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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)

#51

Post by iamerroragent »

syzarian wrote: post #47
iamerroragent wrote: post #44

Earlier quoted context omitted.

"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…

Okay so if you can get to a ring without a multiplicative inverse and then applying that operation to the ring forms it into a field then wouldn't it be fair to say that division is not really the opposite of multiplication the same way that subtraction absolutely is for addition?

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

#52

Post by momentoftop »

brookst wrote: post #49
momentoftop wrote: post #35

Earlier quoted context omitted.

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 say…

The theorem a/b * c/d = ac/bd doesn't hold if x/0 = 6, though.

The theorem prover HOL Light is a close cousin of Isabelle/HOL and doesn't adopt this, and just says that x/0 is some unspecified number. You can't prove much interesting about it. You can prove, say, that x/0 * 0 = 0, but you can't prove whether or not x/0 is, say, positive or not.

If you prefer null, there was a logic that allowed for undefined terms and partial functions that became the basis of the IMPS theorem prover. I found it most notable for the fact that it doesn't have reflexivity of equality: 1/0 = 1/0 is false in IMPS.

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

#53

Post by momentoftop »

ndsipa_pomu wrote: post #46
momentoftop wrote: post #24

Earlier quoted context omitted.

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 mu…

Ah, thanks for the link. I suggested the reason that Isabelle/HOL does this is because it requires total functions and you don't have a convenient way to do refinement types. But that's not an adequate explanation, because Lean does allow such refinements, but it still turns out to be inconvenient for division.

I will note that setting a - b = 0 for a <= b is pretty standard, and is often called "partial subtraction."

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

#54

Post by syzarian »

iamerroragent wrote: post #51
syzarian wrote: post #47

Earlier quoted context omitted.

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…

Okay so if you can get to a ring without a multiplicative inverse and then applying that operation to the ring forms it into a field then wouldn't it be fair to say that division is not really the opposite of multiplication the same way that subtraction absolutely is for addition?

The definition of division is multiplication by the multiplicative inverse. It may be the case that some elements don’t have such an inverse but the definition is analogous to that of subtraction. The analogy is not perfect because every element has an additive inverse while not every element had a multiplicative inverse.

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

#55

Post by nextaccountic »

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 classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting.

There are multiple extensions to the real numbers that allow division by zero. One is a real projective line, which has only one infinity so that 1 / 0 = -1 / 0 = infinity

https://en.wikipedia.org/wiki/Real_projective_line

Another is the extended real number line which has positive infinity and negative infinity, so 1 / 0 = +infinity and -1 / 0 = -infinity and they are different from each other

https://en.wikipedia.org/wiki/Extended_real_number_line

Those are all perfectly fine but they still can't define 0 / 0, which is a harder problem.

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

#56

Post by shagie »

rtpg wrote: post #21
AstixAndBelix wrote: post #17

We didn't invent 'i' to "solve sqrt(-1)". This is an extremely common misconception about maths and how it progressed that unfortunately people get led into believing by lazy teachers every day

So what did happen?

There's a good YouTube video on it that includes an epic math battle.

Veritasium - How Imaginary Numbers Were Invented - https://youtu.be/cUzklzVXJwo

Solving the cubic was a physical thing back then. https://www.maa.org/press/periodicals/convergence/solving-th...

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

#57

Post by comte7092 »

nextaccountic wrote: post #55
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 classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting. There are multiple extensions to the real n…

1 / 0 = +infinity Implies that 0 * +infinity = 1, so it does run into make of the same issues.

There are instances that make it useful, but the extended real number line isn’t used heavily in practice.

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

#58

Post by lanstin »

contravariant wrote: post #31
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).

That is why I found algebra to be annoying, unless it was algebra from algebraic topology. Ring of polynomials is too complicated.

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

#59

Post by renewiltord »

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…

Yep, an extension is only interesting if it is a true extension, i.e. retains the properties of the thing being extended. So complex numbers are interesting as an an extension of reals since reals are isomorphic to the subring. Likewise with quaternions and reals / complex numbers.

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

#60

Post by antognini »

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…

An example where this does work quite nicely has to do with Bring radicals or "ultraradicals [1]. One of the most important results from Galois theory is that the quintic equation has no solution using standard radicals. But the introduction of "Bring radicals" allows quintic equations to be formally solved. As far as I'm aware though, Bring radicals only work for quintic equations in general and don't work for 6th order or higher polynomials, so your bang for the buck is a somewhat limited.

[1]: https://en.wikipedia.org/wiki/Bring_radical

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