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> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these repre

> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1), was there ever an attempt to do math with something like y=1/0 or another illegal construct?

msla 8y ago
> was there ever an attempt to do math with something like y=1/0 or another illegal construct? Wheels are a type of algebra where division by 0 is defined. https://en.wikipedia.org/wiki/Wheel_theory https://en.wikipedia.org/wiki/Wheel_theory https://math.stackexchange.com/questions/994508/wheel-theory-extended-reals-limits-and-nullity-can-dne-limits-be-made-to https://math.stackexchange.com/questions/994508/wheel-theory... To call wheels "obscure" is to make them out to be better-known than they are, however.
- cobbzilla 8y ago
  I had not heard of Wheel theory, quite fascinating. But eek! From the wikipedia article: 0x ≠ 0 in the general case x − x ≠ 0 in the general case Like, you get the ability to divide by zero, but at what cost? Multiplying by zero is also not zero! Seems bonkers. Are there any practical uses for this kind of algebra?
  - ajkjk 8y ago
    Likely no (the peculiar distributive laws seem unusable to me). But I have come across similar rules that I think may have a real usage. It sometimes seems to me that it might make more sense to keep 'factors' attached to 0s, such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of 0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc. If you keep factors like this, then you could implement L'Hopital's rule without the result only being true if considered under a limit: say, lim(x->0) (5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3. This is not like Wheels though; it requires that any power of 0 be a distinct number. I of course have no idea if it is sound or meaningful, but I do find myself thinking about it a lot.
    - hopler 8y ago
      Those "factor" zeros look like infinitesimal forms, which are well studied.
      - ajkjk 8y ago
        There is a similarity, but no one claims infinitesimals are _literally 0_, and certainly no one would claim that 2-2 = 2 epsilon.
roywiggins 8y ago
Surreals and Hyperreals have lots of "extra" numbers that don't exist in the reals. They're a bit niche though.
empath75 8y ago
You can consider almost anything a number if you can do some internally consistent operations with it. Hamilton attempted a theory of triplets before setting on 4d quaternions, so that would be one example of a failed number system off the top of my head.
btilly 8y ago
The original version of Calculus by Newton used "fluxions". That doesn't correspond to any number system that we use today. Leibniz's (re?)invention of Calculus used infinitesmals. Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today. (Yes, yes, something else called infinitesmals do show up in nonstandard analysis and the notation deliberately looks the same. But the underlying concepts are more..complicated.)
- msla 8y ago
  > But the underlying concepts are more..complicated. Can you expand on this?
  - btilly 8y ago
    Can you expand on this? Yes, maybe more than you want. :-) Abraham Robinson's version of non-standard analysis went like this. You start with the standard model of the real numbers and associated concepts like sets, functions, and so on. Using something called the ultrafilter construction, you construct a new, larger model of the real numbers+stuff, and a mapping from the standard model to the nonstandard model. So we have nonstandard numbers, nonstandard sets, nonstandard functions and so on. Some of which are just mappings of the standard ones, and others of which are new. Thanks to something called the transfer principle, all statements in first order logic about things involving real numbers remain true about the nonstandard versions of the same. In other words within the nonstandard model, the world looks the same as within the standard model. But there is a key feature. From the construction we know that the nonstandard model has numbers in it that are closer to 0 than any real number except 0. That set is the infinitesmals. The infinitesmals are a set identifiable from the construction, but are NOT a nonstandard set. And 1/infinitesmal gives you infinite numbers whose absolute value is larger than any standard real. Now here is the point of the whole construction. Suppose that (f(x + dx) - f(x))/dx only varies by an infinitesmal from a non-standard version of a real across all possible infinitesmals. Then it turns out that that real is the derivative in the usual notation. Similarly if a Riemann sum that breaks an area into N pieces always gives the same answer to within an infinitesmal for all infinite integers N, then it turns out that the function is Riemann integral, and that answer is the actual integral. And with these two insights, most of the handwavy arguments that people used to use can be rescued. Some mathematicians have found that the infinitesmal notation helps them think through problems, and a some important new theorems were proven using this approach. However those proofs can be translated back to more standard notation. Many students intuitively prefer the infinitesmal notation over limits. But when you unpack it, does it make sense to invoke the axiom of choice to define the derivative? (The ultrafilter construction uses the axiom of choice. Alternatives exist, but they use complex model theoretic mechanics under the hood. Really understanding them requires a lot of machinery.)
andrewla 8y ago
Infinitesimals [1][2][3] are probably the best example. Though there's a whole range of theories, the basic version is just a magic number, d, such that d^2 == 0. So (x+d)^3 = x^3 + 3dx^2 + 0, giving us a more "natural" path to get to the notion of derivatives without resorting to limits. They get hard to manage, though, so they haven't really made it into the mainstream. [1] https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis [2] https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis [3] https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number
DRMacIver 8y ago
I think mathematicians have mostly moved past the idea of fighting over whether something is or isn't a number. There are lots of interesting number systems that never made it into widespread use in the same way that the "core" number systems did, and there are lots that see niche usage, but people are pretty relaxed about whether they are "actually" numbers - they might or might not typicaly be referred to as numbers, but the strongest negative answer to "Is this a number?" you'd typically see would probably be to shrug and say "sure I guess, if you like?" One example of previous attempts which we'd now consider to be invalid is a lot of operations on infinite series. In early days of analysis you'd get people concluding things like "1 - 1 + 1 - 1 + ... = 1/2", which gets you into more hot water the more you look at it. The problem here isn't that they are philosophically unsound per se - you can define all sorts of notions of "infinite sum" that make this work, like cesaro summation - but they don't behave as nicely as people intuitively expected them to and the naive versions of them don't really work.

