A Conversation for Bigger and Bigger Infinities

looking the other way

Post 1


How many fractions are there of the number 1? Once you get past 8ths they are a bit small but you CAN keep dividing. One quintillionth can be divided into a quintillion sub sections easy - peasy. So long as you are doing it mathematically and not using a hacksaw on a piece of 2x4.

So we have an infinity which exists within 1.

So is the infinity within 2 bigger? I would guess it is, but a mathematician may have something to say on it.

looking the other way

Post 2


1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 +1/1024....

I think I get the point here, however far you go down the sequence there is always going to be that last tiny evasive little piece, and the equation never quite balances at 1=1.

Is there a similar sort of infinity within Pi?

1/22 = 3.14156.... and however far you slice the division, you'll never get to the end of the fraction?

looking the other way

Post 3


Ah, yes but if you keep dividing pi you eventually just end up with crumbs.

looking the other way

Post 4


The short answer is that there are as many fractions "inside 1" as there are natural numbers--and no more. If fact, there are as many fractions total as there are natural numbers; still fewer than |P(n)|.

A nice way to show it is to arrange all the rational numbers into a grid (which goes on forever downward and to the right), with the numerators as the rows and the denominators as the columns, so that the entry in the ith row and the jth column of the grid is i/j, like so:

1/1 1/2 1/3 1/4 1/5 …
2/1 2/2 2/3 2/4 2/5 …
3/1 3/2 3/3 3/4 3/5 …
4/1 4/2 4/3 4/4 4/5 …
5/1 5/2 5/3 5/4 5/5 …

. Of course you'll get many repeats (1/1, 2/2, 3/3, etc.) but that's not a problem, and you'll certainly hit each rational number at least once. Now you can list all the entries in the grid by starting at the top left corner, then moving right one and listing all the entries diagonally down and left, then moving down one and listing all the entries diagonally up and to the right (I wish I could just draw a picture here), and so on. So you'll have the list:
1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, 3/2, 4/1, …
Since you can get all the rational numbers into a list, the set of rational numbers must be no bigger than the set of natural numbers.

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