# Why is a 0 equal to 1

## Forum: Offtopic Why is the factorial of 0 equal to 1?

by A. A. (Company: ...) (leo888)

Hi everyone, there is a valid and explainable reason why the faculty of 0 equals 1. 1! = 1 2! = 1 * 2 = 2 3! = 1 * 2 * 3 = 6 4! = 1 * 2 * 3 * 4 = 24 ... 0! = 0 * ??? = 1 ??? Thank you very much, greetings

by A. A. (Company: ...) (leo888)

Hi, or could you just argue that there is exactly "one" way of not arranging anything ??

Agit A. wrote:> Hi, >> or could one simply argue that there is exactly "one" possibility> nothing to arrange ?? Correct. https://de.wikipedia.org/wiki/Leeres_Produkt

Otherwise the recursion does not work, then everything would be a multiplication by 0. But it is not the strongest argument either. I think the strongest argument is: that's how it's defined (oops!) And it works out well.

In addition, the recursive definition does not work otherwise. So n! = (n-1)! * n for n = 1

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With the reason one could also set the faculty to 1 for all negative numbers;)

If I add 1 element to n elements, then the number of possible permutations is multiplied by n + 1. So if I start with 0 elements, the result is 0! = 1. Willi Wacker wrote:> I think the strongest argument is: This is how it is defined (Oops!) And it> works well. We had a similar discussion recently for division by zero. There it came down to the fact that a different definition of x / 0 would be conceivable depending on the perspective. Ultimately, however, the viewing angles cannot be merged, which is why x / 0 is most sensibly defined as "undefined". But here I am of the opinion that it is not a definition, but results logically.

P. M. wrote:> If I add 1 element to n elements, then the> number of possible permutations is multiplied by n + 1. So if I start> with 0 elements, the result is 0! = 1. but faculty was once invented with natural numbers and according to the old definition 0 is not a natural number, recently yes, today both perspectives are considered permissible, which is astonishing, depending on how it fits better? http://de.wikipedia.org/wiki/Nat%C3%BCrliche_Zahl

Agit A. wrote:> Yes, but that doesn't say why it is like that, but simply that it is like that. There is a link "empty product" - you should follow that instead of just crowing around ...

by Frank M. (ukw) (Moderator)

In addition, it must be 0! = 1, otherwise e = 1.71828 ... and not 2.71828 ... ;-) P.S. Something nice occurs to me: Take a wheel of fortune that contains all the numbers between 0 and 1. There you turn until the sum of the results exceeds 1. Then a new game begins. How often do you have to spin the wheel on average per game?

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Frank M. wrote:> In addition, it must be 0! = 1, otherwise e = 1.71828 ... and not> 2.71828 ... >> ;-) Oh man, that's a typical idiot argument. Note: Mathematical definitions are not from the Holy Scriptures and also not given by God ...

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by Frank M. (ukw) (Moderator)

Uhu Uhuhu wrote: 'Oh man, that's a typical idiot argument. That's exactly why the smiley is under there. Or do I have to write an SCNR for you quick marker?

Joachim B. wrote:> but faculty was once invented with natural numbers and according to the old> definition, 0 is not a natural number, recently yes, today both> perspectives are considered permissible, but zero is a meaningful value for a number. You can have 0, 1, 2, n books on the shelf. If you have 1 book on the shelf and you take one away, you have 0 books. So the zero fits in there. This is no longer possible for negative numbers: If you have 0 books, you cannot take any more away.

If you consider the factorial as a special case of the gamma function (for integer arguments greater than 1), according to and then consider the integral representation of the gamma function for positive arguments: then you see that it comes out for.

Frank M. wrote:> Exactly for the reason the smiley is under there. Or do I have to write an SCNR for> you Schnellmerker? The faculty is a typical example of one pragmatic Definition: it is constructed in such a way that as many different calculations as possible, in which chain products of consecutive whole numbers occur, can be handled consistently and marginal cases that do not hurt anywhere are simply included. Anyone who just wants to multiply the numbers from 1..n will not - like Agit A. - come up with the idea of ​​why 0! 1 should be. Incidentally, it is very common to take a mathematical model unseen for reality and to derive any conclusions from it regardless of whether the model covers the concrete case at all. Your clever objection is of the kind that forms the basis for such errors ... S. E. wrote:> If you consider the faculty as a special case of the gamma function (for integer> arguments greater than 1), according to That it is not. Leonhard Euler found that the gamma function for positive integer arguments - 1 gives the same result as the factorial. This offers the possibility of defining a real extension for the faculty. But to draw the conclusion from this that because of Gamma (0) == 1 also 0! == 1 must be, is not mandatory at first. Only when you saw that not only did you not deal with contradictions, but that the whole thing was actually quite practical, did the definition of the faculty become the special case 0! = 1 extended. If you don't see mathematics as an unhistorical collection of beliefs ("definitions"), but also look at how the whole building has evolved, things become clearer and more vivid. Unfortunately, certain pranksters love to spread mathematics as God's word and to inflate themselves as the high praisers.

