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- Problems with defining 0! using n!=n(n-1)! formula

I am not a Mathematician so the terms I use are probably not correct in Mathematical language. My concern is with the equation n!=n(n-1)! should only to be used for numbers > 2. I contend that using the ! 'function' on the right hand side applies ONLY when one wants to make a shortcut: ie 6!=6(6-1)(6-2)(6-3)(6-4)(6-5) = 6x5x4x3x2x1 = 720. To work out 7! one could go through the whole process again but why? Just simplify the equation to 7!=7(7-1)! or 7 x6! = 5040.

Now, what seems to happen in almost all the 'proofs' I see is that the fact that the process can be simplified is then applied retrospectively for want of a better word. I have heard it said in some of these explanations that 'We can now apply n!=n(n-1)! for ALL other numbers or at least to 1. And this 'logic' is always claimed after some numbers bigger than 4 are looked at- to me it resembles a magic trick. Why should we apply it smaller numbers?

My contention is that this is terribly flawed and contend that if this time saving tool of using the ! 'function' is not used at all (as I said it is only a device for convenience) we get not a different result but an honest and consistent one.

Using the 'tool' of 1! on the right hand side of the equation allows one to make a seemingly rationale inference that 0!=1 by what appears to me to be circular and faulty reasoning - faulty because, as I have said, the usage of 1! on the right for 1! is wrong. The so called reasoning goes: 1!=1(1-1)! = 1!=1 (0)! Since 1! cannot be 1x0=1, it must follow that 0! is 1. To me, this is almost Pythonesque.

So: why use an 'equation' at all? Obviously, for convenience when calculating increasingly large numbers. For smaller numbers it is simply not needed.

1!=1, 2!=2x1 =2, 3!=3x2x1=6, 4!=4x3x2x1=24, 5!=5x4x3x2x1= 120 and so on. As far as 0! goes, there is a way of showing its possible truth - as I said, I am a layman so don't know if this constitutes a 'proof', That is the pattern 120 - 24 - 6 - 2-1 - ie the divisors from 5! are 5 (120 divided by 24, the next number in the descending sequence) ,4,3,2 and finally 1 to get (logically according to the sequence) to 0.

As I said, I'm not sure if this constitutes a proof and if it does not, the 0! question may only be resolved in an essentially philosophical basis: I won't bore with you all the 'arguments' about 0 and its properties or lack of...Or: (and this is my thinking but I'm sure someone else has said this in the same or different ways): sometimes when there are a number of choices we have to choose the one that makes the most sense... It seems to me that 0!=1 makes the most sense in a Mathematical sense at least...

Finally: it drives me to distraction every time I see the above n!=n(n-1)! equation used to justify the 0! =1 proposition. If it is, for example, being taught in schools, it is maybe tantamount to ... I hesitate in using too strong language ... at least misleading. I make comments on You Tube channels that insist on repeating what I see as a fallacy but I have had no responses. I hope this forum can offer some insights because my 'proof' that using the equation is wriong may indeed be flawed... Comments PLEASE!

Now, what seems to happen in almost all the 'proofs' I see is that the fact that the process can be simplified is then applied retrospectively for want of a better word. I have heard it said in some of these explanations that 'We can now apply n!=n(n-1)! for ALL other numbers or at least to 1. And this 'logic' is always claimed after some numbers bigger than 4 are looked at- to me it resembles a magic trick. Why should we apply it smaller numbers?

My contention is that this is terribly flawed and contend that if this time saving tool of using the ! 'function' is not used at all (as I said it is only a device for convenience) we get not a different result but an honest and consistent one.

Using the 'tool' of 1! on the right hand side of the equation allows one to make a seemingly rationale inference that 0!=1 by what appears to me to be circular and faulty reasoning - faulty because, as I have said, the usage of 1! on the right for 1! is wrong. The so called reasoning goes: 1!=1(1-1)! = 1!=1 (0)! Since 1! cannot be 1x0=1, it must follow that 0! is 1. To me, this is almost Pythonesque.

So: why use an 'equation' at all? Obviously, for convenience when calculating increasingly large numbers. For smaller numbers it is simply not needed.

1!=1, 2!=2x1 =2, 3!=3x2x1=6, 4!=4x3x2x1=24, 5!=5x4x3x2x1= 120 and so on. As far as 0! goes, there is a way of showing its possible truth - as I said, I am a layman so don't know if this constitutes a 'proof', That is the pattern 120 - 24 - 6 - 2-1 - ie the divisors from 5! are 5 (120 divided by 24, the next number in the descending sequence) ,4,3,2 and finally 1 to get (logically according to the sequence) to 0.

As I said, I'm not sure if this constitutes a proof and if it does not, the 0! question may only be resolved in an essentially philosophical basis: I won't bore with you all the 'arguments' about 0 and its properties or lack of...Or: (and this is my thinking but I'm sure someone else has said this in the same or different ways): sometimes when there are a number of choices we have to choose the one that makes the most sense... It seems to me that 0!=1 makes the most sense in a Mathematical sense at least...

Finally: it drives me to distraction every time I see the above n!=n(n-1)! equation used to justify the 0! =1 proposition. If it is, for example, being taught in schools, it is maybe tantamount to ... I hesitate in using too strong language ... at least misleading. I make comments on You Tube channels that insist on repeating what I see as a fallacy but I have had no responses. I hope this forum can offer some insights because my 'proof' that using the equation is wriong may indeed be flawed... Comments PLEASE!

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