Because of the way our minds work, we identify certain apparently separate parts of our experience as having something in common and “count” the corresponding “objects” with the natural numbers.
We also experience some aspects of our experience as having sizes or magnitudes which can sometimes be compared by matching one object against a number of copies of another. If that matching always worked exactly, then the rational number system would suffice for giving us a way of labelling all such magnitudes with numbers. But in fact it does not.
When we do the comparison (eg by counting the number of cm on a ruler) we almost always find that if we look closely enough the match is not exact. And when we repeat with a finer ruler (eg in mm) we still see a slight over or underage. In fact, we can prove that if the geometry of the world has certain simple properties (which our intuition finds it almost impossible to deny), then there are situations where such counting-based measurements can never succeed (eg for finding a common unit to exactly measure the side and diagonal of a square).
Thus, to model our experience of the real world we need to go beyond the rationals and invent the real numbers.
Even if spacetime is based on some discrete geometry at the fundamental level, on the scale at which we live it has apparent rotational symmetry which we need the real numbers to model effectively. The fact that trig and log tables give us rational approximations does not alter the fact that we use the theory of real numbers in order to calculate them.
“I am willing to bet you whatever sum you wish that you cannot construct that rectangle with exactly those dimensions.” – or any other dimensions (whether rational,integer, or unit!)
If anyone can construct an exact square of side exactly 1cm then someone else can construct one of side exactly sqrt(2) cm.
Why *do* so many people think that “we live in a world of rational numbers”?
I think it’s more a matter of the way math(s) is taught than anything else because I dare to say that despite the fact that people are drilled to compute with them more than any others, the rationals are really the least “real” of all (in the sense of modelling something fundamental aspect of our experience).
As I said in an earlier comment, the natural numbers arise from our ability to recognize approximate repetitions (either spatial or temporal) in our experience, as when we recognize and count apples separately from oranges, sheep, and our children.
We can divide sets of such things depending on factorizability but not otherwise; half an apple is as impossible to create exactly as half a child, and even then it only works because the object has rotational symmetry.
Gerry refers to money as an example of where we see rationals in the “real” world, but (without getting into the question of how “real” the world of money really is) I would say that that is not the case. Every monetary system is based on a some indivisible unit so eg when we talk of a quarter as a fraction of a dollar that is only because the basic unit is really the penny. (Show me the bank that can exactly handle a third of a dollar and I’ll show you one that can handle Pi dollars.)
The real numbers, on the other hand, do model a real aspect of our experience, namely the possible relationships between similar geometric magnitudes. And for that the rational numbers certainly do not suffice.
So this leads us to a second question.
If it is wrong to think that “we live in a world of rational numbers”, what can be done to correct that?
Some of us really are working on finding an answer to that.
When a measurement of anything is recorded, what is shown is the closest number we can read off a scale (with maybe a guess at the next digit) even though we know that we have *not* got it exactly (which is why scientists are taught to include error bounds whenever they record a measurement). Later we may come back with a more accurate process and/or a finer scale. So the “thing” we can measure actually corresponds not to just one rational number but rather to an unending sequence of successive approximations. This actually leads to the real number system as what is necessary in order to describe any real measurement process.
“There is no atom or electron that represents that root 2 or pi on the number line” – or 1!
(No matter how hard we try we will never succeed in creating a length, or anything else for that matter, that is exactly equal to another such thing.)
In a single measurement we essentially count markings on a scale, so although in this sense “‘The best we can get’ appears always to be a rational number”, that is only true because it is in fact a natural number – which we always know is really a bit different from the actual quantity we are after. What we actually learn from the measurement is that our quantity is given, in terms of whatever minimum sized unit our scale is marked in, by that number plus a (not necessarily rational) part of a unit.
I agree that the Natural numbers are in some sense especially real to us because we are hard-wired to recognize approximate repetitions in our experience (even though no two apples are identical, we can consistently count a pile of apples – at least until they start to rot and mush together), but what I don’t see is why you assign some kind of greater “reality” to the Rationals than to the Reals. Certainly, in the natural number system, SOME divisions are possible but that is not the same as introducing the rational number system in which any number can be divided by any non-zero number.
How can you divide a dollar into thirds?
Of course, even in the natural number system we can divide *some* things into thirds, Natural numbers have factorization properties so 9/3 is a whole number. But 1/3 is not. Unless we can divide something into thirds which is not actually itself a whole number multiple of three we are not really using the rational number system. And any physical situation which really requires the rationals also actually requires the reals.
There’s always a difference between the actual data and what we can express about it by assigning a number. So when I measure the diagonal of a unit square as 1.4 cm I know that is not exactly 1.4 and the real information in that datum is that it is somewhere between 1.4 and 1.5 (though admittedly quite a bit closer to 1.4), and when I measure it as 1.414214 I am just expressing the fact that the best I know is that it is between 1414213 and 1414214 micrometers (but in this case just a little bit noticeably closer to the latter).