Why 0/0 is not 1

The set with zero elements is called the empty set. Even though there is nothing to put in an order, there is one way to do this. Thus we have 0! Another reason for the definition of 0! This does not explain why zero factorial is one, but it does show why setting 0! A combination is a grouping of elements of a set without regard for order.

No matter how we arrange these elements, we end up with the same combination. There are other reasons why the definition of 0! The overall idea in mathematics is that when new ideas and definitions are constructed, they remain consistent with other mathematics, and this is exactly what we see in the definition of zero factorial is equal to one.

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Measure content performance. So basically zero divided by zero can be anything, in my theory. Here is an argument for the kindergarten notion of division, i. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more.

Seeking elegant proof why 0 divided by 0 does not equal 1 Ask Question. Asked 6 years, 11 months ago. Active 2 years, 11 months ago. Viewed 14k times. MJD Doug Kennedy Doug Kennedy 1 1 gold badge 1 1 silver badge 4 4 bronze badges. This is how we know it is impossible to define it in any reasonable way. To say, it's simply undefined so this is invalid is not the way mathematics is done.

OP is interested in why it can't be defined, not in blindly accepting authority. However, saying it's undefined cause it's undefined is a poor argument if it's an argument at all. Add a comment. Active Oldest Votes. Ittay Weiss Ittay Weiss Show 1 more comment. That is correct, but does not answer OP's question.

You did not answer that question. I know why it does not work. 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.

He aimed to show "that there is a good deal of motivation for defining '0 0 ' to be a numeral for 1. Example 1. Vaughan gave the infinite geometric progression. As stated by Vaughan, if 0 0 is not defined, this summation is senseless. Example 2. This example arises from the infinite summation for e x , which can be written as. Example 3. A third example given by Vaughan involves the cardinal number of a set of mappings.