Not necessarily true. A good rule for introductory math students, but some advanced math requires dividing by zero. (As mentioned, that’s what a derivative is, a division by zero.)
Limits are a way of getting information out of a division by zero, which is why derivatives involve taking the limit.
Division by zero is kind of like the square root of a negative number (something introductory mathematics coursework also tells you not to do). It’s not an invalid operation, it’s just an operation you have to be aware of the ramifications of. (If it seems like zero has unusual behavior, well, the same is true of negative numbers with respect to zero and positive numbers, and again the same is true of positive numbers with respect to zero and negative numbers.)
You’ve got it the wrong way round. “A derivative is a division by zero” is the pedagogical lie for introductory students (probably one that causes more confusion than it solves), and advanced maths doesn’t require it.
What are you expecting me to update on? None of what you’ve sent me contradicts anything except the language I use to describe it.
A derivative -is- a division by zero; infinitesimal calculus, and limits, were invented to try to figure out what the value of a specific division by zero would be. Mathematicians threw a -fit- over infinitesimal calculus and limits, denying that division by zero was valid, and insisting that the work was therefore invalid.
So what exactly is our disageement? That I regard limits as a way of getting information out of a division by zero? Or that I insist, on the basis that we -can- get information out of a division by zero, that a division by zero can be valid? Or is it something else entirely?
Incidentally, even if I were certain exactly what you’re trying to convince me of and it was something I didn’t already agree with, your links are nothing but appeals to authority, and they wouldn’t convince me -anyways-. They lack any kind of proof; they’re just assertions.
The definition of limit:
“lim x → a f(x) = c ” means
for all epsilon > 0, there exists delta > 0 such that for all x, if 0 < |x-a|<delta then |f(x) - c| < epsilon.
The definition of derivative:
f’(x) = lim h → 0 (f(x+h) - f(x))/h
That is, for all epsilon > 0, there exists delta > 0 such that for all h, if 0 < |h| < delta then |(f(x+h) - f(x))/h—f’(x)| < epsilon.
At no point do we divide by 0. h never takes on the value 0.
mstevens’ links have several demonstrations that division by zero leads to contradictions in arithmetic.
my link (singular) demonstrates that the definition of a derivative never requires division by zero.
Qiaochu’s proof in a sibling thread that the only ring in which zero has an inverse is the zero ring.
So what exactly is our disageement?
That you continue to say things like “A derivative -is- a division by zero” and “division by zero can be valid”, as if they were facts. Yes, you may have been taught these things, but that does not make them literally true, as many people have tried to explain to you.
Incidentally, even if I were certain exactly what you’re trying to convince me of and it was something I didn’t already agree with, your links are nothing but appeals to authority, and they wouldn’t convince me -anyways-. They lack any kind of proof; they’re just assertions.
Whose authority am I appealing to in my (singular) link? Doctor Rick? I imagine he’s no more a doctor than Dr. Laura. (I actually knew one of the “doctors” on the math forum once, and he wasn’t a Ph. D. (or even a grad student) either; just a reasonably intelligent person who understood mathematics properly.) The only thing he asserts is the classical definition of a derivative.
Or maybe you were just giving a fully general counterargument, without reading the link.
EDIT: It’s simply logically rude to ask for my credentials, and then treat every single argument you’ve been presented as an argument from authority, using that as a basis for dismissing them out of hand.
I am treating your links as arguments from authority, because they don’t provide proof of their assertions, they simply assert them. As I wrote there, I didn’t ask for your credentials to decide whether or not I was wrong, but to provide a prior probability of being wrong. It started pretty high. It declined; my mathematics instructor provided better arguments than you have, which have simply been assertions that I’m incorrect.
My experience with infinitesimal calculus is limited, so I can’t provide proofs that you’re wrong (and thus have no basis to say you’re wrong), but I haven’t seen proofs that my understanding is wrong, either, and thus have no basis to update in either direction on. At this point I’m tapping out; I don’t see this discussion going anywhere.
The law of cancellation requires that all values being cancelled have an inverse. The inverse of 0 doesn’t exist in the set of real numbers (although it does exist in the hyperreals). This doesn’t mean you can’t multiply a number by the inverse of 0, but the product doesn’t exist in real numbers, either. (Hyperreal numbers don’t cancel out the way real numbers do, however; they can leave behind a hyperreal component [ETA: Or at least that’s my understanding from the way my instructor explained why removable discontinuities couldn’t actually be removed—open to proof otherwise].)
0 doesn’t have an inverse in the hyperreal numbers either (To see why this it true, consider the first-order statement “∀x, x*0 != 1” which is true in the real numbers and therefore also true in the hyperreals by the transfer principle). From this it obviously follows that you can’t multiply a number by the inverse of 0.
Not compared to somebody who specializes in the field of mathematics, no.
But I don’t expect to change paper-machine’s mind, where paper-machine expects to change mine. I expect more than appeals to authority. I have some prior that paper-machine might be right, given that this is their field of expertise. My posterior odds that they have a strong knowledge of this particular subject, however, are shrinking pretty rapidly, since all I’m getting are links that come up early in a Google search.
Limits and calculus isn’t what I think of, at all, when I think of division. I pretty much limit it exclusive to the multiplicative inverse in mathematical systems where addition and multiplication work like you think they ought to. There are axioms that encompass all of “works like you think they ought to”, and a necessary one of them is the multiplicative inverse of zero is not a number.
Not necessarily true. A good rule for introductory math students, but some advanced math requires dividing by zero. (As mentioned, that’s what a derivative is, a division by zero.)
