# Are all numbers real numbers?

If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego bricks. It's also conceivable that I could get any real number. My question is, can there mathematically exist a universe in which these ratios are not real numbers? How do we know that the real numbers are all the numbers, and that they dont have "gaps" like the rationals?

I want to know if what I (or most people) intuitively think of as length of an idealized physical object can be a non-real number. Is it possible to have more then a continuum distinct ordered points on a line of length 1? Why do mathematicians mostly use only R for calculus etc, if a number doesnt have to be real?

By universe I just mean such a thing as Eucildean geometry, and by exist that it is consistent.

• Your question is predicated on an idea of "measurement" that seems much stronger than any I am familiar with in the real world. Any form of physical measurement I know will (at best) give the following answer: the quantity $X$ you are trying to measure has value $A < X < B$, where $A$ and $B$ are rational numbers. With such a form of measurement it is not even possible to tell whether $X$ is rational or irrational real (at least not by any finite sequence of measurements, and physically speaking I don't know any other kind). So what do you have in mind here? – Pete L. Clark Jul 14 '11 at 0:11
• I knew physicist Bob Mills (as in Yang-Mills). Once I suggested doing physics using some number system other than the real numbers. He considered that to be a worthless suggestion (even though he was too polite to say it that way). – GEdgar Jul 14 '11 at 0:12
• Also, speaking as a mathematician, I am utterly baffled at what "can there mathematically exist a [physical, presumably] universe" might mean. We are not in the business of deciding whether universes can exist or not! – Pete L. Clark Jul 14 '11 at 0:13
• @vittulainen: It is possible to imagine that a mathematical structure that involves infinitesimals will be useful to physicists. In a certain sense, it already happened (maybe Newton, certainly the Bernoullis, many others). Objects that are distant from simple notions of measurement, such as Hilbert space, are certainly useful to physicists. – André Nicolas Jul 14 '11 at 0:35
• The rationals don't have gaps, they are dense. – starblue Jul 14 '11 at 7:27

## 6 Answers

These ratios are not even numbers! After a certain point, perhaps $10^{-3}$ meters, you would find that you have to make lots of choices about how to measure the length of the sticks. Do you measure along the length of a stick, whatever that means, even if they bend a little, or do you just take the two points on a stick furthest apart and measure the distance between them?

(Regarding measuring along the length of a stick, see coastline paradox.)

What if the sticks are a little springy, so their lengths will vary depending on how you hold them? Okay, so suspend them in a magnetic field or something fancy like that. Even if you are really really careful, beyond a certain point you have to decide what counts as a "point" on a stick, since after all a stick is just a collection of atoms and atoms are mostly empty space. Do you look at the distance between the two nuclei in the stick furthest apart? Do you take the electron shells into account? How do you do either of those things given the uncertainty principle?

Basically any given physical quantity (other than ones which are uniquely determined by some mathematical theory, e.g. $\pi$) cannot be meaningfully measured past a certain number of decimal points, so the question of exactly what kind of number it is is moot.

On the other hand, if we are really only talking about a mathematically idealized model of the real world, let's make some classical assumptions about measurement. Namely, allow me to assume that we can tell if one length is longer than another length. Allow me to further assume that we have a sequence of ideal rulers of lengths decreasing to zero, for example a sequence of ideal rulers of lengths $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, ...$. Then given any object we can eventually tell what rational lengths are less than its length and what rational lengths are greater than its length, and this gives us a real number by the construction of the real numbers using Dedekind cuts.

What I'm saying above is that neither of these assumptions is physically realistic beyond a certain precision.

I want to know if what I (or most people) intuitively think of as length of an idealized physical object can be a non-real number. Is it possible to have more then a continuum distinct ordered points on a line of length 1? Why do mathematicians mostly use only R for calculus etc, if a number doesnt have to be real?

This is possible, but what's the point? I can't conceive of a way in which the distinction between two points which are infinitesimally close could matter in, for example, physics, at least in the obvious way. If two possible sets of initial conditions of a system are infinitesimally close, I would expect them to stay infinitesimally close for all time given any reasonable constraints on the system.

There are mathematicians who use extended real number systems for non-standard analysis, but I don't think there are any people who think seriously about using non-standard analysis in physics (although this might change in the future).

By universe I just mean such a thing as Eucildean geometry, and by exist that it is consistent.

That is not what anyone I know means by either "universe" or "exist." Anyway, the answer to this version of the question is that such "universes" "exist," and as I have said people study them in non-standard analysis.

