Old-School Slashdotter Discovers and Solves Longstanding Flaw In Basic Calculus (mindmatters.ai) 222
Longtime Slashdot reader johnnyb (Jonathan Bartlett) shares the findings of a new study he, along with co-author Asatur Zh. Khurshudyan, published this week in the journal DCDIS-A: Recently a longstanding flaw in elementary calculus was found and corrected. The "second derivative" has a notation that has confused many students. It turns out that part of the confusion is because the notation is wrong. Note -- I am the subject of the article. Mind Matters provides the technical details: "[T]he second derivative of y with respect to x has traditionally had the notation 'd2 y/dx 2.' While this notation is expressed as a fraction, the problem is that it doesn't actually work as a fraction. The problem is well-known but it has been generally assumed that there is no way to express the second derivative in fraction form. It has been thought that differentials (the fundamental 'dy' and 'dx' that calculus works with) were not actual values and therefore they aren't actually in ratio with each other. Because of these underlying assumptions, the fact that you could not treat the second derivative as a fraction was not thought to be an anomaly. However, it turns out that, with minor modifications to the notation, the terms of the second derivative (and higher derivatives) can indeed be manipulated as an algebraic fraction. The revised notation for the second derivative is '(d 2 y/dx 2) - (dy/dx)(d 2 x/dx 2).'"
The report adds that while mathematicians haven't been getting wrong answers, "correcting the notation enables mathematicians to work with fewer special-case formulas and also to develop a more intuitive understanding of the nature of differentials."
The report adds that while mathematicians haven't been getting wrong answers, "correcting the notation enables mathematicians to work with fewer special-case formulas and also to develop a more intuitive understanding of the nature of differentials."
Seems quite a lot larger... (Score:2, Insightful)
I appreciate the new form is technically more accurate but the expansion is pretty large compared to the original form... I wonder if the extra length doesn't wash out the understandability gains you get out of the original form.
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That said, I haven't taken ordinary differential equations in eons and I'm not gonna
Re:Seems quite a lot larger... (Score:5, Informative)
This is my thought as well. Interestingly, I developed this while writing a book (Calculus from the Ground Up [amazon.com]) to use for my homeschool co-op calculus classes. I was trying to find a good way to explain the notation, and I literally had 20 calculus books that I read through trying to find a good explanation for the standard notation in any of them. None of them even attempted an explanation, just "this is the way it is, but don't treat it as a fraction." So, I tried to deduce the notation myself. That's when I realized that it was not just limited, it was actually wrong. So I wrote the paper and finished the book (it's Appendix B in the book).
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The problem with e-book math books is trying to make it look right on a small screen. If you just want a PDF of it, send me an email and I'll send you one, especially if you consider telling other people how great it is. Unfortunately, you can't just tell Amazon to take your PDF and make it an e-book :(
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like trying to write better poetry by using more perfect spelling and grammar
Let's run with your analogy.
The current situation - with the misleading notation - is as if all poetry in history was written using future-tense.
The new recommended notation is like adding the ability to write the correct tenses in poetry.
Sure, you can still use the rough approximation old notation (and many teachers will); just as you can still write poetry that only uses future tense.
This guy empowered us to now use the more correct notation that implies the meaning we intend.
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because the old way ways "you can treat the derivative as a fraction,
Except the second derivative notion isn't a fraction. It's a way of writing "the second derivative of Y with respect to X" in a short form. Not all '/' create "fractions". Unless, of course, you want to argue that I'm putting a lot of "< divided by quote>" fractions in my /. postings.
The error is not in the notation, it's in the inability to overload the / operator when dealing with more complex and abstract mathematical concepts. It's like not being able to differentiate between "e as a variable"
Re:Seems quite a lot larger... (Score:4, Interesting)
Except that, in the first derivative, it *is* used as a fraction. Otherwise you couldn't reformulate your equation for integration (i.e., you have to multiply both sides by dx, which is treating it as a fraction). So, to say that in one case, it is a fraction, but this next case it isn't, but still written as a fraction, even though it *could* be written as a fraction, but we just decided not to, seems strange, at least to me.
