Home Math Teaching About Me

2. What is a Number?

≪ 1. What is a Proof? | Table of Contents | 3. The \(\sup\) and \(\inf\) ≫

Consider the following ontological guiding question:

As a disclaimer, the contents of this discussion and this question are more or less completely irrelevant to this course. However, the lack of a precise and consistent answer would bear considerable philosophical ramifications on many fields of mathematics.

(Stimulating conversation follows.)

Some rather poor definitions of “number” are circular. For instance, my common core education defines a real number as anything that appears on the number line. Naturally, the number line is the set of all real numbers, arranged in order… Likewise, one may define a real number as any number that’s rational or irrational. This begs the question of what an irrational number is, which is often declared to be any real number that’s not rational.

Another very plausible definition is that a real number is anything that can be written as a finite or infinite decimal expansion. That is, \(1.2345678901234567890\ldots\) is a number, \(0.1234567910111213141516\ldots\) is a number, etc. But this is misleading! My name is Hunter, but the characters appearing on your screen right now are not in fact the same as the person it refers to. Likewise, decimal expansions only represent numbers, and just as a name has no meaning if there’s no human behind it, just as currency has no value if governments don’t back it, a representation of a number has no meaning if no number behind it really exists. (More on this later…)

To hammer home the idea that these are representations of numbers and not in and of themselves “numbers”, consider thet \(\frac{1}{2}\) and \(\frac{2}{4}\) are “different representations of the same number”. With infinite decimal expansions, \(0.9999\ldots\) and \(1.0000\ldots\) are famously different representations of the same number.

We thus conclude that the definition “a real number is anything that can be represented by a finite or infinite decimal expansion” is (again, in my opinion) not a good definition. We are saying that a real number is something that can be represented by something that represents a real number.

These definitions share the flaw that they are trying to describe numbers as members of some set of mathematical objects without defining the characteristics of that underlying set: the number line, rational and irrational numbers, decimal expansions.

Hopefully someone comes up with the following idea:

A number is something that precisely describes a physical quantity.

This allows us to accept some obvious things as numbers: the counting numbers \(1,2, 3, \ldots\); rational numbers \(\frac{4}{3}, \frac{10}{7}, \frac{19}{84}\), etc.; even negative numbers and zero can be considered numbers.

Something to highlight is the concept of negatives. Having \(-1\) of something is meaningless in a vacuum, but it makes sense to say that the difference between two quantities is \(-1\). This is distinct from saying their difference is \(+1\), and thus we can interpret the sign as an “orientation” of a number. The point is that numbers are really describing quantitative relationships, and I want you to keep this in mind later.

\(\sqrt 2\) can be physically realised as the length of the hypotenuse of a 45-45-90 triangle with side length \(1\). One may naïvely argue that \(\sqrt 2\) is the side length of a square of area exactly \(2\), but one must first demonstrate that such a square exists. In contrast, it’s very clear that a 45-45-90 triangle with side length \(1\) exists. This thus poses a dilemma: do the quantities \(\sqrt[3] 2\) and \(\sqrt \pi \) exist if we can’t find a way to “construct” them physically? What about something like \(\ln 2\) (or \(e\), for that matter)?

This is the foundational issue that we face when we first initiate the study of real numbers: there are lots of numbers that we can describe and believe in our heart of hearts to exist (like \(\sqrt[3] 2\)), but cannot produce physical evidence in support of this claim.

In the past few centuries, very intelligent mathematicians have discovered definitions of the real numbers that do not rely on the idea that they should have a demonstrable physical meaning. You can find a list of different constructions of the real numbers on Wikipedia. One of these constructions is similar to decimal expansions:

A real number is a sequence of rational numbers \(q_1, q_2, q_3, \ldots\) such that \(\left\lvert q_n - q_m \right\rvert \to 0\) as \(n\) and \(m\) both get bigger and bigger. If \(q_1, q_2, q_3, \ldots\) and \(q_1^\prime, q_2^\prime, q_3^\prime, \ldots\) are two sequences of rationals such that \(\left\lvert q_n^\prime - q_n \right\rvert \to 0\) as \(n\to\infty\), the two sequences are the same real number.

This “\(\to 0\)” can be made rigorous, but I’ll omit the details for brevity. This is a very abstract definition: it says a real number is simply a collection of “equivalent” representations, and one can then interpret certain real numbers physically and make sense of the number system from then on. One can define familiar operations like addition, subtraction, multiplication, addition, division, etc. all very painstakingly.

Another definition I quite like is the “Dedekind cut”:

A real number is two disjoint sets of rational numbers \(L, R \subseteq \mathbb{Q}\) such that \(L\) is “closed downwards” (if \(l \in L\) and \(x < l\), then \(x\in L\)); \(R\) is “closed upwards” (if \(r\in R\) and \(x > r\), then \(x\in R\)); \(L\) does not contain an upper bound of itself; and \(\mathbb{Q}=L \cup R.\)

In words, “a real number is all the rational numbers less than it and all the rational numbers bigger than or equal to it”. Once again you can painstakingly define arithmetic and all that.

The answer is yes. Mathematicians have isolated the following properties of \(\mathbb{R}\) that make it unique:

1. \(\mathbb{R}\) is a field.
2. \(\mathbb{R}\) is totally ordered, and the ordering is compatible with arithmetic.
3. \(\mathbb{R}\) is complete: every set of real numbers that’s bounded from above has a least upper bound.

If Alice has a number system \(A\) and Bob has a number system \(B\) such that both \(A\) and \(B\) satisfy the above three conditions, then there is a way for Alice and Bob to systematically create a “translation guide”: a way to translate a number \(a\in A\) into a number \(b\in B\) that’s consistent with arithmetic and ordering. In this way the real numbers are unique: it doesn’t matter if you use Dedekind cuts or sequences of rational numbers or something much more exotic, we are morally speaking all using the same set of real numbers.

This is thus the starting point of real analysis: we are studying the one and only totally ordered complete field, and the central characteristic is the existence of the \(\sup\) and the \(\inf\).