This question is the one of the excellent doubts I had ever got . I think this will be the most mysterious question ever . It is as follows .

** What is the value when \(\color{blue}0\) is divided by \(\color{blue}0\) ?** and also

\(\color{green}\frac{0}{0} = ?\)

\(\color{orange}0 \times \infty = ?\) (A question by Vaibhav Priyadarshi)

See all the comments below to get a clear picture of these mysterious questions .

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Wiki page : What is 0 divided by 0

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## Comments

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TopNewest\[\begin{array} ~0 \times 1 = 0 \implies \dfrac00 = 1 \\ 0 \times 2 = 0 \implies \dfrac00 = 2 \\ 0 \times 3 = 0 \implies \dfrac00 = 3 \\ \large \quad \quad \vdots \\ 0 \times 1000 = 0 \implies \dfrac00 = 1000 \\ 0 \times 100001 = 0 \implies \dfrac00 = 100001 \\ 0 \times 9999999 = 0 \implies \dfrac00 = 9999999 \\ \large \quad \quad \vdots \\ 0 \times n = 0 \implies \dfrac00 = n \quad \text{where n is finite number} \\ \end{array}\]

When \(0\) is multiplied to any finite number it results in \(0\). So \(\dfrac00 = 1,2,3,... ,10001,10002,......,99999......so~on\). So, we cannot find the exact value of \(\dfrac00\). Hence, it is in in determined form.

\(0 \times \infty\) too is in an in determined form so we cannot tell its exact value. But there is a silly way for finding \(0 \times \infty\) : \[0 \times \infty = 0 \times \dfrac10 = 1\]

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Question 1 Answer: 0 items per 0 boxes yields a case where 0 = undefined.

Question 2 Answer: 0 items per infinite boxes yields complete 0, also known as 0 infinity. Basically means complete, total, and absolute entropy.

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Any number multiplied by 0 gives 0, so 0/0 = any number! So, 0/0 = 0 is true,...........(1) 0/0 = 1 is true,...............(2) 0/0 = 2 is true...............(3) Then from eq. (1), (2), (3), we have, 0 = 1 = 2 which is Contradiction. So, we can't use the original definition of division for 0/0 which says result of y/b equals a number which when multiplied by b gives y. 0/0 does not obey definition of division, so let's say 0/0 is not defined!

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My view is that because the algebraic law of division will not be valid for 0 as the answer to it cannot be determined . So \(\frac{0}{0}\) should be made an exception for algebraic division .

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But how did you get 0/0 = 2 ,3 ,4 and so on .

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Division of p by q means to get a no. which when multiplied by q gives p.

For example: 8/4 means what number that must be multiplied by 4 to give 8. Clearly the answer is 2.

Similarly, 0/0 means what should be multiplied by 0 to give 0. Since every number multiplied by 0 always give 0, so the result can be any number.

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Then what do you think is the correct answer . Is it infinity in any case .

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As \(\frac{0}{0}\) = infinity then \(infinity \times 0 = 0\) (cross multiplication)

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I think so .

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If we make t closer and closer to 0, we see that something like this happens,

0.1*10 = 1

0.01*100 = 1

0.00001*100000 = 1

So this product is coming closer to 0 * Infinity, but the product always remains 1, so 0*Infinity =1

If you take [t*(5/t)], and do the same thing to it, you will see result is always 5.

So, 0*Infinity is also undefined for the same reason!

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Although 0.1, 0.01 , 0.00001 are nearly equal to 0 you can not take them as 0 .

Also, if we assume 10, 100, 100000 as infinity and took 0.1, 0.01, 0.00001 as 0 the result will be less than 1 and greater than 0 \((0 \neq x \neq 1)\) . If we round off that value we will get 0 .

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Also I didn't take them equal to 0, I only take them closer to 0.

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Your explanation is in this manner : Assume that \(2.91 = 3\) and \(3.12 = 3\)

So the answer will be \(3 \times 3 = 9\)

What I am saying is that you just assumed that the 2.91 = 3 and 3.12 = 3 . So you got answer as 9 . But actual answer is : \(2.91 \times 3.12 = 9.0792\)

So in your case the answer will not be 1 and it will be either less or more that that .

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I take 0.1, 0.01, 0.001,......... Closer to zero. I DIDN'T assumed them to 0.

Now finally, what do you think about 0 * INFINITY?

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Wait I will write this question in the main note so that all will see .

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In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value; when it is the form of a limit, it is an indeterminate form.

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Comment deleted 2 days ago

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1) Any number divided by zero is NEVER infinity

2) The answer cannot be zero nor 1. Watch this TED video.

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@Mohmmad Farhan, why Ram Mohith is your idol?

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