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(i.e., you have to multiply both sides by dx,
I cannot remember EVER having to multiply "both sides" of anything by "dx" to do an integration. Maybe "new math" forces this.
Re: Seems quite a lot larger... (Score:2, Insightful)
Yes, you do, to integrate dy/dx = x, you would multiply both sides by dx, cancelling out the denominator in dy/dx to form, dy = x dx. Throw both sides under an integral sign and go! You just memorized the rules with no understanding of why it worked...
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No... if I wanted to solve that DE, I would integrate both sides with respect to $x$. On the right, the integral is easy enough to compute. On the left, it comes down to an application of the fundamental theorem of calculus. The Leibniz notation is convenient in this case, since it lets you treat the differential as a number, but the usual exposition relies on actual theorems which justify this kind of manipulation. It should also be noted that Abraham Robinson went over this in the 60s...
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For every complex problem there is an answer that is clear, simple, and wrong.
That this basic calculus equation was wrong is my new excuse for why I suck at calculus.
Re: Seems quite a lot larger... (Score:1)
The way I studied it - it was never a notation. The second order derivative was just a derivative of the derivative, there was no expression for it. Though that was in the former eastern block.
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The problem with both these forms is that while this may be how the theory-side thinks of it, its not how the applied side thinks of it. Trying to get from a relation stated in the original notation, to an applied calculation via algebra, leads nowhere. The second "notation" is clearly a minefield for any applied guy (*) even if algebraic manipulations are now valid.
(*) most of the
Standard form just as accurate (Score:3)
I appreciate the new form is technically more accurate
It's only technically more accurate if you read the standard form as a fraction. If you actually read the standard form as intended - a notation indicating the second derivative of y with respect to x - the standard notational form is just as accurate as is the Newtonian notation of dots to denote derivatives with respect to time.
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From the abstract on the paper:
"This leads to an overall simplification in working with calculus for both students and practitioners, as it allows items which are written as fractions to be treated as fractions. It prevents students from making mistakes, since their natural inclination is to treat differentials as fractions.Additionally, there are several little-known but extremely help"
and
"Since many in the engineering disciplines are not formally trained mathematicians, this also can prevent professionals
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Damnit... I wish comments could be edited on this...
"...Additionally, there are several little-known but extremely helpful formulas which are straightforwardly deducible from this new notation."
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It's not a new word, it's just part of a confusing notational system that arose from the philosophical attitudes of English teachers to not give such words the same ontological status as words from a dictionary.
Summary's accuracy seems questionable (Score:5, Informative)
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Well I could argue there was a component missing (d2y/dx2 was there but not -(dy/dx)(d2x/dx2)), hence it was a fundamental error that was causing d2y/dx2 to be a notation instead of a mathematical object, while looking like a mathematical object.
If it was called something like (dx/dy)" maybe I would have agreed this is only a notation. But d2y/dx2 is weird enough that it pretends to say something .. which happens to be wrong.
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Considering that, if actually defined, dx/dx should be 1, I would have thought that d^2x/dx^2 = d(dx/dx)/dx = d(1)/dx, which would be the limit of 1/dx as dx goes to zero, which would be undefined or infinity.
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Not quite. d(1) *is* zero. The differential of a constant is zero, basically by definition. If e is an infinitesimal, 0/e is still zero. However, d^2x/dx^2 != d(dx/dx)/dx. d(dx/dx)/dx, using the new notation, is "d^2x/dx^2 - (dx/dx)(d^2x/dx^2)", which is obviously zero by inspection.
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Sounds like it is well known and he came up with a patch that covers all/more of the special cases.
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It's well known to most students who took a calculus course and then took the next level calculus course.
Suddenly your d and dx start doing weird shit, and they can't do other shit you've been told to do with them, then you find yourself questioning WTF the d or the x actually mean, and then you wonder why you never thought d/dx was just 1/x, and you realize your teacher / professor can't explain it either.
