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

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

\(\color{orange}0 \times infinity = ?\) (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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TopNewestAny 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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I think there will be two answers for this question :

Answer 1 :

Any number divided by 0 is not defined or infinity .

Answer 2 :

0 divided by any number gives us 0 so the answer can be 0 .

Note : I think there will be another answer to this question . Any number divided by the same number gives 1 . So 0 divided by 0 can also give us 1 .

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