msla 8y ago

> was there ever an attempt to do math with something like y=1/0 or another illegal construct? Wheels are a type of algebra where division by 0 is defined. https://en.wikipedia.org/wiki/Wheel_theory https://en.wikipedia.org/wiki/Wheel_theory https://math.stackexchange.com/questions/994508/wheel-theory-extended-reals-limits-and-nullity-can-dne-limits-be-made-to https://math.stackexchange.com/questions/994508/wheel-theory... To call wheels "obscure" is to make them out to be better-known than they are, however.

cobbzilla 8y ago
I had not heard of Wheel theory, quite fascinating. But eek! From the wikipedia article: 0x ≠ 0 in the general case x − x ≠ 0 in the general case Like, you get the ability to divide by zero, but at what cost? Multiplying by zero is also not zero! Seems bonkers. Are there any practical uses for this kind of algebra?
- ajkjk 8y ago
  Likely no (the peculiar distributive laws seem unusable to me). But I have come across similar rules that I think may have a real usage. It sometimes seems to me that it might make more sense to keep 'factors' attached to 0s, such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of 0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc. If you keep factors like this, then you could implement L'Hopital's rule without the result only being true if considered under a limit: say, lim(x->0) (5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3. This is not like Wheels though; it requires that any power of 0 be a distinct number. I of course have no idea if it is sound or meaningful, but I do find myself thinking about it a lot.
  - hopler 8y ago
    Those "factor" zeros look like infinitesimal forms, which are well studied.
    - ajkjk 8y ago
      There is a similarity, but no one claims infinitesimals are _literally 0_, and certainly no one would claim that 2-2 = 2 epsilon.

roywiggins 8y ago

Surreals and Hyperreals have lots of "extra" numbers that don't exist in the reals. They're a bit niche though.

empath75 8y ago

You can consider almost anything a number if you can do some internally consistent operations with it. Hamilton attempted a theory of triplets before setting on 4d quaternions, so that would be one example of a failed number system off the top of my head.

btilly 8y ago

The original version of Calculus by Newton used "fluxions". That doesn't correspond to any number system that we use today. Leibniz's (re?)invention of Calculus used infinitesmals. Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today. (Yes, yes, something else called infinitesmals do show up in nonstandard analysis and the notation deliberately looks the same. But the underlying concepts are more..complicated.)

msla 8y ago
> But the underlying concepts are more..complicated. Can you expand on this?
- btilly 8y ago
  Can you expand on this? Yes, maybe more than you want. :-) Abraham Robinson's version of non-standard analysis went like this. You start with the standard model of the real numbers and associated concepts like sets, functions, and so on. Using something called the ultrafilter construction, you construct a new, larger model of the real numbers+stuff, and a mapping from the standard model to the nonstandard model. So we have nonstandard numbers, nonstandard sets, nonstandard functions and so on. Some of which are just mappings of the standard ones, and others of which are new. Thanks to something called the transfer principle, all statements in first order logic about things involving real numbers remain true about the nonstandard versions of the same. In other words within the nonstandard model, the world looks the same as within the standard model. But there is a key feature. From the construction we know that the nonstandard model has numbers in it that are closer to 0 than any real number except 0. That set is the infinitesmals. The infinitesmals are a set identifiable from the construction, but are NOT a nonstandard set. And 1/infinitesmal gives you infinite numbers whose absolute value is larger than any standard real. Now here is the point of the whole construction. Suppose that (f(x + dx) - f(x))/dx only varies by an infinitesmal from a non-standard version of a real across all possible infinitesmals. Then it turns out that that real is the derivative in the usual notation. Similarly if a Riemann sum that breaks an area into N pieces always gives the same answer to within an infinitesmal for all infinite integers N, then it turns out that the function is Riemann integral, and that answer is the actual integral. And with these two insights, most of the handwavy arguments that people used to use can be rescued. Some mathematicians have found that the infinitesmal notation helps them think through problems, and a some important new theorems were proven using this approach. However those proofs can be translated back to more standard notation. Many students intuitively prefer the infinitesmal notation over limits. But when you unpack it, does it make sense to invoke the axiom of choice to define the derivative? (The ultrafilter construction uses the axiom of choice. Alternatives exist, but they use complex model theoretic mechanics under the hood. Really understanding them requires a lot of machinery.)

andrewla 8y ago

Infinitesimals [1][2][3] are probably the best example. Though there's a whole range of theories, the basic version is just a magic number, d, such that d^2 == 0. So (x+d)^3 = x^3 + 3dx^2 + 0, giving us a more "natural" path to get to the notion of derivatives without resorting to limits. They get hard to manage, though, so they haven't really made it into the mainstream. [1] https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis [2] https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis [3] https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number

DRMacIver 8y ago

I think mathematicians have mostly moved past the idea of fighting over whether something is or isn't a number. There are lots of interesting number systems that never made it into widespread use in the same way that the "core" number systems did, and there are lots that see niche usage, but people are pretty relaxed about whether they are "actually" numbers - they might or might not typicaly be referred to as numbers, but the strongest negative answer to "Is this a number?" you'd typically see would probably be to shrug and say "sure I guess, if you like?" One example of previous attempts which we'd now consider to be invalid is a lot of operations on infinite series. In early days of analysis you'd get people concluding things like "1 - 1 + 1 - 1 + ... = 1/2", which gets you into more hot water the more you look at it. The problem here isn't that they are philosophically unsound per se - you can define all sorts of notions of "infinite sum" that make this work, like cesaro summation - but they don't behave as nicely as people intuitively expected them to and the naive versions of them don't really work.