Let us examine numbers as distances from some number.
If we choose as starting point the zero (0), we have the usual case.
0 = |0 – 0|
1 = |0 – 1|
2 = |0 – 2|
n = |0 – n|
Each distance can be divided into parts, let’s think the number space as the familiar set R. For example the distance from zero to two can be divided in n parts, we get distances 2 / n between zero and two.
The point from which we started isn’t a distance but — like said — a point, that can’t be divided into smaller parts. Philosophically speaking this point is now in words of Democritus “atomos”, the smallest unit.
In the book “The way to geometry” it is said: “The magnitude of a point is zero.”
When we founded the numbers as distances in relation to zero, the zero point is indivisible; because in question is a point not a distance, dividing zero (a point) by any number, we get zero. Zero doesn’t get smaller by dividing.
Of course zero can’t be divided by itself. In distance interpretation philosophically speaking indivisible would be divided by indivisible.
Now zero is different from other numbers so that it is a point, the other numbers are distances from this point.
As distance interpretation we could start the numbers also in relation for example to number eight (8).
Now eight is a point, we get the other numbers as distances to number eight.
0 = ||8 – 8| – 0||
1 = ||8 – 8| – 1||
2 = ||8 – 8| – 2||
n = ||8 – 8| – n||
Here from number eight is “created new zero”. First we determine the distance of number eight from itself, then the distance of examined number from the determined distance earlier..
We could get the numbers (distances) and the zero point also by denoting:
0 = |8 – (8 + 0)|
1 = |8 – (8 + 1)|
2 = |8 – (8 + 2)|
n = |8 – (8 + n)|
When we see the numbers as distances from some number, we notice, that the distance can be divided into smaller parts, but the point from which we started, we can’t divide into smaller parts.
In our example when we started from eight, we examine first the distance of eight from itself, |8 – 8| = 0, which is and remains as zero.
The distance can be even or uneven, but the point from which the distance is determined can’t be even or uneven, that is the whole point of this post.
If we want also the negative numbers as distance interpretation, we simply put the minus sign before the determined distance, for example -2 = -|0 – 2| and this is how we can get the entire set R so that the distances from zero point can be divided into smaller parts, but zero point is and remains zero and is not a distance but a point. In the words of Euclid that which has no part. In my own words: “Indivisible”.
In this way the specialty of zero can be seen better.