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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)

#91

Post by topaz0 »

syzarian wrote: post #87
topaz0 wrote: post #84

Earlier quoted context omitted.

I am not at all concerned with what is or isn't possible in a computer for the purposes of this discussion. My only point with the link is that dealing with inverses in particular situations (i.e. where multiplication has or doesn't have certain properties) frequently requires particular considerations, and the properties of division defined as multiplication by the inverse will have different properties as a result.…

This is getting very far from where the original question came from. When talking to a layman one would say division is always multiplication by the inverse. There are nuances involved that a lay person simply can’t appreciate or understand. Had I known you knew about the extended complex numbers I would have answered differently. The extended complex numbers are not a ring, not a group, not an algebra, and so…is it…

Giving the answer that is relevant to the situation is very sensible. Saying that "this is the only thing division can mean in mathematics" is evidently false, though (I take it that you agree with me on this now?), and false in a way that is very relevant to the original question, which did not specify a context of abstract algebra and seemed to me to be very interested in expanding the mathematical horizons of the questioner, not restricting them.

The extended complex plane is a great example in my opinion, because it shows that yes there are reasons to extend the numbers in various ways, that can give useful structure, but you may have to give up something else in order for that make sense. In my opinion that is a much more complete answer to the deeper question. (Similarly for the reals mod 1, which do have the property that x + 1 = x).

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

#92

Post by syzarian »

topaz0 wrote: post #91
syzarian wrote: post #87

Earlier quoted context omitted.

This is getting very far from where the original question came from. When talking to a layman one would say division is always multiplication by the inverse. There are nuances involved that a lay person simply can’t appreciate or understand. Had I known you knew about the extended complex numbers I would have answered differently. The extended complex numbers are not a ring, not a group, not an algebra, and so…is it…

Giving the answer that is relevant to the situation is very sensible. Saying that "this is the only thing division can mean in mathematics" is evidently false, though (I take it that you agree with me on this now?), and false in a way that is very relevant to the original question, which did not specify a context of abstract algebra and seemed to me to be very interested in expanding the mathematical horizons of the…

The one point compactification of the complex plane is not a number system in the normal sense of what that means. Calling the use of the notational convenience 1/infinity a true division operation defies the common usage of the term in mathematics. You may call it whatever you want to though.

The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would break consistency and that is of paramount importance when doing new things in mathematics. There are agreed upon usages of terms and symbols in mathematics. Why call something division in the true sense of the word when it breaks the conventional usage of what that term means? But, also, why invent a new symbol to denote what is analogous to division? So we abuse notation. This is done all the time. So on the one had we’ll say to calculus 1 students 1/infinity is 0 but also say infinity is not a number. Things are done for convenience but when asked, “Is this really division?” the answer is no.

Of course you can redefine all terms you desire and say things like: A circle can be squared, I just mean something different when I say circle than when you say it. But why do that? All of this is my opinion. You disagree and that is ok.

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

#93

Post by topaz0 »

syzarian wrote: post #92
topaz0 wrote: post #91

Earlier quoted context omitted.

Giving the answer that is relevant to the situation is very sensible. Saying that "this is the only thing division can mean in mathematics" is evidently false, though (I take it that you agree with me on this now?), and false in a way that is very relevant to the original question, which did not specify a context of abstract algebra and seemed to me to be very interested in expanding the mathematical horizons of the…

The one point compactification of the complex plane is not a number system in the normal sense of what that means. Calling the use of the notational convenience 1/infinity a true division operation defies the common usage of the term in mathematics. You may call it whatever you want to though. The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would br…

I didn't realise this, but apparently it is also possible to do good algebra on this kind of structure by adding an element 0/0: https://en.wikipedia.org/wiki/Wheel_theory (which someone pointed to in one of the discussions -- I forget which one).

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

#94

Post by topaz0 »

topaz0 wrote: post #93
syzarian wrote: post #92

Earlier quoted context omitted.

The one point compactification of the complex plane is not a number system in the normal sense of what that means. Calling the use of the notational convenience 1/infinity a true division operation defies the common usage of the term in mathematics. You may call it whatever you want to though. The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would br…

I didn't realise this, but apparently it is also possible to do good algebra on this kind of structure by adding an element 0/0: https://en.wikipedia.org/wiki/Wheel_theory (which someone pointed to in one of the discussions -- I forget which one).

Honestly saying you can't have a number 1/0 because it breaks the ring axioms seems exactly analogous to saying 0 can't exist because it breaks the group axioms for multiplication on the non-zero reals. Is ring multiplication "not really multiplication" because it doesn't satisfy group axioms? That doesn't seem consistent with normal usage to me, but you could imagine a pedantic student coming out of their first group theory course and trying to make that argument.