If you don't grok the fundamental theorem of calculus, you'll never grok d/dx and the bullshit that ha
Re:Summary's accuracy seems questionable (Score:5, Interesting)
It's a bit of both. Some of the facts of the matter were known, but it was assumed that this was just "the way it was". That is, no one considered it an open problem. For instance, we view the inability to divide by zero just a fact of mathematics, not a flaw. Likewise, this was not known to be a flaw, it was just assumed that this was the way things worked.
If you need to point to a definitive flaw, it was in our understanding of how it was supposed to work - the relationship between our understanding and the notation. Once *that* flaw was discovered, the actual notation just spilled right out. That is, the flaw was that people were *not* treating dy/dx *sufficiently* as a fraction, due to 19th century preferences against infinitesimals. Once you realize that dy/dx really is a fraction, and has to be treated accordingly, everything automatically works.
It's almost humorous because there was no real advanced work to do. Literally everything needed is available in intro calculus. The problem was (a) the mathematics community had a habit of *not* treating dy/dx as a fraction, and (b) new students who didn't know better were simply taught *what* to do, not *why* to do it, and continued to repeat the mistake for over a century.
Congrats (Score:5, Interesting)
Figured I'd better say congratulations before the inevitable flood of people shitting on your contribution to math gets up to speed.
Re:Congrats (Score:5, Interesting)
Thanks! I appreciate it. Given that this was my first peer-reviewed mathematics paper, I had no idea how long the process was. I submitted the paper over a *year* ago. The necessary changes were minor. But the actual time it took to go through the process was excruciating. I'm happy to finally be on the other side :)
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You bastard, now every student will need to buy a new textbook next year! ;)
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My first paper in chemistry was submitted to the Journal of the American Chemical Society.
The review took over a year. This was before the internet and the editor dropped the ball. At that time,
it was bad form to ask the editor about the status of your submission as the process was already slow and
you did not want to pester the editor and get it treated more slowly.
Email has changed the whole process, but some journals are still slow.
Congratulations on the acceptance of your paper.
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Can you give some examples of special cases that are avoided by your notation?
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Anywhere where you have a second derivative where the variable with which you are taking the derivative with respect to is dependent on another variable. You would previously have to use Faa di Bruno's formula to properly take care of this situation. Now you can just do algebraic manipulations.
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I submitted the paper over a *year* ago. The necessary changes were minor. But the actual time it took to go through the process was excruciating.
Apart from a few exception, yeah submitting papers takes aaageeess. Basically (and this has nothing to do with you or the quality of your work), no one wants t oread your paper. I mean people accpet they need to read and review papers as part of the academic papers, but no one WANTS to do it.
Half of it is they want to be doing their own research. The other half i
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I recently had another paper which sat for 4 MONTHS in the editors inbox, before he decided he just wasn't interested.
What needs to happen is to have a small change in policy like this:
1) You can submit to multiple journals at once
2) A journal makes an offer to send it for review
3) Accepting an offer @2 requires that you remove your submission from other journals
Then the procedure goes on as before. This will prevent editors from wasting everyone's time.
What's super-super frustrating is that I had a *diffe
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This is indeed very cool. I just skimmed the paper (so far), but I fundamentally like the idea that not treating dx/dy as a fraction is just a historical artifact.
Bought your book.
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My coauthor has been doing this to good effect. His book "Controllability of Dynamic Systems: The Green's Function Approach" utilizes it. My role in mathematics is primarily in teaching high schoolers, so I don't spend a lot of time with differential equations. That's also the reason I *have* a co-author. I needed someone to tell me I wasn't crazy :)
Ugh (Score:3)
The d, dx, dy, etc. are not things to be generally operated on.
Writing the second derivative as d^2 / dx^2, or worse, d^2y / dx^2 is doubly absurd. (I'm using the ^ to denote supersripting, not exponentiation.)
d represents the instantaneous rate of change (which itself is a flawed concept - a rate of change cannot be instantaneous as a rate depends on the passage of time), dx represents that instantaneous rate of change of x. d/dx represents the instantaneous rate of some value, possibly some value dependent on x, with respect to the instantaneous rate of change of x. dy/dx, dv/dt, etc. are all the same deal. That rate of change of some variables with respect to other variables.
