It's a big day when you first follow Cantor beyond Aleph Null. Infinity in itself fascinates, and indeed remains unexamined for most -- despite common playground reference. It is the "and so on..." necessary in establishing Algebra, the drive behind Limit Theory (and thus Calculus); and its conceptual presence paves Xeno's road to paradox. To contemplate uncountability as establishing excess to the endless is thus a truly broadening experience. It's a sort of mathematical rite of passage.
The Rational Numbers [Q] are all that any person not vocationally attached in some way to mathematics will ever use on a regular basis. Appropriately, these are the numbers that make sense to us; intuition regarding the behavior of the rational number set is ingrained: Many people find fractions daunting, but presumably few conceptually object to their presence. Inituition more frequently points astray after the introduction of numbers fundementally inexpressable as fully-reduced fractions, when [Q] expands to [R], the Real Numbers.
A general first step in introducing a number so inexpressible is examining the number Radical(2). The proof, like many in mathematics, seems strikingly simple once it is presented; care must be taken afterwards to establish its import, lest it understate itself:
A rational number is simply a ratio, a fraction. More precisely, it is a number that can be expressed as a fraction in simplest terms. 6, for example, can be written as 6/1; while 4/6, 6/9, and 222/333 can all be written as 2/3. The important thing to note here is that the cited representation cannot be any further reduced. (This is what is meant by "simplest terms".) You know something can be expressed as a fully-reduced fraction if you can write it as m/n (replacing m and n with appropriately-selected integers) and m shares no factors with n.
Suppose for a moment that Radical(2) *is* expressable in such a way as m/n. We can establish the following equalities, in quick sequence:
Radical(2) = m/n
[*] 2 = m(sqd)/n(sqd)
2 * n(sqd) = m(sqd)
n(sqd) = m(sqd)/2
We just demonstrated that m(squared) is divisible by 2 -- and, after brief analysis, that m itself must be so as well.
It m is divisible by 2, then it stands to reason that I should be able to pick some number p such that:
m/2 = p
and thus that:
m = 2p
Substititing "2p" in for "m" in the equation marked with [*] above yeields the following:
2 = (2p)(sqd)/n(sqd)
2 = 4p(sqd)/n(sqd)
1 = 2p(sqd)/n(sqd)
n(sqd) = 2p(sqd)
n(sqd)/2 = p(sqd)
Now we've also shown that n(squared) is dividible by 2 -- and again it follows that n itself is also dividible by 2.
Here's the catch: If m is divisible by 2, and n is divisible by 2, then m/n is not a fully-reduced fraction. Since we supposed it was earlier, we've hit a snag.
Since our math above used only the stand-ins m and n, what we just showed will hold true for any inserted integers. That is to say: There simply is no way to write Radical(2) as a fully-reduced fraction. Since it can't be written as a ratio, Radical(2), like many important numbers, is not in fact a member of [Q].
Of course, we can place Radical(2) on the numberline. In fact, we can calculate its decimal representation arbitrarily precisely. This can be restated with somewhat of an illuminating effect: We can produce a finite series of sums (of negative powers of ten) that approximates the value of Radical(2) arbitrarily closely; but if you spent an eternity narrowing this precision at lightning speed you'd never actually get to the number itself. It is the first sight of foreign land on a vast and expansive mathematical horizon.
Mathematics' beauty is that to study it is to expand one's intuition. Little that is interesting is rote.
