I ask because thye seem so slodely related:
-division of a finite number by zero yields infinity
-both are physically unrealizable: in temperature/energy, one cannot reach absolute zero; nor is there any infinte nergy state
-zero marks the transition from negative to positive numbers; is there an analogous role for infinity?
Just wondering
Define “two sides of the same coin.”
Define “infinity,” for that matter.
Because when you ask a mathematician, “Is [something] [something]?” he’s going to go back to the definition and see if the something fits that definition.
But yes, in certain contexts (such as the extended complex plane) there are certain senses in which zero and infinity could reasonably be considered “two sides of the same coin.”
IANAM, but I once expressed similar ideas in the presence of one (a geometry professor):
In geometry, infinity is a real number. It’s not a real number in algebra, but it is one in geometry. In algebra, the point of a circle where the tangent reaches infinity does not exist; in geometry, it bloody well does (this was an important point in a problem I’d just solved using geometry instead of calculus). Infinity is also a point: all infinities are one and the same; the origin is a node, infinity is another. Note that, while for each direction there is only one line that goes through the origin, all the lines in every direction go through infinity, so those two nodes aren’t exactly equivalent.
The geometry professor told my calculus professor (not a mathematician but an engineer) that I was right and me that if I ever got tired of engineering and wanted to study math he’d love to have me in his class. Sadly, anything you do in math has to be expressed in algebraic terms… I’ll have to pass. But that was cool! (That calc prof hated my guts for reasons he’d dreamed up, I found out after my undergrad thesis’ viva)
They are not both unrealizable. It depends what realm you’re talking about. In finance, you can have $0 to your name. You can’t have an infinite amount of dollars to your name. So they are different there.
It’s not true that dividing a finite nonzero number by 0 gives you infinity. It doesn’t have a solution. Your analogies don’t work.
I hope it’s not off-topic to say, but I’ve always marveled at the idea that the universe of numbers lies between 0 and 1 in the sense that 1/any-number yields a fraction between 0 and 1.
If that’s the case, and I’m not asserting such, then there’s a “same side of the coin” going on with 1 and Infinity.
As for the parallel between zero and infinity and the real number scale, there are functions that go from positive infinity to negative infinity at their asymptotes.
However the limit of 1 divided by x as x approaches 0 from the positive side is infinite.
And the limit of 1 divided by x as x approaches infinite is 0.
And as x approaches zero from the negative side, the limit is negative infinity. What’s your point?
The biggest obstacle to regarding infinity as the counterpart to zero is that under the ordinary rules of arithmetic, zero is a number and infinity is not. Zero is one less than one; it’s an integer. Infinity cannot be treated the same way, you cannot add or subtract from it and get a meaningful answer.
I have 0 apples in my left hand (also right, for that matter). I can never have infinitely many in either hand. Now “infinity” covers many concepts not all particularly related. For example, there is the "one point compactification of the natural numbers with consists of 0,1,2,… and an artificially added point at infinity. There is the extended real line with a point at +infinity and another at -infinity. Then there are the cardinal infinities. The first infinite cardinal is the set of the natural numbers. The next infinite cardinal is unknown, but it is consistent with the axioms of set theory that it be assumed to be the set of all subsets of natural numbers. You can take all subsets of that set to give a bigger cardinal, then a bigger, then a bigger and there is no reason to stop there. There are really gigantic cardinals, inaccessible cardinals, measurable cardinals, something more gigantic than all the rest called compact cardinals, and even larger. Then there are infinite ordinals, but therein lies madness, so I will stop. None of these have the slightest resemblance to 0, which is just a plain ordinary number. I might point out that while 0° K is inaccessible, 0° C or 0° F are perfectly ordinary temperature.
Mathematicians: Is Zero and Infinity Two Sides of the Same Coin?
English Majors: Are Zero and Infinity Two Sides of the Same Coin?
If I hit you over the head with nothing, you won’t feel it. If I smack you with infinity you aren’t getting up again.
0° K is not only inaccessible, it doesn’t even exist. 
Moving from MPSIMS to General Questions.
There’s certainly a sense in which it’s true, even if it’s not the only sense in which one might wish to speak of division. For example, in the sense of the natural operation of division upon the projective reals, for example (as Thudlow Boink essentially pointed out as well).
Indeed, quite literally, 0 and infinity are on opposite sides of the circle of projective reals on the standard visualization (where the angle theta corresponds to arctan(theta/2)).
It seems I am somehow in disagreement with all the other mathematicians here except Thudlow Boink. But I think it’s perfectly reasonable to consider 0 and infinity two sides of the same coin in some sense, in some contexts; e.g., they’re reciprocal ratios. And unsigned infinity does, as the OP realizes, mark the complementary transition from positive to negative on the other side of the projective reals from the transition at zero. Really, the only thing I’d take issue with is the OP’s claim that they are both physically unrealizable; that depends, as always in linking mathematics and physics, on what is to count as a physical realization. Certainly, both can be given physical realization: 0 in all the obvious ways, unsigned infinity as, say, the slope of a vertical line. The question is only what interpretations of abstract concepts as descriptions of the physical universe are useful, but I think that’s a useful enough one.
(Shades of the usual argument as to whether sqrt(-1) can be “physically realized”: “No, of course not!”, “Well, look, here’s a 90 degree rotation. Doesn’t that count?” “But you can’t have i many apples!” “Nor can you have sqrt(2) many children, but so what? Complex numbers aren’t supposed to be used to count apples. Different kinds of things describe different kinds of things”. And so on for infinity rather than i.)
IANA mathemematician, but my understanding is that division of a quantity by zero is undefined, not infinity. OTOH, the limit of a quotient as the divisor approaches zero is infinity.
It could be defined. It could be defined as unsigned infinity. Indeed, within some useful formal abstract systems, it is defined as precisely that, and with good reason, including essentially the same reasons that led the OP to suppose so.
People make a big brouhaha about infinity being only a limit of other, more “genuine” quantities. Well, one could say sqrt(2) is only a limit of rational approximations if one likes. But there’s no problem reifying the conceptual limit as a first-class entity in its own right, as seen by the fact that few people say such things about sqrt(2). (One could even say there’s no such thing as a rational number, just pairs of integers manipulated according to certain rules. And that’s both true and meaningless…)
Like anything in maths it depends on defintitions. You might want to construct a mathematical structure that contains the reals and infinity for it’s algebraic or topological qualities, however what you’ll genereally find is that you’ll also lose some of the structure of the real numbers that are considered useful.
Umm, which infinity? There’s a lot of infinities out there. Like an infinite number of them in fact.
Things can’t be two sides of the same coin if there’s one of the first and an infinite number of the other.
And, dude, forget the division by zero thing. Just … forget … it.
Plus, you are thinking rather one-dimensionally. There’s a lot of mathematical entities in two or more dimensions with zero-like properties. (Any of which blows that whole plus/minus uniqueness away.) And you can set up limits of inverses of them that might be infinity-like. So you’re going to need another (infinite) set of coins there.