What is that instantaneous rate of change? The slope of a line (plane, or whatever if you've got more free variables) tangent to your function at a given point, presuming such a thing exists.
How do you determine that tangent line? You take the target point and some point h past it ((f(x) vs f(x+h)) (or before it!) and determine what the line does when you consider h approaching 0. You make sure you can define that shit from both ends and both ends agree. If that works out, have a limit, you've got a derivative, and baby, you've got the fundamental theorem of calculus goin'.
Whoever tried to slap that shit together as a fraction or take shortcuts and try to manipulate those symbols in a way that looks sort of like algebraic manipulation is a clown. Trying to fix that is going to be an uphill battle, but using more of the busted notation isn't really the solution.
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Whoever tried to slap that shit together as a fraction or take shortcuts and try to manipulate those symbols in a way that looks sort of like algebraic manipulation is a clown. Trying to fix that is going to be an uphill battle, but using more of the busted notation isn't really the solution.
100% agree.
The thing is, superior notations are right in front of us. Programmers use a variety of them every day.
Re:Ugh (Score:5, Insightful)
dy/dx doesn't represent instantaneous rate of change. That would be nonsense. The d in dx and dy means "small difference that will eventually go to zero". This is why dy/dx is a fraction. It represents the limit of a small change in y divided by small change in x, as the changes go to zero. This is why we teach students about the limit definition of the derivative as being what the derivative really is. As far dy and dx being tricks of notation, they're really not. They really are small changes. There's no instantaneous rate of change. dy and dx are always finite real numbers. They never actually become zero. dy/dx is the ratio that is approached as they get smaller and smaller.
As far as this guy's new version of the second derivative, I call bullshit. I seriously doubt that this is correct. And the notation d^2y/dx^2 actually makes sense when you think about. It's really just d(dy/dx)/dx, that is, a small change in dy/dx divided by a small change in x, where dy/dx is a small change in y divided by a small change in x. Writing it in the other way is just a good way of doing it. If you draw out what this means graphically, is becomes clear that it's really a small change between two consecutive small changes in y divided by two small changes in x, that is d(dy)/dx^2, hence d^2y/dx^2.
This guy's new version, on the other hand, doesn't make sense at all. I mean, how do you get that from taking the derivative of the first derivative. Let's take a pretty standard function: x=1/2*t^2+2*t+12. x'=t+2; x''=1, whereas his version would be x''=1-t, which doesn't make any sense, unless he has completely redefined everything. I mean, d^2y/dx^2 would have to be something like 2t+5 and d^2x/dx^2 would have to be something like 2, and then we get x''=2t+5-(t+2)*2. I didn't read the paper so I don't know what it would actually be, but there's no doubt that x''=1, so if his method is to make any sense at all it would have to give the same results in the end. I just don't see how it could.
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embrace the suck, apply powers to other math operators, eg: 1 +^2 3 = 7
Except the notation isn't "dy/dx", it's "d/dx" (Score:1)
"d/dx" is an operator on functions that has about a half dozen tidier, less-confusing alternate notations, while "dy/dx" is a limit of a ratio of nonzero numbers that is misleadingly written as a fraction because people in the 18th century weren't as bothered about the whole 'dividing by zero' thing.
The fact that algebraically treating dy/dx as a fraction works in any situation at all is a minor stroke of luck that honestly should be concealed, since thinking that way already hurts the progress of a huge nu
(d 2 y/dx 2) – (dy/dx)(d 2 x/dx 2) (Score:2)
Call differentiation "quark" instead. The new form for d2y/dx2 could be called a double quark or "fred" for short or f for really short. For really rigorous treatment call it f(x). In the UKoGBnNI it shall be known as noddy on Tuesdays unless the year is 2022.
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In the UKoGBnNI it shall be known as noddy on Tuesdays unless the year is 2022.
In your notation, what's Big Ears?
(And what's Mr. Wobbly Man?)
this is actually useful (Score:4, Informative)
It took me a few minutes to get to the nub of the matter.
