# Is zero the center of the numeric sequence?

The numeric sequence has symmetry on zero, with equal infinities of cancelling out + and - values on either side.

Can numbers be said to have different centers of symmetry than zero? Is it possible that something like the numeric quantifying tool that we use, may have a center or linear structure, what are it's major caracteristics in the sense of sequences, centres, and symmetries?

• What do you call "numbers"? Integers, rationals, reals, complex numbers? Also, all this number types have a symmetry centered in any number, under suitable definition of symmetry. – lisyarus Oct 3 '15 at 16:55

## 3 Answers

You have an intuitive notion of the symmetry of the numbers (let's say the integers for concreteness) around $0$, and we can formalize this in mathematics. A “symmetry” is usually understood as a mapping from a set to itself which preserves some sort of structure; typically the “distance” between the elements of the set.

For example, a symmetry $f$ of the square, as normally understood, must take each point $p$ of the square to a single other point $f(p)$, and if two points $p$ and $q$ were at some distance $d$ from one another, then $f(p)$ and $f(q)$ must be at the same distance $d$ from one another. Understood in this way, the only mappings that are symmetries of the square are the ones you expect: rotations by $90$ or $180$ degrees, or reflections across one of the square's four axes.

Your intuitive idea of the symmetry of the integers around 0 corresponds to the observation that the function $f(x) = -x$ preserves distances between integers. The distance between two integers $p$ and $q$ is $|p-q|$. If $|p-q| = d$ then $|f(p) - f(q)| = |(-p) - (-q)| = |-p + q| = |q-p| = d$ also. 0 is the center because is is a “fixed point” of the symmetry, since $f(0) = -0 = 0$.

But in this sense, one could pick any integer to be the center, say 17. The symmetry around 17 corresponds to the function $f(x) = 34-x$. This mapping also preserves distances, as you can show. But instead of leaving $0$ fixed, it leaves $17$ fixed.

In this sense we can categorize the symmetries of the integers as follows:

1. Reflection symmetries of the type $f(x) = n-x$; these symmetries have a center at $\frac n2$. (Note that $\frac n2$ may not itself be an integer; the symmetry $f(x) = 1-x$ is a reflection around $\frac12$.) Your symmetry $f(x) = -x$ is of this type, with $n=0$.

2. Translation symmetries of the type $f(x) = n+x$; these symmetries correspond to sliding the entire number line in one direction or the other, and don't have a center.

• Unlile $x\mapsto -x$, both $x\mapsto n-x$ and $x\mapsto n+x$ don't preserve the additive structure of the number line. Distance is not the only thing symmetry can preserve. – celtschk Oct 4 '15 at 10:41

It depends how much structure you want to impose on numbers.

If you consider the reals just as an affine space, or as a totally ordered set, then there is no good sense in which they can be considered to be "anchored" anywhere. You could pick anywhere and find just as many numbers on one side as the other.

If you want to define a monoid structure (or a group structure) on the reals, then you're going to have to have a privileged element to use as the identity. That imposes a "centre". When the operation is the standard addition, then you get 0 as the privileged element. When the operation is multiplication, 1 is the privileged element, so in a sense, the reals can be viewed as having 1 as the "centre".

If you want to define a normed space structure (that is, something endowed with the usual idea of "distance" derived from being able to measure the "size" of a number), you'll define a "centre" in exactly the same way as with the monoid structure, and you'll end up with 0 as your "centre".

If you want to define a metric space structure (that is, something with an idea of "distance" but in which numbers might not have a "size"), you might not have a "centre" at all: the discrete metric on the reals is entirely homogeneous with no privileged numbers.

• This question had the most depth and concepts to research and learn from, and the other answer was the most pragmatic and clear. I only ticked the other one up because it had 10 approvals, I found this one more mysterious, the theory answer here has the most theoretical information and solutions and the formalized version is great as well, i couldnt decide which is best. – com.prehensible Oct 4 '15 at 2:55
• @comprehensible Absolutely fine - mine is pitched at a higher level, perhaps at someone who has finished two years of a university mathematics course. – Patrick Stevens Oct 4 '15 at 8:29

I found a symmetry line in natural numbers :

                            1 + 2 = 3
4 + 5 + 6 = 7 + 8
9 +10 +11 +12 = 13+14+15
16 +17 +18+19+20 = 21+22+23+24