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You can add a non-zero number which squares to zero to your number system [edit: but this doesn't let you divide by zero]. This results in the "dual numbers" an

You can add a non-zero number which squares to zero to your number system [edit: but this doesn't let you divide by zero]. This results in the "dual numbers" and is practically useful for "automatic differentiation", where we represent quantities in the form x + x'ε, and have the rule that ε² = 0. https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers https://en.wikipedia.org/wiki/Automatic_differentiation#Auto... Quantities which square to zero are also implicit in spacetime. A "lightlike" vector which has equal displacements in space and time between two spacetime “events” (e.g. the displacement between two points along the path of a photon) has a squared length of 0, compared to "timelike" vectors with negative squared length and "spacelike" vectors with positive squared length. (conventions about signs vary from source to source) https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval * * * In other contexts it makes sense to define 1/0 to be the quantity ∞. There are two relevant models here, with different practical applications. One model where we add a single number ∞ which is also equal to –∞, and makes the number line "wrap around" into a circle. Another model where we add two separate numbers +∞ and –∞ at the two ends of the number line. https://en.wikipedia.org/wiki/Projectively_extended_real_line https://en.wikipedia.org/wiki/Projectively_extended_real_lin... https://en.wikipedia.org/wiki/Extended_real_number_line https://en.wikipedia.org/wiki/Extended_real_number_line

lisper 4y ago
The problem with any finite number of infinities is that you can't maintain the identity (a/b)*b = a. To make that work you need a different number to represent a/0 for every value of a, including all of your infinities, so you end up with an infinite hierarchy of infinities.
- jacobolus 4y ago
  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.
kragen 4y ago
Adding some kind of infinitesimals to the reals can be useful, as is done with hyperreal numbers, dual numbers, and other kinds of hypercomplex numbers, but it doesn't allow you to divide by zero; there is no hyperreal or hypercomplex number x that solves x = 1/0. As I understand it, if you add 1/0 = ∞ you can't treat that ∞ as a quantity in the usual algebraic ways. In particular you can't multiply ∞ by 0 and get back some well-defined quantity such as 1; if you allow that, you quickly find that you can prove that all finite numbers are equal. The standard tricky example of this is in https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.h...: Let a = b. Then a² = ab, a² + a² = a² + ab, 2a² = a² + ab, 2a² - 2ab = a² + ab - 2ab = a² - ab = 2(a² - ab) = 1(a² - ab). If we then divide both of these last expressions by a² - ab we get 2 = 1. This is only invalid because a² - ab = 0. Adding a different ∞ₙ for each value of n/0, as lisper suggests, doesn't help. So, if you want to extend your number system with 1/0 = ∞, you either need to throw out some of the standard laws that permit algebraic manipulations like that, or you end up with all finite quantities being equal.
- LadyCailin 4y ago
  I am not at all a mathematician, but I just can’t wrap my head around how 1/0 isn’t infinity. 1/1=1, 1/0.5=2, 1/0.25=4, etc. The closer we get to 0, the larger the number. I just can’t accept that if we actually get to 0 in our progression, it is anything besides infinity. Perhaps it becomes its own class of number, like what happened with imaginary numbers. That is, infinity times zero is 1n (where n is a bookkeeping marker like i in the square root of negative numbers), or like with limits, where you have a “+ C” in there.
  - kragen 4y ago
    Sure, I definitely agree that 1/0 doesn't give you a finite quantity. What I'm saying is that it doesn't give you a number at all unless you're willing to sacrifice major parts of algebra. And I'm not much of a mathematician either, so I might have gotten something wrong.
  - codeismath 4y ago
    Okay, but then you'd show that 1/-1=-1, 1/-0.5=-2, 1/-0.25=-4, etc. The closer you get to 0, the more negative the number. Then you just wouldn't accept that if you actually got to 0 in your progression, it is anything besides negative infinity. And then you've shown that negative infinity is equal to infinity.

lisper 4y ago

The problem with any finite number of infinities is that you can't maintain the identity (a/b)*b = a. To make that work you need a different number to represent a/0 for every value of a, including all of your infinities, so you end up with an infinite hierarchy of infinities.

jacobolus 4y ago
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.

kragen 4y ago

Adding some kind of infinitesimals to the reals can be useful, as is done with hyperreal numbers, dual numbers, and other kinds of hypercomplex numbers, but it doesn't allow you to divide by zero; there is no hyperreal or hypercomplex number x that solves x = 1/0. As I understand it, if you add 1/0 = ∞ you can't treat that ∞ as a quantity in the usual algebraic ways. In particular you can't multiply ∞ by 0 and get back some well-defined quantity such as 1; if you allow that, you quickly find that you can prove that all finite numbers are equal. The standard tricky example of this is in https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.h...: Let a = b. Then a² = ab, a² + a² = a² + ab, 2a² = a² + ab, 2a² - 2ab = a² + ab - 2ab = a² - ab = 2(a² - ab) = 1(a² - ab). If we then divide both of these last expressions by a² - ab we get 2 = 1. This is only invalid because a² - ab = 0. Adding a different ∞ₙ for each value of n/0, as lisper suggests, doesn't help. So, if you want to extend your number system with 1/0 = ∞, you either need to throw out some of the standard laws that permit algebraic manipulations like that, or you end up with all finite quantities being equal.

LadyCailin 4y ago
I am not at all a mathematician, but I just can’t wrap my head around how 1/0 isn’t infinity. 1/1=1, 1/0.5=2, 1/0.25=4, etc. The closer we get to 0, the larger the number. I just can’t accept that if we actually get to 0 in our progression, it is anything besides infinity. Perhaps it becomes its own class of number, like what happened with imaginary numbers. That is, infinity times zero is 1n (where n is a bookkeeping marker like i in the square root of negative numbers), or like with limits, where you have a “+ C” in there.
- kragen 4y ago
  Sure, I definitely agree that 1/0 doesn't give you a finite quantity. What I'm saying is that it doesn't give you a number at all unless you're willing to sacrifice major parts of algebra. And I'm not much of a mathematician either, so I might have gotten something wrong.
- codeismath 4y ago
  Okay, but then you'd show that 1/-1=-1, 1/-0.5=-2, 1/-0.25=-4, etc. The closer you get to 0, the more negative the number. Then you just wouldn't accept that if you actually got to 0 in your progression, it is anything besides negative infinity. And then you've shown that negative infinity is equal to infinity.