What exactly is real number?

Question

This question may sound philosophy, but it has been bothering me for a very long time, therefore I have to ask it here.

The story goes back when my first time reading Apostol's Calculus, then I had learned what real number is by the way Apostol defined it as "undefined objects" with some axioms.

Then I had read Spivak later on(or maybe Courant? I don't remember well. Anyway, that's irrelevant to my question), he used a different approach to define it. Then again I had read other books on how they constructed real numbers. Many authors used their own cool ways.

Then sadly, I found myself do not understand what real number is. I see the tree, but not the forest.

My question is : What is real number at the end of the day?

More generally: What exactly is a mathematical object, if I can construct it in different ways? Does that mathematical object totally depends on the properties I give it? or it has its own very meaning that the definitions we give it are bounded to its very nature? Is that just like we modeling nature with different models in science?

Answer 1

Basically, mathematicians don't care at all what a mathematical object is, we only care about what we can do with it (what operations are defined and what are their properties). So one mathematician might construct real numbers as equivalence classes of Cauchy sequences of rationals, another might prefer Dedekind cuts. Since there is a one-to-one correspondence between those sets of "real numbers", preserving all the structures that we want to define on the real numbers, the disagreement between the two is inconsequential.

Answer 2

I would like to add two things to the discussion:
1. First of all, if you consider standard sets of numebrs, such as natural, integer, rational, real and finally complex number, being very formal, you don't have ANY of inclusions $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$. Each such inclusion should be understood as a natural embedding: integers are equivalence classes of pairs of natural numbers, subject to relation: $(m,n) \sim (p,q)$ iff $n+p=m+q$ (then the embedding is $n \mapsto [(n,0)]$), a rational number of the form $\frac{p}{q}, q \neq 0$ is an equivalence class of pair of integers (the second integer is assumed to be nonzero), subject to the relation $(m,n) \sim (p,q)$ iff $np=mq$ (then the embedding is $k \mapsto [(k,1)]$), any real number is the equivalence class of some Cauchy sequence of rationals (as explained in the answer above: the embedding is $q \mapsto [(q,q,q,...)]$) and any complex number is a pair of real numbers (with the natural embedding $x \mapsto (x,0)$).
2. As you may have already noted, the most dramatic is the passage from rational numbers to the reals: to define integer you need a pair of natural numbers, to define a rational number you need a pair of integers, to define a complex number you need a pair of reals but to define a real number you need infinitely many rationals. As a result of this, the set of all real numbers has no longer the same cardinality as the set of rational numbers. But the main property which is new is the completeness of the reals: roughly speaking you cannot do any analysis without this property so the countability is the price that you pay in order to do any serious analysis.

Answer 3

Throwing my hat into the ring: To begin with, the natural numbers represent the number line extended in the positive direction. They behave as we expect with respect to addition and subtraction:

• The + operator is defined in such a way that if we have $a$ objects, and bring over another $b$ objects, then the quantity given by $a+b$ is how many objects we end up with.

• The $\times$ operator is defined in such a way that if we have $a$ rows of $b$ objects each, then the quantity given by $a \times b$ is how many objects we have altogether.

We can extend the natural numbers into zero and the negative domain in a straightforward way (so that subtraction can be viewed as addition with negative numbers). We have to define multiplication in a way that confused some early mathematicians, but from our modern perspective, it is usually clear that it is the "natural" way to do it.

The integers have gaps in them—gaps that only become apparent if we try to measure continuous things (in some as-yet ill-defined sense) rather than count discrete things. That is to say, if we count apples, then the integers have no gaps that we can discern, but if we try to measure the length of a pencil, it may be more than $6$ inches but less than $7$.

The rationals are a first attempt to fill in these gaps. With them, we can say that a pencil is $6\frac{5}{8}$ inches long, and that (and the rationals in general) will be good enough for any practical purpose we like. The rationals complete the integers in the following sense: If you apply the operations $+, -, \times, \div$ in any finite combination (that does not involve division by zero) to the integers, you get the rationals. Nothing you do in that direction will ever yield anything that is not rational.

However, of course, people (by which I mean mathematicians) eventually became interested in numbers not only for their practical value, but for their own sake. Consider a square one unit on a side. Its diagonals are obviously longer than one unit, but shorter than two units. How long are they, exactly?

They are not integers, clearly, but as you surely know, they are not rationals, either. They are literally irrational—not expressible as the ratio of two integers. The reals (the rationals and irrationals together) therefore fill in gaps in the rationals in the same way that rationals fill in the gaps in the integers. They do this by extending those four operations $+, -, \times, \div$ to infinite combinations. For instance, the number $\sqrt{2}$ can be represented as

$$ \sqrt{2} = 1+\frac{4}{10}+\frac{1}{100}+\frac{4}{1000}+\frac{2}{10000}+\cdots $$

where the ellipsis indicates that the addition and division operations extend to infinity in the appropriate way. That transition from finite to infinite is crucial.

There are, as you discovered, many ways to define/construct the reals, and so the question may arise: Which reals are the "real" reals? Fortunately, the problem resolves itself if we constrain ourselves to worrying only that the reals behave in the way that we expect them to—by adding, multiplying, and dividing as expected—because it turns out that each of the constructions of the reals are equivalent in that way.

One might despair, though, that once again, the reals have gaps in the same way that the integers and rationals did. That turns out not to be true: There are no numbers between reals that are not real themselves.

