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You already can’t maintain the identity (a/b)b = a when you have a zero element, except by declaring "division by zero is illegal". Adding ∞ is not fundamentall

You already can’t maintain the identity (a/b)b = a when you have a zero element, except by declaring "division by zero is illegal". Adding ∞ is not fundamentally different, but just takes a slightly modified list of exceptions. In many contexts this is worth the hassle.

lisper 4y ago
Well, yeah, but the whole point of inventing a new kind of number to represent a/0 analogous to inventing a new kind of number to represent sqrt(-1) is to eliminate these you-are-not-allowed-to-do-that kind of exception. With sqrt(-1) it turns out to be straightforward. With a/0 it isn't, which is why sqrt(-1) is a thing in math and a/0 isn't.
- jacobolus 4y ago
  For me, the whole point of modifying the number system is modeling different kinds of structures. Both complex numbers and projective numbers are interesting in their own right and also useful for making models of both physical reality and other kinds of mathematical patterns. Each system has its own internal logic with its own quirks.
- yucky 4y ago
  >to eliminate these you-are-not-allowed-to-do-that kind of exception. I'm not a mathematician by any stretch, but I never understood the inability to divide by zero to be some arbitrary rule, rather that it's just simply not possible to proof (because you can't multiply any number by 0 to get a non-zero answer). Then again I tapped out after Calculus so maybe that's just ignorance on my part.
  - shadowofneptune 4y ago
    How can something be divided by nothing? If that would give the same result as division by a whole, does that mean that nothing is something? Issues like these are why the concept of zero had trouble gaining acceptance.
    - t-3 4y ago
      We can divide by negatives though? They are just as unreal as zeros, so the only difference seems to be that division by zero is undefined, while division by negatives is well-defined. I think the real difference is that dividing by zero is not useful, so it has never needed defining.
      - wruza 4y ago
        Yet another not a mathematician here. Division by zero never converges (as a limit of any generating process, numerical or symbolic) in contrast to division by anything else. There is no amount of zeroes that can add up to a whole number, by definition of zero. Not an infinite amount, just “no” amount. If you want to handle infinities as numbers, you have to maintain a “progress” somehow, or you end up comparing different stages of processes and get something like 2=1. Much easier to not dig a pit to fall into.
      - SideburnsOfDoom 4y ago
        > I think the real difference is that dividing by zero is not useful, so it has never needed defining. Incorrect. The result of dividing by zero is provably "not well defined" - for standard mathematical definition of what is and what isn't "well defined". Specifically, Limits (1). for 1/x, as the x approaches zero from above and below, the two limits are not the same, therefor the limit is not well-defined (2) As x tends to zero from x > 0, 1/x tends to a limit of +infinity. Calculate 1/x for values of x = 2, 1, 0.5, 0.1, 00.1 and so on as you approach zero: the results are larger and larger positive numbers, without bound. As x tends to zero from x < 0, 1/x tends to a limit of -infinity. Calculate 1/x for values of x = -2, -1, -0.5, -0.1, -00.1 and so on as you approach zero: the results are larger and larger negative numbers, without bound. Even allowing "infinity" as a valid number, this isn't converging to the same answer. (2) I suppose that you could "define" the answer in much the same way that we define i as "the imaginary square root of -1" and thus get complex numbers. The question that would follow though, is: If we e.g. define "q" as the limit of 1/x as x -> 0, this is a number that is both positively and negatively infinite at the same time, what then? What can we do with it? It turns out that complex numbers are useful (3) and look a bit like quarter-circle rotations. But this q, the "self-contradicting infinite number" doesn't really seem to be useful. 1) https://byjus.com/maths/limits/ https://byjus.com/maths/limits/ 2) https://www.mathway.com/popular-problems/Algebra/200474 https://www.mathway.com/popular-problems/Algebra/200474 3) https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755 https://issuu.com/harrowhongkong/docs/final_scientific_harro...
  - Scarblac 4y ago
    Say you have one apple and want to divide it over zero people. It can't be done - give everyone one, fifty, a million, infinity apples: you always have one apple left over.
    - jacobolus 4y ago
      The concept of "division" or "ratio" has different interpretations depending on context. The Euclidean (from The Elements) concept of ratio can certainly admit a 1:0 ratio, which we can write as 1/0 or ∞ if we like, and which is equivalent to the ratio of x:0 for any non-zero number x. There’s no inherent reason why the ratio of "apples per person" should be allowed to be 0:1 but not 1:0. From first principles it’s just as problematic to divide 0 apples among x people as to divide x apples among 0 people. When we e.g. write the values of trigonometric functions, these should ordinarily be interpreted to be among the extended reals, so that tan(π/2) = 1/0 = ∞ (i.e. the ratio 1:0) is entirely reasonable.
      - SideburnsOfDoom 4y ago
        Is that +∞ or -∞ though? The problem with division by zero is not that it's infinite, it is that it limit could be either +∞ or -∞ depending on which side of the limit you look at. Therefor, it is not well-defined.
        jacobolus 4y ago
        In some contexts +∞ = –∞ (e.g. tan ½π = tan –½π; these both represent the orientation of a line at a right angle). In other contexts +∞ ≠ –∞ (e.g. log ±0 ≠ log ±∞; these quantities can be thought of as the two opposite ends of an infinite cylinder)
        SideburnsOfDoom 4y ago
        Well, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?

