Posted:

**Apr 22, 2016 3:30 pm**Yup. Anyway I could shorten my description now, which I shall do. After using a lot of arrows on paper to try and find the most efficient description.

Integers go on infinitely, we could just say that rationals go on infinitely too, but then again, were not giving a 'concrete figure' to either of them. So if we take an increment of 1 in the integers chain, we can then express all of the rationals in this chunk as

Y/X (Y over X). X is any number from 1 to infinity. Y is any number from X * the lower end of our 'chunk' of the integer chain, to infinity. So we can say X and Y both have a range of infinite size.

So to find every combination of them we find all of the possible values of X, which we know to be infinity. We then find all the possible values of Y for all the possible values of X, as we know Y can have a range of infinite size, but can also have a range of say 1, we can do X * Y, which is Infinity * (Infinity/2) as halfway between 1, and infinity is a half of infinity, which is still infinity (as infinity is infinitely large) so we could say that its infinity * infinity or infinity^2

So for a range of 1 in integers, we have a range of infinity^2 (infinity squared) of rationals.

If we were to choose a range of 2 in integers, we can simply extrapolate the pattern found, to be 2*infinity^2 in rationals. So for a range of infinite integers, we multiply infinity^2 by infinity, giving us infinity*infinity*infinity or infinity^3

So for the total amount of integers to rationals we can say the ratio is infinity:infinity^3

Simplifying this ratio down would divide it by infinity, as that is a common factor of both of them. So we then get 1 : infinity^2 (as anything divided by itself = 1, and dividing a^x by a results in a^(x-1))

I would personally be happy to leave it at that, but as we (theoretically) can't have a number bigger than infinity, we can handle this now at the end (just like rounding in a trig question, or such, you leave it till the end to gain the best accuracy).

So we can say, infinity^2, is still infinity as nothing can be larger than infinity, giving us the answer of 1:infinity.