If you're mentally reading the notation d^2 y / dx^2 as the second derivative of y divided by dx squared, you're doing it wrong.
Because what this notion really intends to mean is d(d(y)/dx)/dx, which as the paper points out is a different order of operation.
A more compact notation less misleading than the traditional d^2 y / dx^2 might be (d/dx)^2 dy, which expands via two repeated function applications to d(d(y)/dx)/dx, with the underlying operations now in the right order.
Calculus was never my best thing, so I might be all wet, but it seems to make sense.
I never liked the dx/dy notation much, regarding it more as a cryptic code than anything conceptually helpful (when its not cryptic, it's not helpful, because that's the common case you already know).
With the right lambda notation (riffing on what I proposed above) the fundamental operator nature of d() could be correctly expressed, even if you don't want into these algebraic manipulations, which mostly strike me as far too detailed and tedious.
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You twisted deviant.
I'm so glad your kind never gained a foothold in modern computing.
Well thank (Score:3)
god that's settled. Now we can figure out that P=NP problem that nobody can give a coherent answer on why its even a thing.
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Now we can figure out that P=NP problem that nobody can give a coherent answer on why its even a thing.
I think that problem is you tbh, if you don't understand it.
Linear regression stumper (Score:4, Interesting)
I have a "math issue" that has stumped most of my professors and online math forums. Linear regression typically uses the "least squares" algorithm. However, the power of 2 seems arbitrary to me, and possibly over-emphasizes outliers.
One professor at first said that the power of 2 makes the "best fit" in an objective sense, but later admitted that he doesn't really know, and couldn't find an answer before the end of the semester.
While it is true that the power of 2 may simplify the computation process*, that doesn't necessarily means it produces a better result in terms of line or curve fitting. Now that we have computers to do the number crunching, perhaps it's time to embrace arbitrary or different powers (superscripts).
(Disclaimer, I'm not a math expert.)
* In other words, power-of-2 produces the simplest known algorithm. But my question revolves around best data fit, not computational resources nor algorithm or formula brevity. Note that when using other powers, one may have to add an absolute value function because power-of-2 automatically provides the equivalent. I actually did a simulation that tested different powers; "blurring" known datasets and seeing which power best matched the original. I couldn't find any significant difference, but probably didn't try enough samples. I tested with fractional powers also, such as 1.5, 2.5, etc.
Re:Linear regression stumper (Score:5, Informative)
It's not arbitrary. There's actually a good reason for minimizing (y-yobs)^2, assuming that your observations have a Gaussian distribution. The resulting estimators provide a maximum likelihood estimator of the parameters of the distribution, if and only if it really was Gaussian. Thus, of course, if it isn't Gaussian (outliers of various sorts, et.c), the x^2 may not be the best bet. There is an entire field of 'robust estimators' of quantities, which are more resistant to outliers than least squares. There are also cases in which the underlying distribution is pathologically different from Gaussian; it could be Lorentzian (Cauchy), in which case it is so completely unlike a Gaussian, it doesn't even have a defined standard deviation (it is infinite). There are weighted methods which can fix this too.
So, in short, least squares is the right answer (in the sense that it yields results which provable have the maximum likelihood describing the data at hand) if you have a perfect Gaussian variate; otherwise, it may well not be.
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Good answer.
I'd go a step further and say that the purpose of linear regression is to see if there is a relationship in the data, and not to provide an actual answer to what the relationship exactly is.
In the real world, data relationships are rarely linear and distributions are often not known or are approximated. A linear regression will give you an idea on what is going on. The real relationship maybe too complex to ever know.
So, using least squares, well, it's probably good enough or at least a good sta
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I have a "math issue" that has stumped most of my professors and online math forums. Linear regression typically uses the "least squares" algorithm. However, the power of 2 seems arbitrary to me, and possibly over-emphasizes outliers.
One professor at first said that the power of 2 makes the "best fit" in an objective sense, but later admitted that he doesn't really know, and couldn't find an answer before the end of the semester.
While it is true that the power of 2 may simplify the computation process*, that doesn't necessarily means it produces a better result in terms of line or curve fitting. Now that we have computers to do the number crunching, perhaps it's time to embrace arbitrary or different powers (superscripts).
