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

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

#41

Post by afiori »

xigoi wrote: post #33
afiori wrote: post #14

Earlier quoted context omitted.

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.

True, but the point of giving a "value" to 0/0 is to use it somehow.

For example in the context of limits you define a whole lot of number like values like 0+ or 0- that are useful wrt operations on limits.

I was trying to give an example of how ℝ ∪ {Θ} has almost no advantages compared to just ℝ

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

#42

Post by zeroonetwothree »

Lichtso wrote: post #37
iamerroragent wrote: post #23

Earlier quoted context omitted.

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…

Downside is that now 0⋅∞ is undefined so you’ve introduced a new ‘impossibility’

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

#43

Post by ndsipa_pomu »

amitport wrote: post #10
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…

"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?

The most common definition of division being the inverse of multiplication.

if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0/0

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

#44

Post by iamerroragent »

hansvm wrote: post #36
iamerroragent wrote: post #23

Earlier quoted context omitted.

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 additi…

"but it fails to uphold the most important property division should have -- that it undoes multiplication"

I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication.

Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot.

Apologize if that's not clear as to why that is; it's been a while since I read up on those being defined.

However I recommend One, Two, Three: Absolutely Elementary Mathematics by David Berlinski that gives in my opinion pretty good layman understanding of these nuances and number theory.

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

#45

Post by hansvm »

iamerroragent wrote: post #44
hansvm wrote: post #36

Earlier quoted context omitted.

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 additi…

"but it fails to uphold the most important property division should have -- that it undoes multiplication" I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication. Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot. Ap…

Take a look into division rings as a concept. The usual definition for division in rings and fields is via multiplicative inverses for some subset of the nonzero elements. Not all algebraic spaces have division, but that doesn't change what it is, especially from the "number theory, fields, and rings" point of view.

Unless you're talking about some higher-order concept?

Edit: For a bit of completeness, what's happening with the Riemann Sphere is that the algebraic definition is being extended in a way that has some useful analytic, topological, and quality-of-life properties, but which is no longer wholly compatible with the underlying algebra. The algebraic issues are isolated to the extra point at infinity, so they're not terrible to work around, but the operation in question is a proper extension of the underlying algebraic definitions -- much how the gamma function in no way can be defined as multiplication of integers but is a useful extension of the factorials nonetheless.

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

#46

Post by ndsipa_pomu »

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.

Thanks - I was not aware that theorem provers often allow "division" by zero.

Looking at https://xenaproject.wordpress.com/2020/07/05/division-by-zer... I see that they don't use mathematical division, but define a slightly different operator with an additional condition for handling zero. This appears to be far more convenient for theorem provers.

The trade-off would be that "division" is no longer the inverse of multiplication.

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

#47

Post by syzarian »

iamerroragent wrote: post #44
hansvm wrote: post #36

Earlier quoted context omitted.

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 additi…

"but it fails to uphold the most important property division should have -- that it undoes multiplication" I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication. Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot. Ap…

Division is multiplication by the multiplicative inverse. Subtraction is addition by the additive inverse. Both division and subtraction undo their corresponding operation. Multiplying by a (provided it’s not zero) is undone by dividing by a. Adding a is undone by subtracting a.

In a ring the elements form a group under addition and thus every element has an additive inverse. The additive identity element, let’s call it e, has the property that ea = e and ae = e. For this reason we use 0 instead of e. In a nontrivial ring 0 can’t have a multiplicative inverse because if it did then every element would be equal to the multiplicative identity (which is unique).

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

#48

Post by amitport »

ndsipa_pomu wrote: post #43
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?

The most common definition of division being the inverse of multiplication. if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0/0

Thanks, this definition does seem problematic. In any case, it is not the only possible definition and in a/0=c, c does not have to be defined as a real number. We can define it as similarly to complex number with new rules that do not collide with existing reals.

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

#49

Post by brookst »

momentoftop wrote: post #35
xigoi wrote: post #30

Earlier quoted context omitted.

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…

I appreciate the explanation and I’m in no position to disagree, but ugh. Seems like it would work just as well to define x/0 as 6, or e, or -15. I’m sure that’s not the case. But as a long time tech person who’s always considered underflow/overflow to be a hack to get around limitations of hardware, it offends be a bit to find conditionals in abstract math. Undefined seems cleaner, like null, since it implicitly says “don’t treat this as a normal value that you can operate on”.

I suspect the real math people know what they’re doing more than I do, though.

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

#50

Post by ndsipa_pomu »

amitport wrote: post #48
ndsipa_pomu wrote: post #43

Earlier quoted context omitted.

The most common definition of division being the inverse of multiplication. if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0/0

Thanks, this definition does seem problematic. In any case, it is not the only possible definition and in a/0=c, c does not have to be defined as a real number. We can define it as similarly to complex number with new rules that do not collide with existing reals.

There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/Riemann_sphere) which does define division by zero, but sacrifices the numbers forming a field under addition and multiplication.
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