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Why is 0^0 undefined?

Why is 0/0 undefined?

  • I mean, it shouldn't be undefined, because any number when multiplied by zero is zero. So technically 0/0 = set of real numbers, right?

  • Answer:

    I like to this of this question in terms of measurements. If you measure two numbers to be 0 and 0, then no matter how accurate your instruments were, you still have no information at all about the quotient. I'll elaborate on this a bit. Practically speaking, no measurement is precisely correct. If your ruler says that a rod is 1.3 meters long, that just means it's something close to 1.3 meters; close enough that you can't tell with the tick marks on your ruler. If your ruler has ticks at every tenth of a meter, all you know is that the true length is somewhere between 1.25 and 1.35 (these being the cases where 1.3 is the nearest tick mark). So whenever you do division, both numbers are probably a little bit uncertain. But the reason we don't mind writing 1.3 / 0.4 = 3.25 is that as long as our original measurements were fairly accurate, the quotient will be, too. To illustrate: we measured 1.3 with our ruler, but the actual length might be as small as 1.25 or as large as 1.35. We can't tell with the tick marks on our ruler. Similarly, 0.4 might mean anything from 0.35 to 0.45. But this means that the quotient is at least 1.25 / 0.45 = 2.77, and at most 1.35 / 0.35 = 3.86. This isn't great accuracy, but we certainly know something about the quotient. And if we're not satisfied with our accuracy, no problem: just measure the two numbers with a more precise ruler. We can compute the quotient with as much accuracy as we need. Here's the problem with 0/0: no matter how accurate your instruments are, the ratio could still be anything. To illustrate this: suppose we measure both the numerator and the denominator to two decimal places, and obtain 0.00 and 0.00. It's quite possible that the actual lengths are 0.001 and 0.0001 (making the ratio 10), but the actual lengths could also be 0.000001 and 0.001 (making the ratio 0.001). This isn't a matter of poor accuracy, this is a matter of having literally no idea whatsoever about the actual quotient. Even if our ruler is accurate to a hundred decimal places, if we measure both lengths as 0 then we know nothing about their quotient. In a very real, practical sense, 0/0 is all real numbers, like you say. If you measure two lengths to both be 0 with an imprecise instrument, then the ratio between those lengths might well be any real number at all.

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Other answers

Mathematicians are too stupid too give a definition to 0/0 without breaking the system they had already defined. Maybe you can give a definition by yourself.

Xiao Chen

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Abhinav Khaware

Adding to what has already been said, when you are equating two things, they should be the same 'type'. By definition of the '/' operator, the result of a/b should be 'a' real number for all real values is a and b. Although the direction of your thought is great, what you are proposing equates a number to a set, which bends the rules for the '=' operator and hence is not correct. Going to the next level, lets say what you propose holds true. What operations can you perform on the RHS (the set of real numbers) to make meaningful mathematical conclusions? I know it is uncomfortable to have a feeling of something left undefined and our instinct may urge us to associate 0/0 with something or some name. For all its worth, you can define 0/0 to be a new word called 'zerobyzero' but this definition will not serve any meaningful mathematical purpose other than calming our feeling of discomfort with something left undefined. Hope this helps!

Kedar Gajanan Kale

Suppose Y are the total students in a class and you select X students in the form of X/Y. For eg: In a class of 5 students I select 3 students in the form of 3/5. This means I have selected 3 students from 5 However, In the case of 0/0,you cannot select 0 students from 0 students. This was the example given to us by our maths professor which is quite understandable!

Anonymous

The result of  "=" operator can have only one value and not a set of values unless it is algebraic expression where the value of x take as many values as its power is. Here u r using two fixed values which are "0"s and the result should be a fixed value. But due to the nature of Zero... dividing it by itself , as u mentioned could lead to to a set of values (infinite to be precise). But due to argument in the first few sentences of this post the resultant of this expression cannot be a set of values but should be a single value. Hence we define it as "Undefined".

Raveesh N Kote

If I have 2 choclates which I decide to equally distribute among 2 cute kids. Both will be happy. Both will get 1. But if I don't have any choclates and then I decide to do the same distribution, but this time amongst zero kids! Look at the imaginary unhappy faces of those kids. I am trying to dirtribute nothing among no one. Clearly, I can do it in many ways, but all will be non-realistic. That is, there really is no way to do this distribution. Hence no one can determine what 0/0 would look like. Think of distributing 1 candy amongst no one now. Do you know how to do that?

Lokesh Mishra

0/0 is actually one of the indeterminate forms the set of numbers whose value can not be determined by mathematical logic and rules.There many other like this for example 0/(infinite) . Dividing any other number by zero is undefined and not 0/0 the reason is given above. Hope you understood .

Devarsh Ruparelia

I quote myself from one of my other answers: "The actual value of "0/0" is undefined. A way to imagine this would be to restate the valuation of "0/0" as "how many times must you subtract zero from zero in order to reach zero?". The answer could be just any number be it 0, pi or even infinity. All possible numbers satisfy this operation which make this quantity undefined."

Shobhit Sharma

There are three answers for 0/0: Answer1: Let us assume denominator as 'something', 'zero divided by something is always zero.' Answer2: Let us assume numerator as 'something', 'something divided by zero is always infinity.' Answer3: '0/0=1 like 2/2=1.' As there are three answers for this thing, they concluded it as undefined. PS: I was told by my Mathematics lecturer when I was in 12th grade.

Anoop Chittimalla

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