Why does 0 0 undefined

So, in this scenario the first part doesn't work. So, that means that this is going to be undefined. So zero divided by zero is undefined. So, let's label it as that. Make sure that when you are faced with something of this nature, where you are dividing by zero make sure you don't put an actual number down, or a variable down. Just say that it equals "undefined. We can say that zero over zero equals "undefined. Learning Support. What is Learning Support? How do I know my Placement?

What are the requirements? English Learning Support. Math Learning Support. Placement Test Practice. Study Resources. Learning Support Courses. He writes that it is understood that the base a of the logarithm should be a number greater than 1, thus avoiding his earlier reference to a possible problem with 0 0.

Defining powers is often carelessly done. In this paper [ 1 ], Baron begins the discussion with the following definition:. The powers of any number, are the successive products, arising from unity, continually multiplied, by that number. The first, second, etc. In the same manner, the powers of any number x might be represented as x 1 , x 2 , etc. After stating a few corollaries, Baron writes:. Let us, therefore, next inquire, whether the same definition, will not lead us to a clear and intelligible solution, of the mysterious paradoxes, resulting from the common definition, when applied, to what is denominated, the nothingth power of numbers.

Baron then addresses the rules for dividing powers look back to the argument from the high school text , but he develops a different conclusion:. But since the number x , is here unlimited with regard to greatness, it follows, that, the nothingth power of an infinite number is equal to a unit.

Baron gives credit to both William Emerson [ 3 ] and Jared Mansfield [ 9 ] who wrote on the subject of "nothing. Baron never mentions the term indeterminate form , and he in fact ends his treatise with the following:.

According to Knuth, Libri's paper [ 8 ] "did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether 0 0 is defined. Perhaps Cauchy was developing the notion of 0 0 as an undefined limiting form.

Then the limiting value of [ f x ] g x is not known a priori when each of f x and g x approach 0 independently. And it didn't even matter whether these were positive or negative.

I could make these negative and I'd still get the same result. Negative this thing divided by negative this thing still gets me to one. So based on this logic you might say, "Hey, well this seems like a pretty reasonable argument for zero divided by zero to be defined as being equal to one. Zero divided by 0. Make all of these negatives, you still get the same answer.

So this line of reasoning tells you that it's completely legitimate, to think at least that maybe zero divided by zero could be equal to zero. And these are equally valid arguments. And because they're equally valid, and frankly neither of them is consistent with the rest of mathematics, once again mathematicians have left zero divided by zero as undefined.

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