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number that, when added to the original number, yields zero
My girlfriend and I have discussed this a few times and can't quite decide if there is or not - we are definitely not philosophers.
Initially, we agree that zero is similar nothing and the opposite of nothing is everything. Therefore, infinity is the opposite of zero. But, looking at it numerically, you have negative (or a lack of) values. You could also say that a circumstantial "lack of" anything can be treated at a negative value.
But does this then mean that you could potentially say the opposite of infinity is actually negative infinity? Does this approach not put zero as the only non-value therefore meaning that it isn't even comparable?
At this point it gets a little deep for us, but our only other conclusion is that the opposite of 0 is actually 0. If this is the case, then is this the only example of this?
If by "opposite" you mean "additive inverse" (as -1 is to 1, -2 is to 2, etc.), then 0 is its own opposite.
Initially, we agree that zero is similar nothing and the opposite of nothing is everything. Therefore, infinity is the opposite of zero.
I don't think there's any obvious sense in which infinity corresponds to "everything". There are infinite sets that don't contain everything, for example.
But does this then mean that you could potentially say the opposite of infinity is actually negative infinity?
You could say this is true about the surreal numbers:
https://en.wikipedia.org/wiki/Surreal_number
At this point it gets a little deep for us, but our only other conclusion is that the opposite of 0 is actually 0. If this is the case, then is this the only example of this?
As I said above, if by "opposite" you specifically mean "additive inverse", then 0 is its own opposite, and it's the only real number that has this property.
I'd say "opposite" without further qualification or context is not really a well defined operation. Generally, by "opposite", I think we generally mean something like "reachable by reflecting on an axis of symmetry". But things frequently have multiple symmetries, meaning multiple potential opposites. Eg. we may consider the opposite of 2 to be -2 - if we reflect it through the origin, we get that value. But we might also consider the opposite to be 1/2 - it's multiplicative, rather than additive inverse. These symmetries may be context dependent - we could have different "origin points" that we're reflecting between. Eg. we might view "blue" to be the opposite of "red" if we pick the midpoint of human visible colours as our "point of reflection". But a scientist not attaching any special significance to visible light wouldn't see any reason to view wavelengths of 450nm to be the opposite of wavelengths of 620 nm? But we might also have other, culturally influenced reflection points (eg. "blue" may again be considered the opposite of "red" when viewed through the lens of "colours symbolising particular political viewpoints"). All in all, to speak of "the opposite" of something is to commit an error, in preassuming there could only be one such thing. That may be the case when you have a particular symmetry in mind, but you need to realise that this is what you're doing, and to discard that context is to change the question.
Ie opposites are properties of not just a thing on its own, but of that thing, a type of symmetry, and an axis of symmetry. Change any of those things, and you get different opposites.
In your case, you're looking at 0 using a particular "point of reflection" and symmetry in mind, but in that particular symmetry your point of reflection itself is the same as the point you're "reflecting", meaning 0 would indeed be its own opposite along that same additive symmetry. You can of course bring up different symmetries - but all those symmetries give rise to different "opposites" all the time - they're not just fallbacks for when one symmetry doesn't give you a good answer, making the other one the "real" opposite.
(And as an aside, in IEEE floating point maths, there are actually distinct 0 and -0 values, so in that context, the (additive) opposite of 0 would indeed be -0).