Question

DISCUSSION: Calculus
Due to popular demand for colosseum style entertainment, let us discuss whether calculus is about division by 0...

Answer 1

it is more about dividing 0 by itself

Answer 2

@IrishBoy123 So given a function \(f(x)\), we are discussing whether \(\frac{df}{dx}\) has the idea of division by 0.
Firstly, I would like to bring in the idea that \(\frac{d}{dx}\) is an operator or perhaps a function, since, you input in a function and you get another function out. So given a function input, what function output do you get? So let \(f(x)\) be the input functin of the \(\frac{d}{dx}\) operator and the output function be \(g(x)\). Well, the definition of the operator \(\frac{d}{dx}\) is this:
Given \(x_0\):
\(g(x_0)=\frac{d}{dx}f(x_0) = Lim_{x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}\)

Answer 3

So the intuition of this idea, is the ratio of two things as they become arbitrarily small, by this I mean the ratio between \(f(x)-f(x_0)\) and \(x-x_0\). BOTH things tend to 0 individually, but their ratio is what we're interested in. As @ganeshie8 says, it's more about the idea of the ratio of two infinitesimal things. But at no point do we ever divide by 0. It's fussy detail, but the concepts are quite different. Dividing by 0 and dividing by something infinitesimally small are different.

Answer 4

I completely agree @inkyvoyd ! I'm just trying to go for the more "intuitive" approach for the physicists who don't know the definition or the subtleties of the limit operator :P You may want to check out the context of our discussion here: http://openstudy.com/study#/updates/57252d95e4b0644381e1034c

Answer 5

Also, @inkyvoyd . You talk about infinitesimals not always working... I don't suppose you had non-standard analysis in mind and have studied non-standard analysis? :D

Answer 6

No, what I recall regarding infinitesimals are that they work only when you apply proper rules to them, and iirc this was a top voted math.stackexchange question

Answer 7

In general, dividing by zero is like trying to play with fire. It can do great things but you usually just get burned.

Answer 8

:)

Answer 9

I never considered it that way actually.
I always think of it as part of the real number continuum.
There are infinite numbers that exist between any two distinct real numbers a and b, yet the distance between them is finite.
We apply differential calculus in attempting to find the rate of change. This occurs when when we make the interval as small as we like based on definition of the slope, which converges to the true slope when tangency is reached (derivative). Some might say we can 'zoom' in to the graph as much as we like and see the difference to be minimal.
In integral calculus, we can apply similar with areas by reducing the intervals in rectangular subsections etc. So there is no zero per se.

In fact you cannot exactly define numbers in terms of magnitudes of 0, can you?

Answer 10

@mww When I think of infinite things but finite distance/area etc, I always think that in terms of integration rather than an idea of differentiation. I don't see how "There are infinite numbers that exist between any two distinct real numbers a and b, yet the distance between them is finite." links in with the rest of the things you say though?

Yes, I think the idea of "zooming" in is really great, since it's intuitive and visually obvious - if you have a smooth graph and zoom in, you'd eventually get it looking almost linear, and the slope of the almost linear line is exactly df/dx

And nope you cannot define magnitudes in terms of 0, but it is *possible* that the infinite sum of infinitesimals becomes "substantial", i.e. a finite number. But it is also possible that the infinite sum of infinitesimals is also infinitesimals.

Answer 11

When I say "There are infinite numbers that exist between any two distinct real numbers a and b, yet the distance between them is finite." this is to say we aren't really dividing anything by zero at all.

A number can be very very small and not equal to 0 eg. 0.01 and there will be numbers much smaller than that eg. 0.0000000000001 etc. So our definition of dividing by a 'small number' doesn't quite cut it. We can only say that even though we can continuing dividing by increasingly smaller and smaller numbers, each closer to 0 than the previous, the result might behave very well and converge.

Answer 12

I c. Agreed!

Answer 13

https://m.youtube.com/watch?v=I0JozyxM1M0

Answer 14

https://m.youtube.com/watch?v=SrU9YDoXE88

Answer 15

HAHAHAHA. @skullpatrol That first video is AMAZING. It's hilarious and it's very interesting seeing why it's wrong. So the thing he is touching on, is metaphysics and philosophy of maths. I'm not a philosopher, nor do I have the expertise, so I don't know the name for it, but he is denying the idea of abstraction, saying that maths can only be "logical" and can only be built on things which are directly related to reality and the physical world... If he wants to take that stance, then things which he says can... seem "reasonable", but the stance itself contradicts the existence of maths itself... hence he denied half of maths (~19mins in). He should've just denied all of maths straight off XD

Answer 16

here's how i 'intuitively' understand calculus..
(because that's the only way i know it XD)

i consider it to sort of be an efficient method for ' searching for curves' that fit whatever problem i'm looking at.

for example if the rate at which a quantity is changing is a curve w.r.t to time,
its not too far fetched to see that total quantity present at any time is also a curve,

so the question is which curve?

so i think of infinitesimals like dx,dy or \(\Delta\)t as sort of visual or arithmetic 'aids' that help me find the right curve.

Answer 17

so i guess my way of looking at thing allows me to avoid thinking about 'what are infinitesimals' since i see them as a bit of mental gymnastics to arrive at some tangible information about some curve.

Answer 18

i really hope this thread is not dead; but it can always be resurrected

the idea behind it is really fundamental.

Answer 19

So you agree with me/understand better @IrishBoy123 ? Tbh, i have noticed that several subjects/ideas/definitions which people are often unclear about and it would be great if there was a few sticky threads which we could point people towards whenever they had that problem.