\mathbb{R}$ by Dedekind cut. – Mathematics Stack Exchange is sufficient.

Calculus Ability

Is there a precise sense in which we can say calculus can only be done on $\mathbb{R}$ and not say $\mathbb{Q}$ ? Can we say that the delta-epsilon definition of limit is not well-defined on $\mathbb{Q}$ but well-defined on $\mathbb{R}$ ?

This is a separate question from the construction above, but it can serve as motivation for understanding the importance of $\mathbb{R}$ in future subjects.

Basically, for the limit of a function $f$ to be defined $a$ , $f$ must be defined on an open ball around $a$ (but possibly excluding $a$ .) It is clear that $\mathbb{Q}$ is a metric space, and so limits can be defined on $\mathbb{Q}$ in a similar way. Thus, limits are well-defined on $\mathbb{Q}$ . However, we won’t get certain “nice” properties that are fundamental to calculus, related to completeness. Nevertheless, we may still use calculus notions on $\mathbb{Q}$ , using the standard metric; it doesn’t look like those completeness properties immediately imply inability to do calculus. Thus, to my current knowledge, it is only in an informal sense that calculus is “better” on $\mathbb{R}$ .

Let us ask a precise question, and let’s make it general. If a metric space $M$ is not complete, can we show that the limit definition won’t always be unambiguous? More precisely, if $M$ is not complete, then does there exist $f:M \rightarrow M$ and $a \in M$ such that $L_{1},L_{2} \in M,L_{1} \neq L_{2}$ , both satisfy the $\delta$ – $\epsilon$ definition?

Well, if $M$ is not complete, then there exists $\left\{ a_{i} \right\}_{i = 1}^{\infty}$ that is Cauchy but doesn’t converge in $M$ ; in other words, there doesn’t exist $L \in M$ such that, for any $\epsilon > 0$ , there exists $N > 0$ with

$\displaystyle i > N \rightarrow d\left( a_{i},L \right) < \epsilon.$

We recall the general proof of unambiguity of limit. Assume for any $f,a$ we had

$\displaystyle 0 < d(x,a) < \delta \rightarrow d\left( f(x),L_{1} \right) < \epsilon,d\left( f(x),L_{2} \right) < \epsilon.$

We use the triangle inequality (valid for any metric space) to get

$\displaystyle d\left( L_{1},L_{2} \right) < 2\epsilon.$

Since $\epsilon$ is arbitrary, $d\left( L_{1},L_{2} \right) = 0 \rightarrow L_{1} = L_{2}$ . So this proof works in any metric space regardless of completeness, resolving our question negatively. It really looks like calculus isn’t less doable on $\mathbb{Q}$ than on $\mathbb{R}$ . (EDIT, 02/18/2023: I now know that completeness is not what guarantees uniqueness of limits, but rather that the point is a limit point. It is true that every point in $\mathbb{Q}$ is a limit point, guaranteeing uniqueness of limits on $\mathbb{Q}$ . There are other desirable properties that completeness leads to and is necessary for, but not uniqueness of limits.)

Limits for Decimals

For general rational decimals, we can still define addition and multiplication such that the correspondence to rational numbers is maintained (the number for the sum of two decimals is the sum of the numbers of both, etc.) This is motivated by saying that we can “approximate” the resultant decimal to arbitrary degree with taking finite “truncations” of the operands, and that the “limit” of this procedure is the resultant decimal. We don’t really have a notion of limit yet, but we can define one for this situation. (In fact, this definition is equivalent to the “ $N$ – $\epsilon$ ” definition of limit of a sequence.) We do so as follows: if we have a sequence of decimals where each digit is “fixed” after a point, then we say that the sequence has a limit, which is the decimal consisting of those “eventually fixed” digits:

Definition (Limit of Decimal Sequence). Let $s_{1},s_{2},\ldots$ be an infinite sequence of decimals. If for any $n$ , the $n$ th digits of the $s_{i}$ stabilize at a digit $d_{n}$ — meaning that there exists $N$ such that for all $i > N$ the $n$ th digit of $s_{i}$ is $d_{n}$ — then we say that the decimal $\left\{ d_{n} \right\}$ is the limit of the $s_{i}$ .

We see that this definition is equivalent to the “ $N$ – $\epsilon$ ” definition for the corresponding rational numbers.

Definition (Truncation of Rational Decimals). Let $q \in \mathbb{Q}_{+}$ . The $n$ -th order truncation of $q$ is the finite decimal resulting from taking only $n$ sequence elements after the decimal point (along with of course the part of the sequence before the decimal point.)

Definition (Addition and Multiplication of Rational Decimals). Let $a,b$ be rational decimals. The sum $a + b$ is the limit of the sums of the $n$ -th order truncations of $a$ and $b$ , and similarly the product $ab$ is the limit of the products of the $n$ -th order truncations of $a$ and $b$ .

As mentioned, we see that with this definition, the correspondence between rational decimals and numbers is maintained with respect to addition and multiplication (they are field-isomorphic.)

Now, we can generalize each of these definitions without modification to real numbers, and we can then show that real decimals with these addition and multiplication operations are field-isomorphic to real numbers. Thus, this gives an alternate way of defining addition and multiplication for real numbers, directly for their decimals without immediately having to define Dedekind cuts or show another construction.

(At some point, I might come back to this and write out some of the painstaking details for my own learning.)

Teaching

This is how I would teach this material, assuming no background knowledge. (As I mentioned in another post, I think it’s possible to provide rigorous presentations to students while also giving enough clarity that it won’t be too overwhelming or seem pedantic.)

Start with counting numbers, and assume their basic properties and operations.
Define integers.
Define rational numbers with physical meaning and study their properties, operations, and decimals.
Define real numbers with decimals to “fill the holes in the line.” Study the real number line. Prove completeness (supremum and infimum properties.)
Define operations on real numbers (addition, etc.) both with limits as above and with Dedekind cuts. Show the equivalence of the approaches, and prove the standard field properties (without yet mentioning the word “field.”)

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First written September 26, 2020, then modified in April 2021. Clarified the “Limits for Decimals” section and modified the “Teaching” section on February 18, 2023.