(Disclaimer, I'm not a math expert.)
* In other words, power-of-2 produces the simplest known algorithm. But my question revolves around best data fit, not computational resources nor algorithm or formula brevity. Note that when using other powers, one may have to add an absolute value function because power-of-2 automatically provides the equivalent. I actually did a simulation that tested different powers; "blurring" known datasets and seeing which power best matched the original. I couldn't find any significant difference, but probably didn't try enough samples. I tested with fractional powers also, such as 1.5, 2.5, etc.
Those other exponents probably also work. That term is just to help estimate the slope but any exponent > 1 would likely work there. Its just that its impractical for a variety of reasons including the fact that linear regression is just too simple a model for anything but the simplest use cases. Other techniques which aren't built upon linear regression are used instead so nobody studies this. You very well might be right about outliers for some use cases but it doesn't matter as other techniques ar
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However, the power of 2 seems arbitrary to me, and possibly over-emphasizes outliers.
It's not arbitrary, and yes it does emphasise outliers.
This is going to be hard to explain given slashdot's formatting ability but here goes...
For least squares, the assumption is that the noise is Gaussian, where every datapoint has the same (unknown) noise variance. If you have some linear function of parameters f(a), what you're doing is finding the a that maximises the probability of observing the data.
Assume you have d
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It is not arbitrary!. Using the power of two allows a simple, possible even trivial analytical solution of the problem (Matlab and similar have it built-in and can do it in a single line).
Of course you could use other norms to minimize the regression error - l1 norm, linf norm or any other norm in between, or even any other norm you can come up with. But in these cases, you end up with optimization problems that do not have analytical solutions and require
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But what is the "best data fit"?
That depends on the kind of data, the kind of errors one expects and the properties the fit should have.
Linear regression yields a result with some well known properties, e.g. the resulting linear function passes through the center of gravity. Maybe that's a desirable property. In other cases the y_i could be the result of a measuring process with a gaussian error distribution (where larger errors become more unlikely). Due to the central limit theorem that is often the case,
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Chairman of the FCC? Heck no, don't give him any more power.
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Brute force: shifting the line around incrementally and computationally until an approximate minimum was achieved. Yes, I know it's only an approximation, but it should have been good enough to detect any clear pattern if one existed.
I did assume rough boundaries/limits to avoid problems such as multiple candidates and division by zero. Thus, I wasn't testing every possible slope or line, just those "in the ballpark". (Perhaps "least squares" is handy
It's a complete nonsense. (Score:3)
Leibniz' notation is normally treated as a "suggestive kind", never to be understood literally. The origin of notation d^2/dx^2 goes from applying d/dx to d/dx, but d/dx only means "a derivative w.r.t. x" and nothing else. Sometime taking this notation literally and doing manipulations as if it were the regular fractions work (and that's b.t.w. is attributed to the early discoveries of many differentiation and integration rules), but it doesn't work most of the time. Any decent book on Calculus should point out that fact. Working with fractions helps to discover some rules, yes, but it's never rigorous, it's more like discovering something in a heuristic way, but then you still need a rigorous proof and that involves going back to basic definition of limits, not arguing in terms of "infinitesimals" (yes, I'm aware of Robinson's "non-standard calculus", but IMHO it's not a mainstream approach. Cheers.
Delta square x over delta y square IS retarded... (Score:1)
but Lagrange's notation > Leibniz's notation anyhow.
If you want to do wizardry by manipulating the notation itself, then by all means use '(d 2 y/dx 2) - (dy/dx)(d 2 x/dx 2)'.
Kudos to johnnyb
Old School (Score:3)
Old-School Slashdotter Discovers and Solves Longstanding Flaw In Basic Calculus
Can occasionally be heard yelling at younger mathematicians: "Get off my lambda"
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You never did a second derivative test to determine whether you are at a local minima or maxima?
Most intro calculus books at least show the notation for the second derivative. However, it is true that they rarely take it far enough to hit any problems with the notation.
