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Division by $0$

Everyone knows that $(x/y)\times y = x$
So why does $(x/0)\times 0 \ne x$?

According to Wolfram Alpha, it is 'indeterminate'. What does this mean?

Also, are there any other exceptions to that rule?

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marked as duplicate by Zev Chonoles Oct 9 '11 at 15:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Consider $0\times 2=0\times 3=0\times 5$. This is true, but this does not imply the statement $2=3=5$... – J. M. Oct 9 '11 at 14:38
the expression $x/0$ is meaningless. And $(x/y)*y = x$ is only true when $y \ne 0$. – Mohan Oct 9 '11 at 14:39
Another way of putting it: $0\times 2=0$ and $0\times 3=0$. Do you not think that it is unsatisfactory to obtain $0/0=2$ from the first and $0/0=3$ from the second? – J. M. Oct 9 '11 at 14:44
@user774025 why is $x/0$ meaningless - it may come to $\infty$ but why is it meaningless? – Alex Coplan Oct 9 '11 at 14:50
May be this link will help you. – Mohan Oct 9 '11 at 15:04

1 Answer 1

up vote 3 down vote accepted

$x/y$ means "the unique number such that $y \cdot (x/y) = x$." If $x$ is any number, does there exist a unique number $a$ such that $0 \cdot a = x\;$?

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