Why don't we define “imaginary” numbers for every “impossibility”? (2012) • Hacker News Board

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#31

Post by contravariant » Mon, Apr 10, 2023, 1:32 PM UTC

ubj wrote: post #4

One interesting case of this is the concept of dual numbers [1], where you have the symbol \epsilon !=0 but (\epsilon)^2 = 0. It seems contradictory, but the resulting theory is very useful for automatic differentiation [2] and for mechanics (dual quaternions) [3]. [1]: https://en.m.wikipedia.org/wiki/Dual_number [2]: https://book.sciml.ai/notes/08-Forward-Mode_Automatic_Differ... [3]: https://en.m.wikipedia.org/wiki…

One thing that is interesting to note is that both dual numbers and imaginary numbers arise as quotient of the polynomial ring.

Complex numbers being equivalent to R[X]/(1+X^2) and dual numbers being equivalent to R[X]/(X^2).

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#32

Post by heinrichhartman » Mon, Apr 10, 2023, 1:34 PM UTC

civilized wrote: post #13

heinrichhartman wrote: post #12

For polynomial equations, the construction works in quite some generality, and is known as quotient ring: https://en.wikipedia.org/wiki/Quotient_ring Given any polynomial P (e.g. x^2 + 1) over a filed F (e.g. reals) we can form: `R = F[X]/P` This is an algebraic "set" that supports addition, substraction, multiplication and has 0,1 but not division in general. Elements are elements of F and a new symbol X that satisf…

Thanks for this comment! Quick note - for clarity and conformity with standard notation, it would be good to have parentheses around the denominators of those ring quotients (in those cases like x^2 - 1 where they contain multiple additive terms).

fixed.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#33

Post by xigoi » Mon, Apr 10, 2023, 1:34 PM UTC

afiori wrote: post #14

amitport wrote: post #10

Earlier quoted context omitted.

"it becomes possible to "prove" that any number is equal to any other number." There are multiple ways to define what division by zero means. Which definition leads to this outcome? How?

let ϴ = 0/0 then 1*ϴ = ϴ = 0/0 = (0*0)/0 = 0*(0/0) = 0*ϴ it follows 1 = 0 and thus x = x * 1 = x * 0 = 0 = y * 0 = y * 1 = y for all x and y

This is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#34

Post by poizan42 » Mon, Apr 10, 2023, 1:49 PM UTC

momentoftop wrote: post #24

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

The Isabelle/HOL theorem prover assigns 0 to x/0 for all x, without contradiction.

It's not making a multiplicative inverse of 0 exist though, it just defines a '/' operator that is slightly different from our usual one (i.e. a/b = a*b^(-1))

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#35

Post by momentoftop » Mon, Apr 10, 2023, 1:54 PM UTC

xigoi wrote: post #30

brookst wrote: post #25

Earlier quoted context omitted.

I believe you but that’s kind of mind blowing. How do they avoid the seemingly-obvious corollary that 0*0 = X, for all values of X? That is, just multiplying both sides of “x/0 = 0” by zero.

By specifying that x/y*y is only equal to x if y≠0, I guess?

Exactly.

Functions in these logics are total, so if you want division to be a function (and you probably do), it has to assign something to division by 0.

It would be acceptable to assign an unspecified object from the domain, for which you have no non-trivial theorems, and so all your real theorems must have a precondition about the denominator being non-zero. But if you specify a candidate like 0, you can get some theorems which don't have the precondition. Consider:

a/b * c/d = ac/bd.

This now holds even if one of b or d is 0.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#36

Post by hansvm » Mon, Apr 10, 2023, 2:08 PM UTC

iamerroragent wrote: post #23

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere

They could have been more precise, but they probably shouldn't have to in the space of a comment. The Riemann Sphere defines a value for the expression x/0, and it's often useful, but it fails to uphold the most important property division should have -- that it undoes multiplication. Division by 0 (with some assumptions about not being in a trivially small space and how those operations behave with respect to addition) does lead to contradictions in that latter sense.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#37

Post by Lichtso » Mon, Apr 10, 2023, 2:10 PM UTC

iamerroragent wrote: post #23

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere

You don't even need the complex part for this. You can do the infinity-projection trick on the real numbers alone as well: https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

A similar trick (point at infinity or ideal point) is used in projective geometry to distinguish between directions (vectors) and places (points) by using coordinates only: https://en.wikipedia.org/wiki/Projective_geometry

But if you actually want to do calculations with infinities and infinitesimals the surreal numbers might be better suited for that: https://en.wikipedia.org/wiki/Surreal_number

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#38

Post by 2muchcoffeeman » Mon, Apr 10, 2023, 2:10 PM UTC

iamerroragent wrote: post #23

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere

This just goes to show that you really have to be careful when slinging out math facts. I've done some under grad maths and the only line on that page that I understand is

"The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1 / 0 = ∞ 1/0=\infty well-behaved."

It clearly does not satisfy a primitive understanding of 1/0.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#39

Post by caf » Mon, Apr 10, 2023, 2:18 PM UTC

aap_ wrote: post #27

rtpg wrote: post #21

Earlier quoted context omitted.

So what did happen?

Square roots of negative numbers came up when solving cubic equations, even if the final solutions were all real. This meant the square root of a negative number was not something nonsensical the way you might claim for x^2 = -1, but actually...real in some sense.

Specifically I believe it involved a geometric construction for solving the cubics, which in some cases could not find a solution unless you allowed a square with "negative area".

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#40

Post by ndsipa_pomu » Mon, Apr 10, 2023, 2:20 PM UTC

iamerroragent wrote: post #23

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere

That's a good example of where defining division by zero leads to interesting maths, but it ends up sacrificing some of the usual rules of arithmetic, so it comes down to a choice of which is more useful in the relevant circumstance.