• ok, but it doesnt say that the real number is a unique mathematical object satisfying beeing smaller and larger then two sequences of rationals. why cant we apply this method again, such that every real number is larger or smaller then the new object extra-real number – vittulainen Jul 13 '11 at 23:02
• @vittulainen: if I understand you correctly, we can in fact do this, and the study of what happens when we enlarge the real numbers this way is called among other things non-standard analysis (en.wikipedia.org/wiki/Non-standard_analysis). As far as actually measuring the lengths of actual objects, though, I can't see how literally infinitesimal differences could possibly matter in any practical sense. – Qiaochu Yuan Jul 13 '11 at 23:05
• @QiaochuYuan, "If two possible sets of initial conditions of a system are infinitesimally close, I would expect them to stay infinitesimally close for all time given any reasonable constraints on the system." - Poincare' was first to show the phenomenon of sensitive dependence on initial conditions on orbits of ODEs, that was 100 years ago - I find it hard to believe you have not heard of this, so what do you mean? – alancalvitti Dec 23 '12 at 6:47
• @alan: I mean "infinitesimally close" in the literal sense, not as a metaphor for "very close." – Qiaochu Yuan Dec 23 '12 at 6:52
• @QiaochuYuan, understood, that's why I mentioned ODEs, as opposed to discrete chaos of, say, Collatz or Bak-type avalanche dynamics. in ODEs that display sensitive dependence, initial conditions can be infinitesimally close at $t_0$ yet diverge exponentially, albeit being bounded in an attractor, as per Lorentz system which is an ODE with state space $\mathbb R^3$, the smallest dimension supporting such behavior. – alancalvitti Dec 23 '12 at 6:57

The Cantor–Dedekind axiom states that there is a bijection between real numbers and points on a line. In other words, every linear measurement corresponds to a real number and vice-versa.

• The question is precisely if we should accept the axiom you mentioned. This is not obvious to everybody. – Mikhail Katz May 6 '16 at 8:23
How do we know that the real numbers are all the numbers, and that they don't have "gaps" like the rationals?

If you allow for infinitesimals then there are more, for example surreal numbers.

Of possible interest is the following paper:

Wendell Melville Strong (1871-1942), Is continuity of space necessary to Euclid's geometry (also digitized here), Bulletin of the American Mathematical Society 4 #9 (June 1898), 443-448.

This appears to the published version of Strong's 1898 Ph.D. Dissertation at Yale University.

Review in "Revue Semestrielle Des Publications Mathematiques":

Dedekind has defined a discontinuous space in which "so far as he sees" the Euclidean constructions can be made. This space consists of all points whose orthogonal rectangular coordinates are algebraic numbers in a given unit of distance. Now the author considers three kinds of discontinuous space: rational, quadratic and transcendental space, the first containing only points with rational coordinates, the second moreover those whose coordinates imply extractions of square roots, the third only points with transcendental coordinates. He shows that in rational space parts of a figure may disappear by a change of position, that quadratic space is the least space allowing the Euclidean constructions, that transcendental space, though possessing an infinitely greater number of points, does not admit all such constructions.

See also:

Review in Nature (in English, lower half right column of p. 310)

JFM review (in German)

Actually, in a way, what the submitter asks about does happen:

In Electrical Engineering, one sees that analyzing resistor-only circuits is easy because the equations require only simple algebra, whereas including capacitors and inductors makes things much harder due to their time-dependent behavior. HOWEVER: There is a cute trick which simplifies things: It turns out that capacitors and inductors can be modeled as resistors with imaginary resistance. And the properties of the resulting alternating currents - wavelength, phase - can be combined by representing the currents as complex exponentials. In that sense, electrical currents are being measured using complex numbers.

With regard to the OP's question Can there mathematically exist a universe in which these ratios are not real numbers? to provide a meaningful answer, the question needs to be reinterpreted first. Obviously the real numbers, being a mathematical model, do not coincide with anything in "the universe out there". The question is meaningful nonetheless when formulated as follows: What is the most appropriate number system to describe the universe if we want to understand it mathematically?

Put this way, one could argue that the hyperreal number system is more appropriate for the task than the real number system, since it contains infinitesimals which are useful in any mathematical modeling of phenomena requiring the tools of the calculus, which certainly includes a large slice of mathematical physics. For a gentle introduction to the hyperreals see Keisler's freshman textbook Elementary Calculus.