I actually figured this out while trying to find a good way to explain the notation to my students, which is a homeschool co-op class (I have a range of 9-12 graders - the 9th grader is an exception, but she is ridiculously smart). I read t
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If you read my paper, I actually suggest this as a shortened form of my own. This notation is Arbogast's, and is woefully underused. I show how to interconvert between Arbogast notation and my own in the paper.
What about partial derivatives? (Score:2)
Very cool! I think the paper will help me understand more deeply problems with the notation I've fought with many times!
However, I'm a bit disappointed that the notion of partial vs. full derivative wasn't raised, which I think is very relevant to the question...
Re:What about partial derivatives? (Score:5, Interesting)
I've actually got a second paper on partial derivatives just about ready to go. It was originally part of this paper, but it got a little long, and I wanted to rethink and clarify a few concepts. Anyway, partial differentials have the same notational problem *plus* one more. The problem is that there are several partial differentials which all go by the same name. Once you name them properly (i.e., give them each a distinct name) the problems go away.
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Could I get that explained... (Score:2)
Sigh. (Score:3)
Often in maths, a mere change of notation, analogous equation in another field, or just looking at things in a slightly different way will open up whole new areas of maths.
Fermat's Last Theorem took forever to prove and the proof relies on translating the problem to a completely unrelated area of maths, solving it there, and then translating the results back.
And if you do things like use polar coordinates, etc. some areas of maths burst open with good sense and nice equations.
Something as simple as a notation change can work wonders. But this is just for convenience of amateurs who don't understand what a derivative actually is and does. It's like saying "Don't use the word multiplication for vectors, because it's not the same as for scalars". We know. Anyone handling it knows. Anyone dumb enough to confuse the notations is going to find out very quickly that nothing works. Sure, it might help if you've literally never done those kinds of equations before, but likely then you'll not be making any ground-breaking mathematical discoveries any time soon.
Things don't tend to survive hundreds of years for no reason, especially when they are one pen-stroke away from being changed, and have themselves gone through several notational iterations in their time.
I got through a degree in maths without thinking "Well, this notation is stupid", including three years of advanced calculus.
If you don't understand the notation, that's the very least of your worries as regards actually doing any calculus.
this is absolute AWESOME (Score:2)
Re:And in a sane curriculum (Score:5, Informative)
The messed up notation by Newton is not used and instead the much saner stuff from Riemann is used. Newton was smart, but a hack and a crank. And he tried to suppress Riemann notation. Mathematics would probably have done better without Newton.
Surely you mean Leibniz (1646-1716), not Riemann (1826-1866).
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I mean the Riemann Integral. But you are correct, differentiation came first.
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He could not have. He was dead at the time all information became available. But you seem to be stupid, so you are probably incapable of understanding causality.
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Thanks for your opinion on the history of Calculus, person who is not even aware of the most basic of facts and provides no real rationale.
Newton and _Leibnitz_ both useful (Score:5, Informative)
The messed up notation by Newton is not used and instead the much saner stuff from Riemann is used.
The advantage of the Newtonian notation is that it is a lot faster and easier for, unsurprisingly, basic Newtonian mechanics where you only really differentiate with respect to time. This is why it is used extensively in this area of physics. Leibnitz's (not Riemann's!) notation is a lot more versatile which is not surprising: Leibnitz was a mathematician who was interested in the abstract concept whereas Newton was a physicist who only developed calculus so he could describe mechanics and so did not really need a broader, more flexible notation.
It is actually quite a common that fundamental physics can find itself ahead of maths. For example String theory today is really a joint venture between maths and physics since they are having to develop the maths needed to describe the physical models they work on.
Finally, Newton was neither a hack or a crank but he was a somewhat evil genius. He could be quite nasty and viscous, sometimes in extremely petty ways. For example he discredited Leibnitz and he fell out with Robert Hooke and had all contemporary portraits of him destroyed which so angered a modern artist that she spent the time an effort painting multiple portraits of Hooke from contemporary descriptions so that, today, there are more portraits of Hooke than Newton!
