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well, as Tanvir said 1/0 simply does not exist, so there can be no question of whether you can multiply something with it or not, because there is no such thing to begin with.

what i think Rafi meant by 0/0 is actually $\lim_{x \to 0} \frac{f(x)}{g(x)}$ where $\lim_{x \to 0} f(x) = \lim_{x \to 0} g(x) = 0$. this limit is indeterminate because it will be different in different cases.

This limit is not indeterminate.
It depends on the functions, that does not make it in indeterminate. If we do not know the function then of course the limit can be anything depending on what the function is. When we know the function we can determine the limit (if it exists).

This concept of indeterminate seems completely nonsense to me. I don't know who makes these things up. I have never seen them in an actual mathematics book.

Tanvir bhai, so you are saying there is no such thing as indeterminate...

This concept of indeterminate seems completely nonsense to me. I don't know who makes these things up. I have never seen them in an actual mathematics book.

Post 43

Here you said that

The word undetermined has a very specific meaning; it means something is definable, but cannot determined.

Post 23

These two posts are conflicting...

post 43 is in response to limits being indeterminate. There is no such thing as an indeterminate limit. If a limit cannot be determined than it does not exist.

post 23, indeterminate is a set theoretic concept about the truth of a proposition (which is not related to this discussion). For example, Continuum hypothesis is indeterminate Zermelo-Frankel set theory. If you don't understand this then don't worry about it. It simply means that if you assume Zermelo-Frankel set theory (which is a list of axioms) then it is proved that Continuum hypothesis cannot be proved or disproved. But the Continuum hypothesis still exists and is well-defined. We just can not know whether it is true or false (under Zermelo-Frankel).
This is a very different concept from saying that a division or a limit is indeterminate. If an algebraic (division) or analytical (limit) value cannot be determined then it does not exist i.e. it is not defined.

Don't worry about the set theory stuff, it will make things more confusing (it is not relevant here anyway). Let's focus on real numbers.

In your proof that $0/0$ is indeterminate you first wrote $a/b$. But $a/b$ means $a \times \frac{1}{b}$ and this is only defined if $b \neq 0$. So once you wrote $a/b$ in the first line, you cannot take $b=0$ later, because in your first line you assumed $b \neq 0$.

And about indeterminate numbers (and limits): A number or limit that can be any number has no mathematical meaning or usefulness. Of course, you can define random things, just like you can define flying cows, but it will have no meaning except in your definition which will be useless anyways. Meaningless/useless things have no place in math or science.

i don't think the word "indeterminate" was used here as a technical term. it was used literally, meaning that the limit can not be evaluated without knowing the specific functions and will not be the same for everything.

One thing has to be made clear. You can't build your arguments on undefined (i.e. sth that is not defined) notion. Perhaps we all agree that division, as we understand it, can't be done if the divisor is zero. You can't talk about distributing 3 apples among 0 people, because you actually cannot do it. The fact that division by zero is undefined in mathematics can be intuitively understood by considering that you can't pour any water, if there's no pot!!!

All we talk about 0/0 or 1/0 or infinity only make sense when we talk about limits. Otherwise, they all are meaningless daydreams. 0/0 is indeterminate because under consideration of different functions the limit may take any real value, can even blow up. 1/0 will never take a real value and can even have no fixed infinite limit. Same as infinity, it does not mean a number greater than any number one dumbhead can imagine. It means that under appropriate limiting considerations, the result might be made as large as you want.

In physics books, infinity and 1/0 are abruptly used, but they hardly mean it!!! Literally! I mean, has anyone ever put an object infinitely far from a lens? Or have you ever put a charge infinitely far from another charge? Is there anything called an infinite resistance? [Believe me, even an open circuit may conduct current if you push it enough!!]
So all these are in fact limits! They just mean 'very very far' or 'very very big' or 'very very large' (Okay, you may put two or three extra 'very's if you aren't satisfied yet)

Avik bhai, I am not sure but as far as i can remember in your book "Mathescop" i have read that , \[0^{-1}= 0\]

but in this discussion I have got that , \[0^{-1}\] is undefined ....Please make it clear...