What happens as we approach infinity on the number line?

[jakrabb]

Please forgive any ignorance on my part because this is a little new to me... I was just thinking that if we examine All of the numbers from .1 <= x < 1 that we see all possible sequences of numbers. when we change powers of 10 all we do is make that sequence of numbers more precise i.e. 12.3 is just the sequence .123 * 10^2. Now when we examine the number line and see 12.0 <= y < 13.0, we see all the same sequences of numbers that we see from .12 <= z < .13 . There are the same number of sequences of numbers for y as there are for z, but on the number line we see them span one full integer for y and one tenth of an integer for z. I am not sure if this is merely because we are operating in a base ten system but it seems like as we look towards larger numbers we get more precise as to the numbers between the integers i.e. 123456789.0 <= s < 123456790.0 only has the sequence of numbers that starts with 123456789 and there are fewer sequences of numbers for s than we have for 1 <= d < 2 . now the unique sequences between larger and larger numbers gets smaller and smaller. so what happens as we approach infinity? is it one unique number? could we not represent infinity as the sequence of numbers that is almost equal to 1 but less than 1? does infinity then start with a 9?

Any thoughts on the topic are much appreciated.

 

[davilla]

I guess I can see some logic to this argument but it's certainly not stated in any standard way. Positive infinity is one unique number that cannot be written as a string of digits (regardless of the base). If we compare infinity to any real number s by your method then s becomes zero and infinity is mapped to one or whatever you want, as defined by the method. If we compare infinity to itself then they are mapped to the same thing.

 

[matt grime]

Positive infinity is not a unique number. At least not with any reasonable meaning for number, infinity is not a number: it is not a nautural, raitonal, real or complex number. The symbols for positive and negative infinity are useful symbols but don't mean any particular number, and indeed if one looks at the one point compactification of the real line (as a sub-space of the Riemann Sphere) they are the same point.