6 ms·

Can another imaginary number do the same thing for division by zero, or am I getting the cause/effect backwards?

deleted 4y ago
[deleted]
vletal 4y ago
Yeah. Great mind teaser. I can imagine an imaginary number (-/+) "inf" for divisions by number approaching zero from left/right, yet the algebra would not be possible to define properly because you can approach zero with a different rate. The undefined cases, where the left/right limits are not equal coulf get imaginary number "shrug" because it would be even less useful. Of is anyone able to define a useful algebra for these? I'm really curious.
jacobolus 4y ago
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.
greendream17 4y ago
No, but there are algebraic structures that allow for this, like the Riemann sphere. The proper way to talk about this is the concept of "zero dividers". For Z the only zero divider is zero, 0/0=0. The closest thing to what you describe is the the dual number (that together with the imaginary and hyperbolic numbers make the geometric numbers), which has zero dividers, and is defined as k^2=0 (where k is not in R). This is a very interesting number and it can help clean up a lot of problems when using complex numbers.
yk 4y ago
No, or at least not in a field. From the field axioms you can directly proof that 0x=(a-a)x = ax - ax =0, and therefore 0 doesn't have a well defined inverse. You can look at other algebraic structure, but those behave less like numbers.
jqgatsby 4y ago
yes, check out Yaglom's "Complex Numbers in Geometry" for a fairly complete treatment of this topic. You can extend the dual numbers and double numbers with their own special infinities, similarly to how the complex numbers can be extended to include a single point at infinity.

deleted 4y ago

[deleted]

Yeah. Great mind teaser. I can imagine an imaginary number (-/+) "inf" for divisions by number approaching zero from left/right, yet the algebra would not be possible to define properly because you can approach zero with a different rate. The undefined cases, where the left/right limits are not equal coulf get imaginary number "shrug" because it would be even less useful. Of is anyone able to define a useful algebra for these? I'm really curious.

jacobolus 4y ago

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.

greendream17 4y ago

No, but there are algebraic structures that allow for this, like the Riemann sphere. The proper way to talk about this is the concept of "zero dividers". For Z the only zero divider is zero, 0/0=0. The closest thing to what you describe is the the dual number (that together with the imaginary and hyperbolic numbers make the geometric numbers), which has zero dividers, and is defined as k^2=0 (where k is not in R). This is a very interesting number and it can help clean up a lot of problems when using complex numbers.

yk 4y ago

No, or at least not in a field. From the field axioms you can directly proof that 0x=(a-a)x = ax - ax =0, and therefore 0 doesn't have a well defined inverse. You can look at other algebraic structure, but those behave less like numbers.

jqgatsby 4y ago

yes, check out Yaglom's "Complex Numbers in Geometry" for a fairly complete treatment of this topic. You can extend the dual numbers and double numbers with their own special infinities, similarly to how the complex numbers can be extended to include a single point at infinity.