Can we discuss this?

[Paprika]

1/x as x->0 doesn't converge to any value.

It depends on how you define what an infinite sum is, but simple progressive addition doesn't produce any convergence.

Indeed. So again, and hopefully for the last time, if infinite sums can be defined as an operation yet not produce any convergence for some values over which it is defined, why not 1/0?

[37818]

There's an extra "e" in the title. Vowels cost money!

Oh, that is right, I ceep on frogetting that I cnat spel. You can remove the unnecessary e. Can you? Please.

[Cow Poke]

Oh, that is right, I ceep on frogetting that I cnat spel. You can remove the unnecessary e. Can you?

Why.... it's .. um... GONE!!!!! ShaZAAAAAAAMMMMMM!

[37818]

Infinity is NOT a number.

It can be expressed by the symbol ∞.

∞ - ∞ can be equal to 0. But did you know -∞=∞=∞i too.

[37818]

Why.... it's .. um... GONE!!!!! ShaZAAAAAAAMMMMMM!

Thank you.

[Paprika]

Okay, I'll try again:

It depends on how you define what an infinite sum is, but simple progressive addition doesn't produce any convergence. There are other kinds of ways to define the summing of an infinite number of terms and interestingly they all give 1/4 for the above sum. Euler demonstrated this by defining a new type of infinite sum of terms as the following

You do realise what my point is, right? I am asking why we can't extend the definition of division to include 1/0. You're telling me that mathematicians have extended the definition of addition numerous times (which, incidentally, I'm not unaware of); that does not form objection but precedent.

[Leonhard]

Indeed. So again, and hopefully for the last time, if infinite sums can be defined as an operation yet not produce any convergence for some values over which it is defined, why not 1/0?

Because a*0 = 0, for all a, no matter what a is. However in order to divide a number you need a*b = 1. And if you want to define 1/0 by the limit of 1/x, you'd want it to converge to one and only one result. You can do this for some of those irregular series, where using Ramanujan, or Abel summation you'd get some finite value... the same isn't true for 1/x for x->0, there you can get any result except that |1/x| -> infinity in all cases.

1/0 is undefined.

[37818]

1/0 could be given the output: NaN ∞ or ∞ NaN or error NaN ∞.

Then everybody can be happy or mad, or what ever.

But really what is the true answer? There is none. Or ∞. Or NaN. [NaN = "Not a Number"]

[pancreasman]

It can be expressed by the symbol ∞.

∞ - ∞ can be equal to 0. But did you know -∞=∞=∞i too.

You're using that symbol as if it were a number, but it isn't.

[37818]

Because a*0 = 0, for all a, no matter what a is. However in order to divide a number you need a*b = 1. And if you want to define 1/0 by the limit of 1/x, you'd want it to converge to one and only one result. You can do this for some of those irregular series, where using Ramanujan, or Abel summation you'd get some finite value... the same isn't true for 1/x for x->0, there you can get any result except that |1/x| -> infinity in all cases.

1/0 is undefined.

That is what has been decided. As I had noted, ∞=-∞. That is not accepted. I have my reasons for believing that it is true.

Do an inverse graph. 1 to the center being ∞. -1 is on one side, and 1 is on the to other. y = 5. The line crosses -∞ +∞ being the same point.

[37818]

You're using that symbol as if it were a number, but it isn't.

So its use should be disallowed.

[Leonhard]

x/(x + 1) = 1 for example.
(multiply both sides by (x + 1))
x = x + 1
(divided both sides by x)
x/x = x/x + 1/x
1 = 1 + 1/x
(subtract one from both sides)
0 = 1/x
(1/∞ = 0)
x = ∞

Again, you're treating infinity as a number which is improper as its only defined in terms of limits, not in terms of algebraic operations.

Here's how you'd do that equation properly. We desire to find an x belonging to the real numbers, fulfilling the following property you've given. This is typically written like



We'll do some algebraic manipulations here, since if the above expression is fulfilled by any x, then these following equations must be fulfilled as well, and be true.

Multiplying by (x+1) we get



However this implies that



Which is always false, but since we've only used valid algebraic operations, then our first sentence must be false, reducto ad absurdum. It becomes then its antithesis



And in fact its trivial to show that this result can be extended to any system of numbers where you can do multiplication and division like in the reals, complex numbers etc... called fields.

If you assert that the equation has a solution, any solution, then you assert that "0 = 1".

However its quite possible to show that as you make x larger and larger there is a definite limit to the equation



However notice the subtle distinction here. You were using infinity as an algebraic object, where as here its a symbol that means that we're constantly picking an ever larger x, and that the equation continously gets closer and closer... in fact arbitrarily close to 1. There's a very precise way of defining such a limit, but it would require a discussion of what continuity means. Safe to say that it avoids ever mentioning the word infinity, and the 'lim' with x -> ∞ just becomes a neat short hand for something that would otherwise be long, dry, repetitive and boring to write out.

OK. I respect your answers. I really do. I have come to some of these ideas that are not accepted as correct, for a long time. It does not mean I cannot learn. Or for that matter change my mind. Honestly, I dislike being wrong. To the best of my ability, as limited as it might even be [do to ignorance], I choose to believe what true, when I can understand it to be true.

That's okay, I don't want you to be convinced just because I say so. I'm kinda hoping you'd see the neat logic of it. There's reason why these ways of doing it became standard. Maybe your way could make more sense, but it seems to me that you keep missing certain things, and you tend to justify your logic by reference to how calculators you've used tend to work, rather than what makes sense.

[klaus54]

There's nothing dumb about Teal, but there definitely is something dumb about this post of yours.

Stop turning this thread into a discussion about evolution.

I didn't say she was "dumb".

Do you know the meaning of "incredulous"?

No? I didn't think so.

Incredible.

K54

P.S. I despise Fundies.

[Leonhard]

1/0 could be given the output: NaN ∞ or ∞ NaN or error NaN ∞.

Then everybody can be happy or mad, or what ever.

But really what is the true answer? There is none. Or ∞. Or NaN. [NaN = "Not a Number"]

NaN

You can't divide by 0, so the result is not a number.

[klaus54]

So its use should be disallowed.

So you think all mathematical symbols represent numbers?

Remarkable...

K54