Seems Silly But Still Tricky

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 What is the value of \(\color{red}0 \times infinity\) ?

  • \(\color{green}\frac{0}{0} = ?\)

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

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

See more of here :

A riddle question number 7

Division of 0 by 0

Wiki page : What is 0 divided by 0

Note by Ram Mohith
4 months, 1 week ago

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

  • 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 \(0 \times \infty\) : \[0 \times \infty = 0 \times \dfrac10 = 1\]

Ram Mohith - 2 days, 11 hours ago

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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.

Brody Burkett - 2 months, 3 weeks ago

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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!

Vaibhav Priyadarshi - 4 months ago

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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 .

Ram Mohith - 4 months ago

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

Ram Mohith - 4 months ago

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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.

Vaibhav Priyadarshi - 4 months ago

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@Vaibhav Priyadarshi Ok

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

Ram Mohith - 4 months ago

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@Ram Mohith What do you think about Infinity * 0 ?

Vaibhav Priyadarshi - 4 months ago

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@Vaibhav Priyadarshi I think :

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

Ram Mohith - 4 months ago

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@Vaibhav Priyadarshi It will be 0 .

I think so .

Ram Mohith - 4 months ago

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@Ram Mohith Let's take [t*(1/t)]

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!

Vaibhav Priyadarshi - 4 months ago

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@Vaibhav Priyadarshi But it seems you took 10, 100, 100000 as infinity and took 0.1, 0.01, 0.00001 as 0 . Is it a correct method .

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 .

Ram Mohith - 4 months ago

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@Ram Mohith I do not take 10, 100, 10000 as Infinity, I only make them closer to Infinity to see what happens to the product.

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

Vaibhav Priyadarshi - 4 months ago

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@Vaibhav Priyadarshi I am telling that if you took the values nearer to infinity and 0 the result will also be less than 1 as you just assumed that they are near to infinity you should assume that the result will also be somewhere near 0 .

Ram Mohith - 4 months ago

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@Ram Mohith I didn't understand what you said. Please explain with example.

Vaibhav Priyadarshi - 4 months ago

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@Vaibhav Priyadarshi For example take \(2.91 \times 3.12\)

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 .

Ram Mohith - 4 months ago

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@Ram Mohith Let I clear it, I assumed 0.0001 = 0, I only take this series closer to 0,

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?

Vaibhav Priyadarshi - 4 months ago

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@Vaibhav Priyadarshi I think the answer will be any real number other than 0 and it will be approximately equal to n where n is the number you took in the formula t*(1/t) .

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

Ram Mohith - 4 months ago

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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.

Marco Afonso - 4 months ago

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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.

Mohmmad Farhan - 1 week, 3 days ago

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@Mohmmad Farhan, why Ram Mohith is your idol?

Vaibhav Priyadarshi - 1 week, 3 days ago

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@Vaibhav Priyadarshi Well yesterday, he spent the entire day patiently teaching me on Brilliant. I think it was like 3-4 hours together

Mohmmad Farhan - 1 week, 2 days ago

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