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I know in calculus the form $\frac{0}{0}$ is indeterminate.but if it is not calculus is it still indeterminate or undefined in the real number field?

P.S. I know that $\frac{1}{0}$ is undefined whether it is calculus or normal arithmetic in the real number system. but in the projectively extensive real number system, $\frac{1}{0}=\infty$ but in this system what would be $\frac{0}{0}$ it becomes $0*\infty$ also an indeterminate form. in that system the problem $\frac{1}{0}$ is solved but $\frac{0}{0}$ remains.

P.P.S some answers in the question which has been identified as a duplicate of my Q state the $\frac{0}{0}$ is indeterminate. but in Wikipedia page(division by zero) states it is undefined. and that question was asked not to clarify this ambiguity(indeterminate or undefined) and there is no clear answer to my problem.


marked as duplicate by GNUSupporter 8964民主女神 地下教會, copper.hat, José Carlos Santos, Delta-u, mathreadler Apr 28 '18 at 22:50

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  • $\begingroup$ idk exactly what it is but i think it is a matter of convenience as both are usually avoided in computations $\endgroup$ – The Integrator Apr 28 '18 at 19:31
  • $\begingroup$ @GNUSupporter It does not answers to my problem $\endgroup$ – thomson Apr 28 '18 at 19:52
  • $\begingroup$ Why this question having eleven answers doesn't answer your question? Your question is "is $\frac00$ indeterminate or undefined". Lehs' answer suggests "In the ordinary number systems division by zero is undefined.", and Bram28's answer suggests that "we say that $\frac00$ is 'indeterminate'" $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Apr 28 '18 at 20:09
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    $\begingroup$ How do you distinguish indeterminate from undefined? $\endgroup$ – copper.hat Apr 28 '18 at 21:16
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    $\begingroup$ I think you are splitting hairs on a red herring... $\endgroup$ – copper.hat Apr 28 '18 at 21:56

Unlike $0\cdot\infty$, which is 0 if it's zero measure times infinite value (or zero value times infinite measure ) but otherwise not defined, one never attributes a value to $0/0$. It's BOTH indeterminate AND undefined.

To be specific it's indeterminate because if you tried to give a meaning to it, you could justify anything. It's undefined because (unlike with $0\cdot\infty$) one doesn't define it by convention either.

  • $\begingroup$ can you give me some recommendation to further reading? and what is your source? $\endgroup$ – thomson Apr 28 '18 at 19:52
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    $\begingroup$ For $0\cdot\infty=0$ in a measure theory context, any good book on real analysis using measure theory will cover that, for example Rudin's Real and Complex Analysis. As for never giving a value to $0/0$, I have a lot of mathematical experience and I've never seen $0/0$ or $\infty - \infty$ given values, whereas it is occasionally useful to define other expressions one doesn't usually define like $0\cdot\infty=0$ in a measure theory context or $0^0=1$ in a power series or polynomial context or $1/0=\infty$ in the case you cited. $\endgroup$ – C Monsour Apr 28 '18 at 21:01

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