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Division by $0$
I've always been inclined to believe that x/0 = NaN is a placeholder for a character or constant that no one has created yet.
I know assume that none of you can tell the future, but is there an expectation that someone will eventually (successfully) define division by zero?
2 Answers
$\begingroup$Division by zero can be defined. It is called Wheel Theory. It's not a very popular set of mathematics and the paper that it originates from is a little difficult to find. Division by zero is left undefined in modern mathematics because it causes a loss of many useful statements. For example, $\frac{a}{b}=c \Rightarrow cb=a$. This is not true when $b=0$ and $a$ is nonzero. ($cb=0\neq a$) So, we lose generality by allowing division by zero to be defined.
Allowing division by zero also leads to proofs such as this which are valid:
$$a=b$$ $$a^2=ab$$ $$a^2-b^2=ab-b^2$$ $$(a+b)(a-b)=b(a-b)$$ $$a+b=b$$ $$2b=b$$ $$2=1$$
$\endgroup$ 7 $\begingroup$My understand of arithmetic division over R is this: a/b = {x in R| b*x = a}. In case of division by 0, the answer would be none unless a = 0.
By studying the limits of the function 1/x we can tell that 1/0 is the biggest number ever(which is known to not exists :))
What I wanna say, is that 1/0 doesn't exists and thus cannot be defined. Unless..who knows :)
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