Limits are a way of getting information out of a division by zero, which is why derivatives involve taking the limit.
Division by zero is kind of like the square root of a negative number (something introductory mathematics coursework also tells you not to do). It’s not an invalid operation, it’s just an operation you have to be aware of the ramifications of. (If it seems like zero has unusual behavior, well, the same is true of negative numbers with respect to zero and positive numbers, and again the same is true of positive numbers with respect to zero and negative numbers.)
You’ve got it the wrong way round. “A derivative is a division by zero” is the pedagogical lie for introductory students (probably one that causes more confusion than it solves), and advanced maths doesn’t require it.
Another link, this time explicitly dealing with derivatives and division by zero, in the vain hope that you’ll actually update someday.
What are you expecting me to update on? None of what you’ve sent me contradicts anything except the language I use to describe it.
A derivative -is- a division by zero; infinitesimal calculus, and limits, were invented to try to figure out what the value of a specific division by zero would be. Mathematicians threw a -fit- over infinitesimal calculus and limits, denying that division by zero was valid, and insisting that the work was therefore invalid.
So what exactly is our disageement? That I regard limits as a way of getting information out of a division by zero? Or that I insist, on the basis that we -can- get information out of a division by zero, that a division by zero can be valid? Or is it something else entirely?
Incidentally, even if I were certain exactly what you’re trying to convince me of and it was something I didn’t already agree with, your links are nothing but appeals to authority, and they wouldn’t convince me -anyways-. They lack any kind of proof; they’re just assertions.
The definition of limit: “lim x → a f(x) = c ” means for all epsilon > 0, there exists delta > 0 such that for all x, if 0 < |x-a|<delta then |f(x) - c| < epsilon.
The definition of derivative: f’(x) = lim h → 0 (f(x+h) - f(x))/h
That is, for all epsilon > 0, there exists delta > 0 such that for all h, if 0 < |h| < delta then |(f(x+h) - f(x))/h—f’(x)| < epsilon.
At no point do we divide by 0. h never takes on the value 0.
Sigh. Consider this my last reply.
mstevens’ links have several demonstrations that division by zero leads to contradictions in arithmetic.
my link (singular) demonstrates that the definition of a derivative never requires division by zero.
Qiaochu’s proof in a sibling thread that the only ring in which zero has an inverse is the zero ring.
That you continue to say things like “A derivative -is- a division by zero” and “division by zero can be valid”, as if they were facts. Yes, you may have been taught these things, but that does not make them literally true, as many people have tried to explain to you.
Whose authority am I appealing to in my (singular) link? Doctor Rick? I imagine he’s no more a doctor than Dr. Laura. (I actually knew one of the “doctors” on the math forum once, and he wasn’t a Ph. D. (or even a grad student) either; just a reasonably intelligent person who understood mathematics properly.) The only thing he asserts is the classical definition of a derivative.
Or maybe you were just giving a fully general counterargument, without reading the link.
EDIT: It’s simply logically rude to ask for my credentials, and then treat every single argument you’ve been presented as an argument from authority, using that as a basis for dismissing them out of hand.
I am treating your links as arguments from authority, because they don’t provide proof of their assertions, they simply assert them. As I wrote there, I didn’t ask for your credentials to decide whether or not I was wrong, but to provide a prior probability of being wrong. It started pretty high. It declined; my mathematics instructor provided better arguments than you have, which have simply been assertions that I’m incorrect.
My experience with infinitesimal calculus is limited, so I can’t provide proofs that you’re wrong (and thus have no basis to say you’re wrong), but I haven’t seen proofs that my understanding is wrong, either, and thus have no basis to update in either direction on. At this point I’m tapping out; I don’t see this discussion going anywhere.
You said ” Dividing by zero doesn’t produce a contradiction”
Several of these links include examples of contradictions. There is no authority required.
For example:
Er, 1⁄0 * 0 != 1.
The law of cancellation requires that all values being cancelled have an inverse. The inverse of 0 doesn’t exist in the set of real numbers (although it does exist in the hyperreals). This doesn’t mean you can’t multiply a number by the inverse of 0, but the product doesn’t exist in real numbers, either. (Hyperreal numbers don’t cancel out the way real numbers do, however; they can leave behind a hyperreal component [ETA: Or at least that’s my understanding from the way my instructor explained why removable discontinuities couldn’t actually be removed—open to proof otherwise].)
0 doesn’t have an inverse in the hyperreal numbers either (To see why this it true, consider the first-order statement “∀x, x*0 != 1” which is true in the real numbers and therefore also true in the hyperreals by the transfer principle). From this it obviously follows that you can’t multiply a number by the inverse of 0.
Further, if you did decide to adjoin an inverse of zero to the hyperreals, the result would be the zero ring.
Going to have to investigate more, but that looks solid.
Since you asked this of papermachine, it seems reasonable to reflect it back:
Are you asserting this as somebody with strong knowledge of mathematics?
Not compared to somebody who specializes in the field of mathematics, no.
But I don’t expect to change paper-machine’s mind, where paper-machine expects to change mine. I expect more than appeals to authority. I have some prior that paper-machine might be right, given that this is their field of expertise. My posterior odds that they have a strong knowledge of this particular subject, however, are shrinking pretty rapidly, since all I’m getting are links that come up early in a Google search.
Limits and calculus isn’t what I think of, at all, when I think of division. I pretty much limit it exclusive to the multiplicative inverse in mathematical systems where addition and multiplication work like you think they ought to. There are axioms that encompass all of “works like you think they ought to”, and a necessary one of them is the multiplicative inverse of zero is not a number.