Zero: Dividing and multiplying [closed]

My question is $\frac{0}{0}*0=?$ I think it should be zero. Beause $\frac00$ can be any number (both real or imaginary). And I think any number multiplied by $0$ should be $0$. I know the proof like this: x*0=y*0 so (0/0)=(x/y) Thus (x/y) can be any number. If this is a common question, please do not downvote?

closed as unclear what you're asking by G Tony Jacobs, TheGeekGreek, egreg, Juniven, NamasteFeb 6 '18 at 0:57

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• Why is $0/0$ any number? Why is it even a number? – Michael Burr Feb 5 '18 at 14:58
• By 'any number' I think he means that form is indeterminate, or as some would say, a variable. – Allawonder Feb 5 '18 at 15:01
• @Allawonder That presupposes that $0/0$ is a number, albeit unknown. – Michael Burr Feb 5 '18 at 15:05
• @Allawonder, “Merely asserting something.... doesn’t make it true”. Physician, heal thyself 😂 – G Tony Jacobs Feb 5 '18 at 17:52
• Maybe in your private mathematics, $0/0$ represents an arbitrary field element, but to the worldwide community of mathematicians, it does not. – G Tony Jacobs Feb 5 '18 at 17:53

It is not $0$. And it is not different from $0$. It is just meaningless.

• It is not meaningless. Any algebraic extension of arithmetic requires that an indeterminate multiplied by zero be zero. – Allawonder Feb 5 '18 at 14:59
• @Allawonder But no algebraic extension of arithmetic can make sense of $0/0$. – Arnaud D. Feb 5 '18 at 15:55
• @ArnaudD. That's not what I'm saying. If $0/0=x$ for some $x\in F$, where $F$ is some field, then $0\cdot x=0$. This holds for all $x\in F$. – Allawonder Feb 5 '18 at 16:01
• I do agree that this implication is logically true, but only because the hypothesis is never satisfied. – Arnaud D. Feb 5 '18 at 16:07
• @ArnaudD. Nice reply! – José Carlos Santos Feb 5 '18 at 16:08

My question is (0/0)*0=?

It's not defined because $0/0$ is not defined

Beause (0/0) is any number

No it's not.

And any $\color{red}{\text{number}}$ multiplied by 0 should be 0

True, but as said before, $0/0$ is not a number.

• I think what he meant to say is that $0/0$ is an indeterminate. In that case he is indeed right. – Allawonder Feb 5 '18 at 15:00
• @Allawonder, Why should an indeterminate times $0$ equal $0$? I can think of situations involving limits where that's not true. – G Tony Jacobs Feb 5 '18 at 15:07
• @Allawonder Then they should replace "any number" with "indeterminate" in his question. "indeterminate" and "any number" is not the same thing, and I prefer to answer the questions that are asked, not those that maybe were and maybe weren't meant to be asked. – 5xum Feb 5 '18 at 15:08
• Are we using the words "intermediate" and "indeterminate" interchangeably? – G Tony Jacobs Feb 5 '18 at 15:09
• @GTonyJacobs It's a case of my fingers being quicker than my brain... – 5xum Feb 5 '18 at 15:10

Common problem in mathematics to think about expressions as computational steps which you carry out in your head, so you can talk about steps being undefined. Mathematics is all about statements and the rules which connect these statements. When you write $\frac{0}{0}$ you say "the number which is 0 multiplied by the inverse of 0" at this instant you introduced an object the existence of which you have not proven (the inverse of zero) so your question can be rephrased as:

"Assuming the existence of the inverse of 0 and denoting it with $\frac{1}{0}$ what $\frac{0}{0}0$ is equal to?" Since your assumption leads to contradiction the answer is everything.

• How is the answer everything? Can you please clearify this? – Asif Iqubal Feb 6 '18 at 4:58