Why don't we define “imaginary” numbers for every “impossibility”? (2012)(math.stackexchange.com) |
Why don't we define “imaginary” numbers for every “impossibility”? (2012)(math.stackexchange.com) |
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.
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.
There are instances that make it useful, but the extended real number line isn’t used heavily in practice.
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
"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.
There are multiple ways to define what division by zero means. Which definition leads to this outcome? How?
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
The closest thing you'd get to it is to
1. define a limit (lim x->a of ƒ(x) exists if and only if given any ε > 0 there exists a δ > 0 such that ...)[1].
2. chose a function ƒ(x) such that on a given "a", ƒ(a) = ƒ(a)/0.
3. prove that the limit exists and is finite.
Now if we defined division by zero it would look like this:
Axiom: For every element x of the real numbers there exists a x' in the real numbers such that x/0 = x'
I advise you to play with this new "rule" to see if it leads to something interesting. Hint: try to prove that 1/0 = 2/0
[1]: https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...
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.
Is the short answer it's not parsimonious or useful?
Infinity and negative infinity can make a lot of sense. You can even allow imaginary infinities.
The caveats:
* You cannot multiply zero by infinity, or divide infinity by infinity
* You cannot add different infinities
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...
Complex numbers being equivalent to R[X]/(1+X^2) and dual numbers being equivalent to R[X]/(X^2).
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 satisfies "P(X) = 0".
Examples:
R[X]/(x^2 + 1) = C
R[X]/x = R
C[X]/(x^2 + 1) = C + C.x
R[X]/1 = 0
- If the polynomial P is invertible, i.e. has degree 0 and is not zero, then the resulting ring is zero R[X]/P = 0. This is what happens in the example x = x-1 (which corresponds to P = x - 1 - x = -1).
- If the polynomial P has degree 1 (i.e. P=aX+b), then the equation P=0 is equivalent to x=-b/a, representing an element already present in R, hence the ring R[X]/P is equal to R.
- If the polynomial P is irreducible (i.e. not a product of two proper polynomials) then the quotient R[X]/P is a field. This happens in the case R[x]/(x^2 + 1) which results in the complex numbers.
- If the polynomial P is a product of two polynomials P1,P2 which don't have common divisors, then R[X]/P = R[X]/P1 + R[X]/P2, this happens in the case that C[X]/(x^2+1), since P = x^2 + 1 factors as (x+i)*(x-i) in C. The equivalent result for integers is known as Chinese Remainder Theorem.
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.
1/0 is maybe a bit trickier and leads you to invent projective spaces.
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, then 0 = 0 * i < i * i = -1, absurd, or if 0 > i and thus 0 < -i, then 0 = 0 * -i < -i * -i = -1, absurd).
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).
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...
every polynomial with algebraic coefficients has 'n' solutions (counted with multiplicity)!
so e.g. x^121 + sqrt(7)x^9 + fithroot(22)x^7 + (1+i)x^3 + 22/7 = 0 has 121 solutions. and they're all algebraic numbers: nothing weird like pi in there.
It's a stupid question, but it's not related to your response.
Most questions asked by beginners in an area are “stupid” and few as insightful as this one. I’ve taught mathematics at a community college for 20 years and I would be delighted to have been asked this. Usually questions are mundane like, “Why did you add x to both sides?”. Here the person is trying to understand what mathematicians do, what the basis of expanding a number system really involves. This is a fantastic question.
Peoples’ curiosity ought not be labeled as stupid.
a/b = c if and only if a = c*b and b!=0
c*b = a if and only if a/b = c and b!=infinity and c!=infinity"/" means "* reciprocal of".
If "infinity" is defined as "reciprocal of 0", what is the problem?
Yes it is an exception to 0*n=0.
It won't work in every setting, but it works in some settings, like inversive geometry.
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.
I will note that setting a - b = 0 for a <= b is pretty standard, and is often called "partial subtraction."
is a kind of sentence that is almost never true, and even if it were, it would be impossible to prove that someone hadn't jotted a valid definition on a napkin somewhere. In this case it is certainly not true (as others have mentioned: https://en.wikipedia.org/wiki/Riemann_sphere ). Now, specifying a definition for division by zero does require you to be careful about how the other operations extend to this new number, but there are perfectly consistent (and useful!) ways to do so.
[1] https://en.wikipedia.org/wiki/Rational_function
[2] https://en.wikipedia.org/wiki/Polynomial_ring
So you also find matrices too complicated?
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
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.
I suspect the real math people know what they’re doing more than I do, though.
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.
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.
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).
Correct. That is why I feel more comfortable asking "stupid" questions to chatGPT. I clarified a lot of concepts in economics through repeatedly asking questions about each concept that pop up in its answers and trying to push it to the limits of what can be defined, explained, etc. One cannot be sure of the truthfulness or soundness of the answers, but they may help.
I mean, you've already gotten it wrong. This can be done in other situations. Where it isn't done, it isn't done because doing it is pointless, not because there's some bar to giving names to opaque labels.
If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because anything you call 0 that has an multiplicative inverse makes it behave like something other than zero. So no one would use 0 to describe such an element. In a ring, or abelian group, the symbol 0 is reserved for the additive identity element.
Similarly, I could say snkwoo is what most people call a chair. A grammarian would say there is no word snkwoo even though I just defined it.
Your original comment was wrong and bad. Instead of just admitting it or moving on you’ve decided to double down and make another bad comment.
The answer (to both of those questions!) is, of course, that we could do that, but it wouldn't accomplish anything. Asking the question just means you have no idea what you're saying. Or in other words, it's a stupid question.
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.
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?
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?
I know. 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?
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.
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.
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).
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.
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.