5 ms·

j/0=1

The idea to define i^2=-1 brought up a lot of interesting math. Could it be similarly useful to define a number j so that j/0=1?

OhNoNotAgain_99 4y ago
j + ou dont know
KeyXiote 4y ago
Have been interested in similar abstractions, take a look at contravarient and covariant tensors. Great in depth on Tensors and relational ML databases: (1) https://bjlkeng.github.io/posts/tensors-tensors-tensors/ https://bjlkeng.github.io/posts/tensors-tensors-tensors/
gus_massa 4y ago
In math, it's common to extend the numbers, but the problem is that most of the times you lose important useful properties. For example in quaternions you add i^2=j^2=k^2=-1, https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion but the problem is that ij=-ji, so the multiplication in new set of numbers is not commutative. It's possible to continue extending the numbers adding more fake "roots of -1", but the properties of the operations are even worse. There are a few ways to add infinite to the operations, and add some operations for them, but the extensions breaks the properties of the operations https://en.wikipedia.org/wiki/Cardinal_number https://en.wikipedia.org/wiki/Cardinal_number Physicist use something like what you are proposing, using Taylor and hand waving. I read a few tries to formalize that, but they are just too weird and no one use them. I think the siplest example is https://en.wikipedia.org/wiki/Graded_ring https://en.wikipedia.org/wiki/Graded_ring that you can use to define ϵ and a number is something like 7.9+3.1ϵ+27.4ϵ^2+... So you have that ϵ is almost like 0, and ϵ/ϵ=1, but 7/ϵ is undefined. It look hard to make an extension where ?/0=1. I think it's better to be happy with something similar enough.
eesmith 4y ago
https://en.wikipedia.org/wiki/Division_by_zero https://en.wikipedia.org/wiki/Division_by_zero says > There are mathematical structures in which a/0 is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the Projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms). ... > This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, ∞ + ∞ is undefined in this extension of the real line. ... > Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.
11235813213455 4y ago
Are there some function f such that f(x) / x tends to 1 as x tends toward 0?
- greydius 4y ago
  sin(x)/x
  - 11235813213455 4y ago
    oh yea, this or just x/x
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j/0=1

The idea to define i^2=-1 brought up a lot of interesting math. Could it be similarly useful to define a number j so that j/0=1?

OhNoNotAgain_99 4y ago

j + ou dont know

KeyXiote 4y ago

Have been interested in similar abstractions, take a look at contravarient and covariant tensors. Great in depth on Tensors and relational ML databases: (1) https://bjlkeng.github.io/posts/tensors-tensors-tensors/ https://bjlkeng.github.io/posts/tensors-tensors-tensors/

gus_massa 4y ago

In math, it's common to extend the numbers, but the problem is that most of the times you lose important useful properties. For example in quaternions you add i^2=j^2=k^2=-1, https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion but the problem is that ij=-ji, so the multiplication in new set of numbers is not commutative. It's possible to continue extending the numbers adding more fake "roots of -1", but the properties of the operations are even worse. There are a few ways to add infinite to the operations, and add some operations for them, but the extensions breaks the properties of the operations https://en.wikipedia.org/wiki/Cardinal_number https://en.wikipedia.org/wiki/Cardinal_number Physicist use something like what you are proposing, using Taylor and hand waving. I read a few tries to formalize that, but they are just too weird and no one use them. I think the siplest example is https://en.wikipedia.org/wiki/Graded_ring https://en.wikipedia.org/wiki/Graded_ring that you can use to define ϵ and a number is something like 7.9+3.1ϵ+27.4ϵ^2+... So you have that ϵ is almost like 0, and ϵ/ϵ=1, but 7/ϵ is undefined. It look hard to make an extension where ?/0=1. I think it's better to be happy with something similar enough.

eesmith 4y ago

https://en.wikipedia.org/wiki/Division_by_zero https://en.wikipedia.org/wiki/Division_by_zero says > There are mathematical structures in which a/0 is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the Projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms). ... > This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, ∞ + ∞ is undefined in this extension of the real line. ... > Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.

11235813213455 4y ago

Are there some function f such that f(x) / x tends to 1 as x tends toward 0?

greydius 4y ago
sin(x)/x
- 11235813213455 4y ago
  oh yea, this or just x/x

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