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If someone has come up to you, and just asks you directly to compute

$$ \frac{1}{0} $$

Would we say that no solution exists or that it equals infinity?

CivilSigma
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    I guess one should be polite. Maybe no, thank you. My calculator is less polite. It flashes a big E and refuses to do anything more until it is reset. – André Nicolas Aug 26 '15 at 18:49
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    Suppose $\dfrac{1}{0}=x$. Then we might be compelled to say this is the same as $1=0x$ (using what we know from algebra). But those same rules tells us that zero times anything is zero, which would suggest that $1=0$. Is this true? – user170231 Aug 26 '15 at 18:50
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    I think that this question has been asked already $\frac10$ times. – AdLibitum Aug 26 '15 at 18:52
  • If you're operating over extended complex numbers, then yes, it is a complex infinity. But as rule of thumb - here. – Kaster Aug 26 '15 at 18:53
  • 1/0 is undefined, because it is asking you to find a number R such that R*0=1 – Socre Aug 26 '15 at 18:54

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To be clear, you don't say $1/0$ either has or does not have a solution, because it is not an equation. If you wrote $1/x=0$ then you it would make sense to speak of whether or not there's a solution.

The proper way to ask the question is "Is $1/0$ defined or should it be left undefined?" And the answer is that it must be left undefined because it can be argued that it should be different things depending on the context.

Gregory Grant
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  • Thank you for the clarification. – CivilSigma Aug 26 '15 at 19:04
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    In some contexts it makes sense to say it is $\infty$ and to make no distinction betwene $+\infty$ and $-\infty$. That is true when dealing either with vertical asymptotes of rational functions or trigonometric functions, or with horizontal asymptotes of trigonometric functions, and in projective geometry. ${}\qquad{}$ – Michael Hardy Aug 26 '15 at 19:10
  • @MichaelHardy Right, in fact it's almost definable, since it has to be some sort of infinity - as compared to $0/0$ which really could be anything. – Gregory Grant Aug 26 '15 at 19:14