# Are there two types of negatives? Or why is there no symmetry between + and -? [closed]

Say, we we take `+` and the opposite of `-`, like `+` is going to the right on the x-axis, while `-` is going to the left of the x-axis, everything is fine so far.

Now note they are opposite, and one is taken to be positive, and one is taken to be negative, just like Yin and Yang.

But what if it is `left` and `right` as the Yin and Yang? Consider `(-1) x (-1)`, it gives you `+1`, or `-(-1)`, it also gives you `+1`, but what about `(+1) x (+1)`? Why doesn't it gives you the symmetry of giving you `-1`? The same with `+(+1)`. Why doesn't it give you `-1`? That is, if `+` and `-` are just in two different directions, then, say, you put the chart on the table. The person sitting across the table, will now see the `+` as `-`, and see `-` as `+`. So to me, if `-` `-` can get the opposite: the `+`, why can't `+` `+` get the opposite: the `-`? Because to the person sitting across me at the table, my `-` is his `+`.

This may explain it more: if you say `left` `left` should give `right`, then why `right` `right` will not give `left`? Because if I stand facing north, versus a person facing south, then my `left` is his `right`. Then my `left` `left` giving `right`, to him, is `right` `right`, giving `left`, but to me, it is `right` `right` giving `right` (just like `+(+1)` is `+1`. So why it is not symmetric?

I suppose one easy argument may be, because `-` is to "negate" something and therefore `-` and `+` are different, but this doesn't explain, what about `left` and `right`? Then how can we define what is the `negate`? What I see going left, the other person sees it as right, so if left is negate, then he will say, no, the direction you are pointing at, is "right" and it is not negate. If I say, `that direction` `that direction` (`-` `-`) will give the opposite (`+`), and he will say, no, `that direction` `that direction` (`+` `+`) WILL NOT give the opposite. Why is that?

P.S. I am asking this question because if math is one way of describing properties in the universe, then we hold it true and accurate that `-` `-` gives a `+`, but `+` `+` gives a `+`. On the other hand, we have a philosophy that you can name one direction one thing, be it left or right, left can be right if another person sees it at a different angle, and we can name one direction as positive or name it as negative, and if you were to do calculations, everything still work as expected. So if these two philosophies, or ideas, or notions are true and accurate, why do they seem to contradict each other? (because if you name it `△`, then `△` `△` will give you `▢`. But if you name it `▢`, then `▢` `▢` will give you `▢`. Supposedly, how you name it shouldn't matter.)

## closed as unclear what you're asking by Keelan♦, virmaiorDec 3 '15 at 22:47

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• I vote for reopen. Even when the post mixes a series of different concepts, the core question is clear: Why are +1 and -1 not symmetrical with respect to multiplication? I tried to give an answer. – Jo Wehler Dec 3 '15 at 23:01
• I do agree with virmaior. You will have an easier time understand the (correct) answer of jo Wehler if you do not think of positive and negative as yin and yang. On the surface, both are "opposites," but if you look any deeper than the surface, you find that the relationship between yin and yang is quite entirely unlike the relationship between positive and negative in mathematics. – Cort Ammon Dec 3 '15 at 23:33
• I tried to understand as a quest of knowledge, or as a love of knowledge. Or to understand the universe. I am even trying to not talk too much about math, as the universe can run without a single person talking about math. – 太極者無極而生 Dec 3 '15 at 23:35
• @virmaior I consider the philosophical content of 太極者無極而生 's question comparable to the philosophical content of some of Plato's thoughts about numbers. – Jo Wehler Dec 4 '15 at 0:19
• One approach may be to think of negation as a mirror, switching the chirality of things. If you reflect something once in a mirror, left and right are swapped, but if you reflect it twice, left and right return to their normal state. This can be repeated any number of times (not just 0 1 or 2 reflections), so we can reasonably say that left and right will be swapped after 31 reflections. – Cort Ammon Dec 4 '15 at 0:44

## 1 Answer

The role of +1 and -1 in mathematics is best understood when considering the set of integers including zero. Integers (Z,+,*) form a ring with respect to addition and multiplication. Integers have to distinguished elements:

1. The neutral element of addition is 0:

x + 0 = 0 for all x from Z

Any x has an inverse -x with respect to addition, i.e. x + (-x) = 0. Hence with respect to addition, x and -x are symmetric.

1. The neutral element concerning multiplication is 1:

x * 1 = x for all x from Z, in particular 1 * 1 = 1

The same does not hold for -1. Hence 1 is a distinguished element concerning multiplication, but -1 is not. There is no symmetry between 1 and -1 concerning multiplication.

From 1 + (-1) = 0 follows be multiplying both sides by (-1) and applying elementary rules:

(-1) * 1 + (-1) * (-1) = (-1) * 0

-1 + (-1) * (-1) = 0

(-1) * (-1) = 1, q.e.d.

• I really am not looking for a "proof" that `(-1) x (-1)` is `1`. I am asking why there is no symmetry, as in the PS: "if you name it △, then △ △ will give you ▢. But if you name it ▢, then ▢ ▢ will give you ▢. Supposedly, how you name it shouldn't matter" just like if you name a duck as woofie, the duck is still a duck and will not bark – 太極者無極而生 Dec 10 '15 at 3:12
• @太極者無極而生 You ask why there is no symmetry between +1 and -1? - What about changing the view and asking: Is there any reason to assume that +1 and -1 are symmetrical? Then the answer is: No, there is no reason. Because +1 is a distinguished number concerning multiplication and -1 is not a distiguished number. – Jo Wehler Dec 10 '15 at 5:37