Counting Rational Numbers
We've gabbed about rational numbers before, but here's a new spin on an old favorite: do the rational numbers form a sequence? In other words, can we write them in a countable list?
We can't exactly "count" all the rational numbers, since there are infinitely many of them. However, we can put enough of them in a list so that we can project what they'd be. Keep this list and your shopping list separate; it would be a major nuisance if you got home from grocery shopping only to find your canvas bags full of fractions.
Let's start by putting the positive rational numbers into a table:
Now list them, going along the diagonals:
And so on. Eventually, every positive rational number will show up on this list. For example, will show up when we get to row 67, column 1234 of the table. (We used some pretty high-level mathematics to figure that one out.) If we continue forever, we'll have listed all the positive rational numbers. Microsoft Excel goes down quite a ways, but even this program has its limitations.
To list all the rational numbers, start with 0 to make sure it gets on the list. Knowing him, he'll throw a complete hissy-fit if we forget about him. Then list all the positive rational numbers as before, but after each number, list its negative:
Eventually, every rational number will show up on this list.
Since we can put the rational numbers into a list, we say the rational numbers are countable. Yes, that means you, too, zero. Chill.