# Zero = infinity [closed]

I am an artist not a mathematician, but have a science background. My no doubt crude understanding is that different types of Maths systems can be constructed if one is allowed to make certain assumptions or start with certain precepts. Eg normally , using Euclidian aussumptions, parallel lines never meet, but if space is curved then parallel lines meet at infinity.

I regard Art, Science (inc Maths) , philosophy and Religion as 4 main approaches humans take to trying to understand ourselves and the Universe. All approaches can be valuable and may possibly not be in competition but can on occasion usefully connect.

As an artist largely ignorant of Maths, but interested in fractals and also video feedback, I have a question I'd appreciate getting a Mathematicians reactions to: Could a system of Maths be constructed if one starts from the assumption that zero = infinity? If so what would the system be like?

Most probably this is a silly question but if anyone can kick it around a bit, it might be fun. yesterday I read an article in New Scientist about Hawkings et al new theory that the cosmological constant might be negative not positive as has been assumed, which would make Space negatively curved, and help string theory out. Maybe there's a connection there.

Anyway grateful for any thoughts on the subject. If this is the wrong place to ask such a wacky question please suggest where any Mathematicians could entertain it. I looked around but couldn't find anywhere that suited better.

Thanks,

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## closed as off-topic by studiosus, Norbert, Paul, Dennis Gulko, AdrianoNov 6 '13 at 7:41

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is not about mathematics, within the scope defined in the help center." – studiosus, Norbert, Paul, Dennis Gulko
If this question can be reworded to fit the rules in the help center, please edit the question.

You need to define what you mean by zero, and what you mean by infinity. Otherwise this question makes no sense. One example I can think of that might fit your criteria is the tropical semiring, where the addition operation (which is not normal addtion) has identity $\infty$, and since the additive identities are normally denoted $0$, in this case one might say $0=\infty$. –  Alex Becker Jul 10 '12 at 23:45
You need to specify what you mean by $0$ and what you mean by "infinity" as a precursor of saying whether one can begin by the assumption "zero=infinity". If you don't care about what those words denote, then "sure, of course you can". If you want them to "mean" something similar to what they mean in the context of real numbers, then at least one of the two meanings has to "give" in order to be able to start with that and not reach a contradiction. –  Arturo Magidin Jul 10 '12 at 23:46
Mathematical and scientific understanding are very different from the other four. Not to sound belittling or anything, I do arts as well, but "Wouldn't this be cool?" can't be left as a statement on its own like it can in the other three. You need a rigorous starting point. And I have no idea why negatively curved space would imply $0=\infty$. Is there a particular source or idea that caused you to link the two? –  Robert Mastragostino Jul 11 '12 at 0:02
This bounty is preposterous. –  Asaf Karagila Nov 24 '12 at 22:54
A nonsensical question. Should be closed. But the bounty prevents it being closed. A pretty sneaky strategy! –  GEdgar Nov 25 '12 at 1:30

There are some situations in higher mathematics where infinity is zero, sort of.

1. The points on the graph of $y^2={\rm\ a\ cubic\ in\ }x$ form an abelian group when addition is defined by saying that three points add to zero if they are collinear. The zero element of this group is the point at infinity. If you want to learn more about this, the keyphrase is "elliptic curve", but I warn you it won't be easy going for someone untrained in mathematics.

2. In the upper-half-plane model of hyperbolic geometry, the "points at infinity" are the points on the $x$-axis, that is, the points with $y$-coordinate zero.

In both cases, it would be misleading to say, "zero equals infinity". In the first case, the point that acts the way you would expect zero to act is physically located at infinity; in the second case, the points that should be infinitely far away have been brought down to height zero.

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+1 very nicely put. –  robjohn Nov 29 '12 at 22:49

Let's say we agree on the following intuitive definitions of "zero" and "infinity":

A number $x$ is zero if adding any other number $y$ to it leaves $y$ unchanged: $$x + y = y.$$

A number $x$ is infinity if adding any finite number $y$ to it leaves $x$ unchanged (still infinity): $$x + y = x.$$

If $x$ is both zero and infinity, both equations must be true. Therefore for any finite number $y$, $$x = y,$$ and so $x$ also equals all finite numbers.

(Notice that the above argument uses some concepts that weren't rigorously defined, such as "number," "addition," and "finite." But the argument is general, and remains true if you "plug in" different formalizations of these intuitive concepts, so long as equality is transitive.)

One mathematical system where 0 is equal to all finite numbers, and also satisfies the "infinity property," is the field of one element, $\{0\}$.

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On the one hand, the field with one element does not really exist. On the other hand, this may be precisely the sort of mysterious, shadowy object the bounty message asks for. On the foot, I'm not sure it's a good idea to encourage that sort of bounty message... –  Rahul Nov 25 '12 at 1:19
There exists a field with one element. Call the element $0$ and define $0\cdot 0=0$ and $0+0=0$. We satisfy all the field axioms. But as I'm sure you're well aware, we usually require that $1$ and $0$ are distinct for a field. But theoretically, there is nothing wrong axiomatically with the one element field. –  mathematics2x2life Nov 5 '13 at 23:37
@mathematics2x2life It is standard practice to define a field as having at least two elements $0\neq 1$. There are good reasons for this convention. You can drop the axiom if you like, but it makes the theory of fields much messier. –  Daniel Rust Nov 6 '13 at 0:20

The bounty asks for a definitive reference on the relationship between zero and infinity. Such an item was written by Arthur Koestler. Highly recommended, and readily available in English translation.

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.> if $\infty=0$ then $0$ would be the largest possible number and every other number would be less than $0$. $1$ would be the second largest possible number, $2$ would be the $3$rd largest possible number as each number is a value taken from $0$. as you count up. you lose value.