Of course, as you may well know, the reals have a gap in a different sense. We pointed out that if we take the integers and permit arithmetic to be performed on them, we end up with the reals. For instance, if we square any integer (or indeed, any real), we end up with another real. We can square any real. However, the reverse is not true: We cannot, by squaring any real, obtain a negative number. We therefore must introduce the imaginary and then the complex numbers. Those, at last, are complete in the sense of algebraic closure.

Answer 4

A real number can either be accepted as a primitive notion, or defined in terms of "simpler" primitive notions. A canonical method constructs the real numbers from the axioms of ZFC set theory. You can read more about this in Enderton's Elements of Set Theory.

Ultimately, this is a question about the philosophy of mathematics, and there are several common doctrines, including the one you mentioned about the "very nature" of mathematical objects.

Answer 5

If there are more ways to define an object, then all these objects share all the important properties we want them to have (although they might be different). In the case of real numbers, the different ways to define them (derived from rational numbers via series or dedekind intersections, etc.) all fullfill the desired axioms of real numbers. since all the math we do with real numbers only depends only on these axioms, it suffices to have one of them (or define them only using the axioms).

Answer 6

The OP stated

I see the tree, but not the forest.

In philosophy there is a thought experiment,

$\quad$ If a tree falls in a forest

Here is another one,

If a real number divided by $2$ is greater than $1$, but no one is around to do the calculation, have any calculations been made?

If you want to examine a number where many calculations have been made, look at the Euler–Mascheroni constant. You might be left feeling very humbled - we don't even know if this number belongs to the field of rational numbers!

Our models say the real line has no gaps, and if we try to observe a new spot on the line our axioms patch things up so it gets filled in with a real number that we can then analyze.

Answer 7

Robert Isreal's answer states that the real numbers can be defined as equivalence classes of Cauchy sequences. I know there's a simple intuitive way to define that once you already know what the rational numbers are but then it's very hard to decide how to define addition and multiplication on them and show that it's a complete ordered field. Also, it's easier to construct the real numbers from the Dedekind cuts of the rational number. It's even easier to construct just the dyadic rational numbers, the numbers that can be gotten by starting from any integer and repeatedly applying the operation of cutting the number in half and then construct the real numbers from the Dedekind cuts of the Dyadic rational numbers. Although it's very hard to construct the real numbers as equivalence classes of Cauchy sequences, once you've already constructed them in the much easier way like I will do, it's so easy to then prove that all Cauchy sequences of real numbers approach a real number.

Some people treat a real number as an undefined concept that they assume is a complete ordered field. How do we know a complete ordered field exists? We can construct one and call the objects of that set real numbers. I will construct the real numbers, which can be shown to be a complete ordered field which is unique up to isomorphism although I won't prove that in this answer.

First we define the natural numbers as finite ordinal numbers. The natural numbers are denoted $\mathbb{N}$. We can define addition, multiplication, and inequality on them in the usual way. Let $S$ be the successor function. There's no solution to $S(x) = 0$ so we can invent a solution -1. There's still no solution to $S(x) = -1$ so we can invent a solution -2. After you keep doing that for ever, the resulting objects are called the integers and the set of all integers can be denoted $\mathbb{Z}$. Again we can define addition, multiplication, and inequality on $\mathbb{Z}$ in the intuitive way.

For each odd integer $x$, there is no solution to $2 \times y = x$ so we can invent a solution $y$. In this new system, we can say every number $y$ is a half integer if and only if $2 \times y$ is an odd integer. For each half integer $y$, there is still no solution to $2 \times z = y$ so we can invent a solution $z$. If you keep repeating this process for ever, the resulting system is called the dyadic rational numbers, which I will denote $\mathbb{D}$. Again, we can define addition, multiplication, and inequality on the dyadic rational numbers in the intuitive way.

There is still no solution to $3 \times x = 1$. For any number $x$ in the system, either $3 \times x > 1$ or $3 \times x < 1$. Maybe we could construct $\frac{1}{3}$ from the set of all numbers $x$ in the system such that $3 \times x < 1$.

I know this is not the real definition of a Dedekind cut but I'll define a Dedekind cut as a subset of the dyadic rational numbers that satisfies the following properties:

• $\forall x \in \mathbb{D}\forall y \in \mathbb{D}$, if $x$ is in the subset and $y < x$, then $y$ is in the subset.
• The subset and its complement are both nonempty.

For some Dedekind cuts, the Dedekind cut has a maximal element and its complement doesn't have a minimal element. For some Dedekind cuts, it doesn't have a maximal element but its complement has a minimal element. For some Dedekind cuts, the Dedekind cut doesn't have a maximal element nor does its complement have a minimal element. For each Dedekind cut such that it doesn't have a maximal element nor does its complement have a minimal element, we invent a number that's larger than all members of the Dedekind cut and smaller than all members of its complement. Now we have a new system, the real number system which is denoted $\mathbb{R}$. Again, we can define addition, multiplication, and inequailty in the intuitive way and show that it's a complete ordered field which is unique up to isomorphism. $\forall x \in \mathbb{R}\forall y \in \mathbb{R}$, $x - y$ is defined to mean $x$ plus the additive inverse of $y$ and if $y \neq 0$, $x \div y$ is defined to be $x$ times the multiplicative inverse of $y$.