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

#95

Post by syzarian »

topaz0 wrote: post #93
syzarian wrote: post #92

Earlier quoted context omitted.

The one point compactification of the complex plane is not a number system in the normal sense of what that means. Calling the use of the notational convenience 1/infinity a true division operation defies the common usage of the term in mathematics. You may call it whatever you want to though. The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would br…

I didn't realise this, but apparently it is also possible to do good algebra on this kind of structure by adding an element 0/0: https://en.wikipedia.org/wiki/Wheel_theory (which someone pointed to in one of the discussions -- I forget which one).

Mathematics is a vast subject and I can’t keep track of all developments. In 2010 there was a paper on meadows. I’ve never heard the term before. In that paper it is written:

As usual in field theory, the convention to consider p / q as an abbreviation for p · q−1 was used in subsequent work on meadows (see e.g. [2,5]). This convention is no longer satisfactory if partial variants of meadows are con- sidered too, as is demonstrated in [3].

So, as I’ve stated many times, I talked about convention and indicated you can use whatever terms you want. In the paper quoted above they acknowledge what the convention is. That is that division is multiplication by the inverse. They are arguing that it is worthwhile in this new algebraic object to change the usual notion a bit. If people agree to a new usage of the word division then definitions will change accordingly. None of this is pertinent to the spirit of the original question given the context under which it was asked. All of this is highly technical.

Definitions and notions change as new mathematics is created (discovered?). This happens all the time. All you have to do is convince other mathematicians to go along with it.

https://arxiv.org/pdf/0909.2088.pdf

EDIT: Regarding what you wrote in your other comment: The analogy is not apt in my opinion. It’s hard to say zero can’t exist because the nonzero…. The moment you say nonzero means it does exist. I think a better way to look at the situation is:

I have an object that is a group under a binary operation f. There is another natural binary operation on that object that operates with f in a consistent way. That operation doesn’t form a group but if I add a symbol to my set and give these rules then both operations interact in a consistent, natural way. I get a group under the new symbol with the second operation while preserving the group under the first operation minus the new symbol.

With extended complex numbers you don’t quite preserve the structures or properties that one normally wants so I’d say it isn’t true division. It is division like.

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

#96

Post by topaz0 »

syzarian wrote: post #95
topaz0 wrote: post #93

Earlier quoted context omitted.

I didn't realise this, but apparently it is also possible to do good algebra on this kind of structure by adding an element 0/0: https://en.wikipedia.org/wiki/Wheel_theory (which someone pointed to in one of the discussions -- I forget which one).

Mathematics is a vast subject and I can’t keep track of all developments. In 2010 there was a paper on meadows . I’ve never heard the term before. In that paper it is written: As usual in field theory, the convention to consider p / q as an abbreviation for p · q−1 was used in subsequent work on meadows (see e.g. [2,5]). This convention is no longer satisfactory if partial variants of meadows are con- sidered too, as…

I'm happy to agree to disagree about where the line between "division" and "division like" should be placed. As you say, it is a question of convention and not really a question of math. But I don't agree that a student with the curiosity to ask about extending the numbers in various ways would not find something "division like" with the properties they're interested in to be relevant to the question (even if it is missing some other properties that most mathematicians consider to be essential to the notion of division).

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

#97

Post by red_trumpet »

hgsgm wrote: post #82
red_trumpet wrote: post #15

Earlier quoted context omitted.

> If the polynomial P is invertible, i.e. has degree 1 Should be degree 0: only constant polynomials are invertible. E.g. x+1 is not invertible, and modding it out doesn't result in the zero ring. The example is a bit confusing, because $x=x+1$ is equivalent to $0=1$, which has degree 0.

What's wrong with negative degree monomials?

You mean like $x^{-1} = 1/x$? That's called a rational function[1], but not a polynomial, so it's not an element of the polynomial ring[2]. Of course you can also consider the algebra of rational functions, but this is a field[3] (almost by definition: you make every polynomial invertible), which means that modding out anything other than 0 yields the zero ring[4].

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

[2] https://en.wikipedia.org/wiki/Polynomial_ring

[3] https://en.wikipedia.org/wiki/Field_(mathematics)

[4] https://en.wikipedia.org/wiki/Zero_ring

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

#98

Post by lanstin »

hgsgm wrote: post #81
lanstin wrote: post #58

Earlier quoted context omitted.

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

"too complicated" is a weird way to say "provides a concise and consistent way to model superficially diverse phenomena and show how similar they really are" . So you also find matrices too complicated?

matrices over reals are ok especially if you keep to SO(n) but you can get very weird maths as polynomial quotients. they do not look to me like they are very similar. complex plane and extensions of all kinds of weird. seems hacky rather than illuminating to me. but then i only really like complex numbers as a field since analytic functions are so nice
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