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by Frank M. (ukw) (Moderator)

Uhu Uhuhu wrote:> The faculty is a typical example of a / pragmatic /> definition: it is constructed in such a way that as many> different calculations as possible, in which chain products of consecutive> whole numbers occur, can be handled consistently and relate to> marginal cases, that don't hurt anywhere just with one. You should have simply solved my wheel of fortune puzzle above a) once in a Monte Carlo simulation b) and then again using mathematical considerations For b) you need the definition and application of the faculty (n!), For a) only a computer with a good random number generator. If you then compare the results, you would find that my "idiot argumentation" is not that stupid at all. I don't want to reveal more. P.S. The Monte Carlo simulation converges badly, but it does :-)

Frank M. wrote:> For b) you need the definition and application of the faculty (n!), For> a) only a computer with a good random number generator. And which computer and which random number generator did Euler use? > If you then compare the results, you would find that> my "idiot argumentation" is not that stupid at all. She's stupid because she just happened to be picked from a bunch of examples.

by Frank M. (ukw) (Moderator)

Uhu Uhuhu wrote:> And which computer and which random number generator did Euler> use? Nothing. He could count [1]. > She's stupid because she was just picked randomly from a bunch of examples. But it is a very nice example and of course only a necessary but not a sufficient argument. Just wait until someone writes the solution ;-) By the way: To mathematically refute a thesis, a single counterexample is sufficient. Where is yours? [1] With "arithmetic" is meant the application and demonstration by means of formulas, not arithmetic with concrete numbers. Just to avoid a common misunderstanding ...

> But that's not what it is. Leonhard Euler found that the gamma function> for positive integer arguments - 1 gives the same result as> the factorial. This offers the possibility of defining a real extension for> the faculty. Of course, this conclusion is not mandatory. That's the nice thing about mathematics in contrast to the natural sciences. According to Pipi Longstocking, everyone can, in principle, define everything as they want, including what value the "faculty" should have for them at 0. The question that mathematicians ask themselves is mostly: Does that make sense, or is there a clear solution (which of course always requires further assumptions). For the gamma function, for example, it is the only possible generalization of the factorial if one wishes for a few properties for such a function (which exactly see Bohr-Mollerup theorem). So this extension (and with it the choice 0! = 1) makes a lot of sense from a mathematical point of view. However, if one does not attach importance to preserving these properties of the gamma function (and thus indirectly of the faculty), one can of course also make another choice. I just wanted to emphasize that it is not just a matter of definition, but that there are definitely strong mathematical arguments for this choice. Incidentally, in my opinion, it has little to do with historical development.

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Frank M. wrote:> By the way: To mathematically refute a thesis, a single> counterexample is sufficient. Where is yours? Now you are probably completely overwhelmed ... Just think about it in peace. S. E. wrote: 'Of course, this conclusion is not mandatory. That's the nice thing about> mathematics in contrast to the natural sciences. Yes, otherwise only theologians have this "luck". And this is the trap into which school leavers who have been religiously indoctrinated for over a decade ("religious instruction") like to fall into and happily confuse archetype and image and cannot help separate cause and effect. > According to Pipi Longstocking, everyone can in principle define everything as they> want, including what value the "faculty"> should have for them at 0. The thought structure has to be consistent in itself - Pipi Longstocking is satisfied that it fits into the imagination. > I just wanted to underline that it is not just a> matter of definition, but that there are definitely strong mathematical> arguments for this choice. That's what I meant by the "pragmatic".

by Yalu X. (yalu) (Moderator)

 1 4! = 5! / 5 = 120 / 5 = 24 2 3! = 4! / 4 = 24 / 4 = 6 3 2! = 3! / 3 = 6 / 3 = 2 4 1! = 2! / 2 = 2 / 2 = 1 5 0! = 1! / 1 = 1 / 1 = 1

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