2. So that number (1.42857142857142857142857...ad infinitum) is an infinite number and has something to do with our concept of infinity, right?
It has an infinite number of digits, but it is not an "infinite number."
It does, however, have an infinite number of digits, which means that, yes, it does have something to do with our concept of infinity. The implication is that no matter how far out we carry the division - no matter where we lop it off - the result is still not exactly equal to 10/7.
As Faldage says below, this is an example of a rational number (called rational presumably because it can be expressed as the ratio of two integers). As Emmanuela pointed out in another thread, while the following:
{even numbers} isASubsetOf {Integers } isASubsetOf {rational numbers}
but that each of these three sets has a the same cardinality. That is, they are all the same size, all equally infinite. But 10/7 is a repeating decimal. There are numbers the number of whose digits go on toward infinity, but never repeat. Those are called irrationals. Examples: pi, e, sqrt(2), (in fact, any square root that is not a perfect square of a rational number).
It turns out that the set of integers (and therefore the set of rationals), while infinite, is nevertheless very small compared to the set of reals. Since the set of reals is just the set of rationals unioned with the set of irrationals, if we take this neglible thing called rationals away, we still end up with a set much bigger than integers. This is about the limit of my understanding on the subject, but I wanted to inject it, as it's something that always intrigued me.
k
