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

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 3

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 4

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 5

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.