Next question: Is it possible for a piece of string to be pi units long? How do transcendental numbers translate to the real world?
Given that any error bar contains some rational numbers, the case can be made that rationals are all you'll ever need for measurements in the real world. Nothing can ever be measured so precisely that it has a definitively irrational length.
However, irrational numbers do serve as a useful abstraction when dealing with the real world, since one is often working in different scales. A rational approximation of pi that works in one context -- say, construction paper projects -- might not be good enough in another -- say, orbital mechanics. Identifying the abstract, transcendental value has the practical application that you can use a rational approximation appropriate to the scale.
In the real world, you always run into limits on how finely you can measure things before you run into quandaries about number theory.
And especially when you attach atoms together the spacing between them varies depending on the strength and type of the bond.
Additionally, molecules are constantly in motion (heat) - one of the motions is changing the size of the spacing between them.
But even if you froze it to absolute zero (which is impossible), it doesn't matter, the size is not determinate, it's only probabilistic.
So a real physical object can never have a fixed size.
Now presumably 1 meter will be equivalent to some integer multiple of base units, though that may turn out not to be the case. Perhaps the current definition of a meter in terms of certain wavelengths of light will turn out to have a remainder of 1/7 of a base unit. ;) In that case you'd just define a slightly adjusted meter as a whole number of base units. So for simplicity let's assume that a meter is a whole number of base units.
Then it's clear that a string of any integer meter length would be possible. A string of length 1/3 meters might be possible, assuming that the number of base units in a meter was divisible by 3. But a string of length pi meters would be impossible.
As a computer programmer I tend to think of transcendental numbers in terms of processes which relentlessly converge, and that of course is the theory of limits. But I also recognize the financial constraints on running processes. So there will always be a "good enough" aspect to any physical measurement.
There is no discrete base unit of space, and on top of that objects do not have determinate sizes.
But it doesn't matter, actual physical object do not have definitive sizes.
So although it's a fun thought experiment to think about strings that that are precisely 4998997308233 base units long, actually measuring such a thing is too expensive or even impossible due to Heisenberg.
Nevertheless, mathematics has demonstrated that it's useful to think of "real numbers" as infinitely precise things, because that way your number system doesn't impose any preordained limits on your measurements -- even though nature itself does.
By far most real numbers are actually non-computable, meaning that there is no finitely expressible procedure for listing their digits to any desired length.
Now it's always seemed clear to me that if a thing is fundamentally unobservable and unidentifiable in any way, you might as well say that thing does not exist at all. Nevertheless, the theory of real numbers implies that the uncomputable numbers "exist" in some sense.
That actually simplifies the theory. Otherwise you'd have to confine yourself to the computable numbers, namely all strings of binary digits that can be produced by some Fexl function (see http://fexl.com/). For example, the number .1010... could be expressed as:
\number == (1; 0; number)
That might make the strict constructivists happy, but it might also hamper the free reigning thought processes of mathematicians.
A physical object does not have a sharp edge.
How does number theory relate to this?
(I think you just misused the term, and are thinking more about the philosophy of mathematics and not about that specific discipline.)
So it doesn't matter if we're talking about lengths that are integer, rational, algebraic, transcendental, computable, or otherwise. The end of the string is fuzzy at many different scales, so even defining its length at high precision becomes a problem, and at very high precisions, you actually change the length when you measure it.
If that sounds like me ducking the question it's because nature itself ducks the question.
That isn't what the field of number theory concerns itself with.
Recall what I said here: "In the real world, you always run into limits on how finely you can measure things before you run into quandaries about number theory."
In other words, when you measure entities and processes in the real world, it is unlikely that you'll ever have to think about the nature of the continuum, or transcendental numbers, or even for that matter precise integers.
For example, back in the '80s when I was using an Apple II computer to collect measurements from a microwave dish, it was obvious that we should measure amplitudes to four decimal places (or whatever, I forgot), and not concern ourselves with the impossible task of counting a precise integer number of energy quanta at each frequency.
Certainly when you're measuring energies in a particle accelerator you'll have different standards, but you will still always come face to face with the economics of "good enough" and even Heisenberg's hard limits on observability in principle.
So to reiterate: the constraints of the real physical world will always bind you long before you ever have to care about the vagaries of number theory.
Nevertheless, the abstract theory of real numbers is definitely useful even in the physical sciences, because that theory transcends all physical constraints and therefore imposes no a priori limits on observations. So it is not wise to fetter your mathematics with the chains of physical constraints.
Blammo! Its diameter is 1 and its length is now pi ;).
When things interact they also do it by probabilities. The lower the probability, the longer it takes to interact. (i.e. bring two deuterium nuclei near each other - will they fuse? Well, it depends on how close they are, the closer they are the less time it takes, since there is a greater chance that they are in the same spot at the same time.)
So in some sense, that's what time is.
You can't fabricate a string that is exactly Pi long anymore than you can fabricate a string that is exactly One long. But you can get as close to a transcendental number as you can to an algebraic number.
[1] I'm ignoring the trivial case.