lisper 4y ago

Well, yeah, but the whole point of inventing a new kind of number to represent a/0 analogous to inventing a new kind of number to represent sqrt(-1) is to eliminate these you-are-not-allowed-to-do-that kind of exception. With sqrt(-1) it turns out to be straightforward. With a/0 it isn't, which is why sqrt(-1) is a thing in math and a/0 isn't.

jacobolus 4y ago
For me, the whole point of modifying the number system is modeling different kinds of structures. Both complex numbers and projective numbers are interesting in their own right and also useful for making models of both physical reality and other kinds of mathematical patterns. Each system has its own internal logic with its own quirks.
yucky 4y ago
>to eliminate these you-are-not-allowed-to-do-that kind of exception. I'm not a mathematician by any stretch, but I never understood the inability to divide by zero to be some arbitrary rule, rather that it's just simply not possible to proof (because you can't multiply any number by 0 to get a non-zero answer). Then again I tapped out after Calculus so maybe that's just ignorance on my part.
- shadowofneptune 4y ago
  How can something be divided by nothing? If that would give the same result as division by a whole, does that mean that nothing is something? Issues like these are why the concept of zero had trouble gaining acceptance.
  - t-3 4y ago
    We can divide by negatives though? They are just as unreal as zeros, so the only difference seems to be that division by zero is undefined, while division by negatives is well-defined. I think the real difference is that dividing by zero is not useful, so it has never needed defining.
    - wruza 4y ago
      Yet another not a mathematician here. Division by zero never converges (as a limit of any generating process, numerical or symbolic) in contrast to division by anything else. There is no amount of zeroes that can add up to a whole number, by definition of zero. Not an infinite amount, just “no” amount. If you want to handle infinities as numbers, you have to maintain a “progress” somehow, or you end up comparing different stages of processes and get something like 2=1. Much easier to not dig a pit to fall into.
    - SideburnsOfDoom 4y ago
      > I think the real difference is that dividing by zero is not useful, so it has never needed defining. Incorrect. The result of dividing by zero is provably "not well defined" - for standard mathematical definition of what is and what isn't "well defined". Specifically, Limits (1). for 1/x, as the x approaches zero from above and below, the two limits are not the same, therefor the limit is not well-defined (2) As x tends to zero from x > 0, 1/x tends to a limit of +infinity. Calculate 1/x for values of x = 2, 1, 0.5, 0.1, 00.1 and so on as you approach zero: the results are larger and larger positive numbers, without bound. As x tends to zero from x < 0, 1/x tends to a limit of -infinity. Calculate 1/x for values of x = -2, -1, -0.5, -0.1, -00.1 and so on as you approach zero: the results are larger and larger negative numbers, without bound. Even allowing "infinity" as a valid number, this isn't converging to the same answer. (2) I suppose that you could "define" the answer in much the same way that we define i as "the imaginary square root of -1" and thus get complex numbers. The question that would follow though, is: If we e.g. define "q" as the limit of 1/x as x -> 0, this is a number that is both positively and negatively infinite at the same time, what then? What can we do with it? It turns out that complex numbers are useful (3) and look a bit like quarter-circle rotations. But this q, the "self-contradicting infinite number" doesn't really seem to be useful. 1) https://byjus.com/maths/limits/ https://byjus.com/maths/limits/ 2) https://www.mathway.com/popular-problems/Algebra/200474 https://www.mathway.com/popular-problems/Algebra/200474 3) https://issuu.com/harrowhongkong/docs/final_scientific_harrovian_issue_vi-i/s/11488755 https://issuu.com/harrowhongkong/docs/final_scientific_harro...
- Scarblac 4y ago
  Say you have one apple and want to divide it over zero people. It can't be done - give everyone one, fifty, a million, infinity apples: you always have one apple left over.
  - jacobolus 4y ago
    The concept of "division" or "ratio" has different interpretations depending on context. The Euclidean (from The Elements) concept of ratio can certainly admit a 1:0 ratio, which we can write as 1/0 or ∞ if we like, and which is equivalent to the ratio of x:0 for any non-zero number x. There’s no inherent reason why the ratio of "apples per person" should be allowed to be 0:1 but not 1:0. From first principles it’s just as problematic to divide 0 apples among x people as to divide x apples among 0 people. When we e.g. write the values of trigonometric functions, these should ordinarily be interpreted to be among the extended reals, so that tan(π/2) = 1/0 = ∞ (i.e. the ratio 1:0) is entirely reasonable.
    - SideburnsOfDoom 4y ago
      Is that +∞ or -∞ though? The problem with division by zero is not that it's infinite, it is that it limit could be either +∞ or -∞ depending on which side of the limit you look at. Therefor, it is not well-defined.
      - jacobolus 4y ago
        In some contexts +∞ = –∞ (e.g. tan ½π = tan –½π; these both represent the orientation of a line at a right angle). In other contexts +∞ ≠ –∞ (e.g. log ±0 ≠ log ±∞; these quantities can be thought of as the two opposite ends of an infinite cylinder)
        SideburnsOfDoom 4y ago
        Well, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?