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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)

#81

Post by hgsgm »

lanstin wrote: post #58
contravariant wrote: post #31

Earlier quoted context omitted.

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.

"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?

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

#82

Post by hgsgm »

red_trumpet wrote: post #15
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…

> 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?

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

#83

Post by syzarian »

topaz0 wrote: post #76
syzarian wrote: post #73

Earlier quoted context omitted.

The word division means something in mathematics. There is general agreement in what that word ought to mean. You can define a binary operation in such a way that it doesn’t look like what we normally think of as division and label your operation division . In the same way you can define the symbol duck to refer to what most people call a chair. You won’t get anyone else agreeing with your new definition though.

We redefine multiplication for new contexts every day in math, I don't see why division should be any different. See also: https://en.wikipedia.org/wiki/Division_(mathematics)#Divisio...

I think I understand better where you are coming from. In computer science I don’t know what they typically mean when they say “division”. I’ll be more precise. In abstract algebra division means multiplying by the inverse. All of the notions of division mentioned in the Wikipedia page come from this idea. Computers can’t work with within the realm of the entire real number system. There they have notions of type. They like to extend common operators like “/“ to things that normally it doesn’t apply to. A computer language will sometimes return a value of int or some other type when the integer 5 is divided by 3. Depending on how the language designer wanted things to work. This isn’t division in a mathematical sense though.

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

#84

Post by topaz0 »

syzarian wrote: post #83
topaz0 wrote: post #76

Earlier quoted context omitted.

We redefine multiplication for new contexts every day in math, I don't see why division should be any different. See also: https://en.wikipedia.org/wiki/Division_(mathematics)#Divisio...

I think I understand better where you are coming from. In computer science I don’t know what they typically mean when they say “division”. I’ll be more precise. In abstract algebra division means multiplying by the inverse. All of the notions of division mentioned in the Wikipedia page come from this idea. Computers can’t work with within the realm of the entire real number system. There they have notions of type. Th…

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.

To be clear, do you disagree that it is commonplace in complex analysis to extend the complex plane by {infinity} and define 1/0 = infinity, 1/infinity = 0? I find it hard to imagine that you can't have encountered that given how much you seem to know about abstract algebra. Or do you just think that it is a bad idea, despite being commonplace? In either case, to say that mathematicians would not call that operation division as a result is contradictory to my experience, even if those two special cases don't fit the category of multiplication by the inverse.

Also to be clear, I know of no counterexamples in abstract algebra and it would make sense to me that in that context division would mean something very particular, in order to be able to talk about it with any generality. But as it happens, abstract algebra isn't all of math.

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

#85

Post by thaumasiotes »

syzarian wrote: post #75
thaumasiotes wrote: post #74

Earlier quoted context omitted.

I'm having trouble following the argument from your premise "it is a stupid question to ask why I referred to a chair as a chair instead of a snkwoo " to your conclusion "it is not a stupid question to ask why, when we have no answer to a question, we don't just say that we do have one". The answer (to both of those questions!) is, of course, that we could do that, but it wouldn't accomplish anything. Asking the ques…

I'm having trouble following… I know. Please don’t become a teacher.

> Please don’t become a teacher.

I've been one!

Interestingly, the most consistent comment I got, from both students and school administration, was "you're so patient with the students".

Try humoring me. Did I describe your premise accurately? Did I describe your conclusion accurately? How do you get from one to the other?

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

#86

Post by thaumasiotes »

scubbo wrote: post #77
thaumasiotes wrote: post #74

Earlier quoted context omitted.

I'm having trouble following the argument from your premise "it is a stupid question to ask why I referred to a chair as a chair instead of a snkwoo " to your conclusion "it is not a stupid question to ask why, when we have no answer to a question, we don't just say that we do have one". The answer (to both of those questions!) is, of course, that we could do that, but it wouldn't accomplish anything. Asking the ques…

> Asking the question just means you have no idea what you're saying. Or in other words, it's a stupid question. So, to be clear, you're saying that the only kind of question that isn't stupid is the one where the querent already has perfect knowledge of the discipline?

I'm saying that to avoid asking a stupid question, you need to know the meaning of your own question. Stringing words together at random isn't going to get you there.

Compare the famous anecdote from Charles Babbage:

On two occasions I have been asked, -- "Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?" [...] I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Is it necessary to have perfect knowledge of the workings of the Difference Engine to avoid asking that question? Of course not. Any knowledge at all would do the trick. If you put gravel into a water mill instead of grain, will you still get flour out of it?

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

#87

Post by syzarian »

topaz0 wrote: post #84
syzarian wrote: post #83

Earlier quoted context omitted.

I think I understand better where you are coming from. In computer science I don’t know what they typically mean when they say “division”. I’ll be more precise. In abstract algebra division means multiplying by the inverse. All of the notions of division mentioned in the Wikipedia page come from this idea. Computers can’t work with within the realm of the entire real number system. There they have notions of type. Th…

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 really division then?

In math often times the answer we give depends on the knowledge of the person asking the question. For instance we tell calculus 1 students 1/x is not continuous as a function from R-{0} to R. Of course in the standard induced topology it is a continuous function but explaining this to calculus 1 students would be very difficult.

https://math.stackexchange.com/a/2524779

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

#88

Post by syzarian »

thaumasiotes wrote: post #85
syzarian wrote: post #75

Earlier quoted context omitted.

I'm having trouble following… I know. Please don’t become a teacher.

> Please don’t become a teacher. I've been one! Interestingly, the most consistent comment I got, from both students and school administration, was "you're so patient with the students". Try humoring me. Did I describe your premise accurately? Did I describe your conclusion accurately? How do you get from one to the other?

A child asked her mother, “Can I put my hand in the fire?”. The mother responded, “That’s a stupid question. Of course you can.”. The child put her hand in the fire and got severe burns on her hand. She learned then that instead asking “Can I…” she should have asked, “Is it advisable…”. Unfortunately for her she lived in a society in which people frequently say things like, “You have to file taxes on or before April 15.” when they mean, “You can file taxes after April 15 but you may incur fees and penalties if you do so.”. She later became a teacher and was very patient with her students.

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

#89

Post by nextaccountic »

marcosdumay wrote: post #61
nextaccountic wrote: post #55

Earlier quoted context omitted.

> 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…

> There are multiple extensions to the real numbers that allow division by zero. Well, the gotcha is that they redefine the operations so that none of addition, subtraction, multiplication or division are total. Those operations just break in a different number than zero.

Addition can be total if you have a single infinity, just make infinity + n = infinity for all n

Subtraction, multiplication and division is harder

But making further operations partial isn't that of a big deal; in fields the division is already partial due to division by zero not being defined.

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

#90

Post by devit »

You can do that, but there's a tradeoff of losing properties that otherwise hold.

For example, by adding the imaginary numbers, there is no longer an ordering compatible with addition and multiplication (ordering compatible with multiplication means that z > 0 and x > y implies x * z > y * z: assuming that, if 0 i and thus 0 You can certainly add a number x such that x = x + 1 (e.g. what is commonly called an infinity or NaN), but that implies no longer having additive left inverses assuming you keep associativity of addition and 0 != 1 (since otherwise 0 = -x + x = -x + (x + 1) = (-x + x) + 1 = 0 + 1 = 1).

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