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Finally, Newton was neither a hack
As a mathematician, he was most definitely a hack. Or he just did not care, which is about as bad. As a physicist he was probably somewhat better.
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As a mathematician, he was most definitely a hack. Or he just did not care...
He was not a mathematician: he was a physicist. He only developed the mathematics needed to describe his physics so why should he care about maths beyond that when that wasn't what he was interested in? Complaining he was a poor mathematician would be like claiming you are a poor journalist for starting a sentence with 'or'.
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Did you miss what the story was about?
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Newton was most definitely a mathematician. He held the Lucasian Chair of Mathematics at Cambridge for a number of years. He also developed mathematics outside of what he needed to describe his physics, such as finding the infinite series necessary to expand (a+b)^n where n is negative, or a fraction.
Also consider this quote about Newton from Leibniz himself: "Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half."
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Would make perfect sense. What a twat.
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The messed up notation by Newton is not used and instead the much saner stuff from Riemann is used. Newton was smart, but a hack and a crank. And he tried to suppress Riemann notation. Mathematics would probably have done better without Newton.
Riemann lived 2 centuries after Newton. And your conclusions aren't correct, they aren't even wrong!!!
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Not that much. Privileged and had time. The adoration some people have for him is not founded on facts.
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Unfortunately we can't go back through history and give every single human who ever lived privilege and time to see what they're capable of.
So we can only assess those who have made prominent contributions. There are many others who had privilege and time and still didn't contribute what Newton did, so it's still fair to call him a genius.
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It is not. His contributions are verifiable not that good in quality (see the the story, for example) and there was a lot of low-hanging fruit around. For calculus, he did only the simple, standard-space version (and did it badly), while Riemann did a far more general and superior version basically at the same time. Calling Riemann a genius may or may not be justified, but Newton does not make it into that group.
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There is a strong indicator: Leibnitz did the same thing independently at the same time. If you factor in that the scientific community was pretty small back then, that means the prerequisites were all there, the question had been asked and it just took somebody to put it together. That makes the results a "good" scientific result, but not a "genius level" one. And I am not talking about his contributions to physics, I am talking about his contributions to calculus, see the original story.
As to Riemann, I c
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And alternatively, I really know what I am talking about and regard ACs as basically scum. However, I cannot see how you come to "pretentious", unless you are an authoritarian follower that things people that have a name may not be criticized. Is that it? Newton was so great, nobody is allowed to criticize him? That would be a pretty bad stance. As for "self-centered", were to you see any evidence for that?
So I got confused on a name in the history of mathematics. But this is /. and I did not look it up to
Re: And in a sane curriculum (Score:1)
Exactly! If gweiher here had more time on his hands, he would have written supercalculus by now. But no, his solemn duty to try to top dead people on Slashdot takes up his otherwise suuuuper valuable time.
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It does not work for almost all cases. (Mathematical "almost all".) And it does not warn you of it. It is basically not mathematics, but clever shifting around of symbols with pitfalls which works purely by accident. Not a good thing.
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Nah. He's right. The current notation is bullshit. It looks like standard algebra and people try to manipulate the symbols as such.
I mean, you could shorten d/dx to 1/x, right?
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Nah. He's right. The current notation is bullshit. It looks like standard algebra and people try to manipulate the symbols as such.
I mean, you could shorten d/dx to 1/x, right?
The current notation is mathematically correct. It express that fact that the second derivative is the incremental limit of the first derivative.
The new notation for d^2/dx^2 loses the corect mathematical meaning and gains nothing of value.
d^2/dx^2 (f) (x) = lim (h->0) (d/dx (f)(x+h) - d/dx (f) (x))/h
Re: Uh (Score:2)
Finally the first commenter who knows what dy/dx really is. It is the limit of an expression containing a fraction, so not a "real" fraction in algebraic sense.
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The current notation is bullshit. It looks like standard algebra and people try to manipulate the symbols as such.
And that's why they call it "learning calculus" instead of "coming up with calculus all on your own". Not understanding how something works or what the terms mean can lead to horrible results.
Re: I have ... (Score:1)
That would make you a jerk.