>>11547550>Explain to me, from bottoms up, why exactly does 0.999... != 1, without invoking your divine scripturesAll real numbers are supposed to represent points on a number line. This is all you need to accept. That's how Greeks perceived numbers - as lengths of line segments.
Integers are like marks on a ruler, this is basically a choice of scale. Closed line segments between the marks (say 1 and 3) are denoted as [1,3], you get the idea.
Fractions are also clear: for example 4/6 means divide the segment [0,1] into 6 parts of same size. The number 4/6 is the fourth mark.
But only very few points on the number line can be described as a fraction. So we need a way to describe EVERY point on the number line. Decimal expansion is the best we have come up with. This is how it works on an example:
0.999... has integer part zero. That means that the point lies somewhere in the segment [0,1].
The first decimal digit is 9. That means that the point lies in somewhere in the segment [9/10,10/10]
The second decimal digit is 9. That means that the point lies somewhere in the segment [99/100,100/100].
The third decimal digit is 9. That means that the point lies somewhere in the segment [999/1000,1000/1000].
And so on.
The only point on the number line which satisfies all of this is 1. This can be proved using high school geometry basically.
The thing is that you need to define the decimal expansion (as explained above) using CLOSED line segments [a,b] as I did, and not open segments (a,b) nor half-open segments [a,b). This is simply a technical issue, it just wouldn't work otherwise. This results in the unpleasant fact that one numbers can have two distinct decimal expansions. Ultimately it's just a consequence of the fact that the closed segments [0,1] and [1,2] share a point in their intersection.
