The nature of number

[jonb]

what do we mean by zero?
For the most part, we insist that an element is "zero" when we have some sort of addition operation on a set such that 0 + a = a =a+ 0 for all a our addition operation can take as an input. The origin takes this role in vector addition, and the 'zero function' (f(x)= 0 for all inputs x) fills this role when we're adding functions. In this sense, there can only be one 'zero element' for a given addition operation (if we have 0 and 0' both acting as zeros, 0 = 0 + 0' = 0').

I am not sure what this means and how it applies to what I think I may have seen, so forgive me an let me try to reiterate what I have done in what I think maybe a clearer way, then If you would pull it apart again I might have a clearer idea of where I am going wrong.

I thought of an equation as a geometric form- for instance 2x3=6





Then thinking of 'x3' as a common focus I could lay out every number that could be multiplied by three and got a consistent result in the order of the numbers which had been multiplied (even though the series that resulted are in the opposite direction to the origin).



It seems to me what I have here is not just a method of comparing individual numbers, but of sets, and series. While at the same time each individual number in the series can be mapped to a number in the resulting series. Thus I am not making a lot of individual calculations, but an infinite set of numbers are all dealt with in one go.

Now when I multiply by '0' I get this result.



The series multiplied by '0' gives me a series of '0' and all the individual numbers in the original series become '0's in just the place where you would expect them to be in the order you would expect them to be in. So is the result a series of distinct '0's? So we have a series of individual parts that do not progress in value, or is the result a single '0' that is not a point, but that has at least some width?

What you have written above I think would be consistent with all the individual lines from origin to result. So is what I have done so far OK?