A Factorial Sum Produces the Factorial Number System (visual proof)

Рет қаралды 10,255

22 күн бұрын

This is a short, animated visual proof demonstrating a finite sum involving products of factorials. The proof exploits the classic inductive proof of the formula in question. As a bonus, we discuss the factorial number system and show how the formula can be used to count up in this system.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoffee.com/VisualPr.... Thanks!
This animation is based on a proofs by Tom Edgar from the December 2016 issue of Mathematics Magazine page 338 (doi.org/10.4169/math.mag.89.5....
If you want to know more about the factorial number system, check out en.wikipedia.org/wiki/Factori...
#mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #factorial #squares #factoradic #discretemath #inductionproof #induction #factorialnumbersystem #finitesums #sum #series
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Пікірлер: 66

@clemdelaclem 21 күн бұрын

I think that's the very first time I've seen only an ! as a subscript and have it not be nonsense

@narfharder 20 күн бұрын

Wait, did you see it somewhere that it _was_ nonsense?

@Dalroc 21 күн бұрын

Since 5! = 120 we know that 77 will be a four digit number in base factorial. How many 4! can we fit in 77? 4! = 24, so we can fit three of those for a total of 24*3 = 72. We're still missing 5 to get 77. 3! = 6 and thus we have none of those. 2! = 2, so we can fit two of those. 2*2 = 4. We're still missing 1. 1! = 1 and that fits exactly once. 77 is thus written as 3021 in base factorial.

@MathVisualProofs 21 күн бұрын

greedy strategy ftw :)

@theseusswore 13 күн бұрын

my though t process was 60% of the way there and reading this comment made me feel like I got that last Tetris block to complete the board. nice!

@chrismay9878 18 күн бұрын

"We can keep counting up by adding one each time" was the only part I understood.

@jakobthomsen1595 20 күн бұрын

Great to see the factorial number system here! Combinatorial number system next?

@MathVisualProofs 20 күн бұрын

Would be fun but I don’t know a good visual to go with it…. You know one?

@jakobthomsen1595 20 күн бұрын

@@MathVisualProofs Hm, not yet. A really nice visualization of the binomial coefficient is as cut through a hypercube, as shown e.g. in the video PBS Infinite Series "in Dissecting Hypercubes with Pascal's Triangle". If I find more I'll let you know.

@RandyKing314 21 күн бұрын

your geometrical animation is nice and shows a neat result, but you took it to “a whole ‘nother level” by introducing a related concept with the symbolic animation. excellent work as usual! perhaps future early childhood education will use animation to motivate and demonstrate concepts … this would no doubt change the way those children will perceive knowledge … not only as static pictures and numbers and words, but as elements of the fluid of reason. could be an interesting research project on learning…

@MathVisualProofs 21 күн бұрын

👍

@leif1075 20 күн бұрын

@@MathVisualProofsThanks for sharing but wouldnt yiu agree I don't see 2jy anyone would ever.tgink of writing the minus 1 likenthat..why wojld they? Thanks

@sdf420 20 күн бұрын

@Proud_Kuffar 21 күн бұрын

You are doing a great job. I don't know why this types of videos are not famous.

@MathVisualProofs 21 күн бұрын

Thanks!

@danielsantrikaphundo4517 18 күн бұрын

Very interesting!! I did not expect the sum to lead to a number system.

@3_14pie 19 күн бұрын

I will have to worldbuild a whole culture to use this number system

@Simpson17866 19 күн бұрын

I'm not sure how practical it would be ;) but there have been plenty of cultures that used straightforward systems alongside idiosyncratic systems :D Like how Romans were expected to be experts in political history in order to know what year something happened in. (WALL OF TEXT INCOMING) If the first digit of is either 0 or 1 And the 2nd digit is either 0, 1, or 2 And the 3rd digit is either 0, 1, 2, or 3 ... and the 9th digit is 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 and the 10th digit is either 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or A etc... then you run into the worst of both problems with a small base system (needing many digits per number) AND a large base system (needing many possible symbols for certain digits), and you also can't learn arithmetic intuitively because different place values play by different rules 1 + 1 = 10, but 10 + 10 = 20 11 + 11 = 100, but 110 + 110 = 220 21+ 21 = 120, but 210 + 210 = 1020

@3_14pie 19 күн бұрын

@@Simpson17866 @Simpson17866 nah, not everything needs to be practical, it's nice to do things just for fun sometimes, but thanks for the advice

@alucs6362 21 күн бұрын

Since the nth decimal position corresponds to a value of 1/(n-1)! (such that, for example, 0.1*10=1) then e = 1x2!+0x1!+1/2!+1/3!+...=10.11111...

@MathVisualProofs 21 күн бұрын

What is 0.1 here though. Is it 1/1! Or 1/2’? It’s a weird system but I think you want 10.011111….

@alucs6362 21 күн бұрын

@@MathVisualProofs 0.1 is generically defined as 1/10 (e.g., in base ten, 0.1=1/(10^1), instead 1/(10^0)). Even if you don't define it like that, base changes shouldn't affect the truth of algebraic relationships; since 0.1*10=1 independently of basis, that means 0.1 would be 1/2!. This also preserves the idea that a purely decimal number is smaller than or equal to 1 and maintains unique finite numerical representations!

@MathVisualProofs 21 күн бұрын

agree. I’ve just seen it a number of places as I mentioned. :)

@Dalroc 21 күн бұрын

@@MathVisualProofs It all comes down to whether we care about redundancy or not. If we used your suggested definition in this comment e could also be written as: 1.11111....., since 1! + 1/1! = 2!. Positional values in regular bases are b^n, where n=0 is the last position before the decimal point and n increases to the left. In base factorial the positional values are (n+1)!. We need to add 1 in the factorial to avoid redundancy since 0! = 1!. If we did not add 1 we'd get 10 = 1. If we were to simply extend this system to the right of the decimal point we'd get that the first decimal would be 0!, again resulting in redundancy. All other decimal values would be ±∞ and if the system wasn't broken before it surely is now. Thus there has to be an alternative definition for decimals in this system. Defining them as (abs(n)+1)!^-1 seems most logical and gives e = 10.1111... as the one and only representation. If we do not care about redundacy we'd have to ask ourselves: why skip 0! and 1/0! ? Depending on how you resolve this issue you can have multiple answers, but only one resolves the redundancy. If we include 1/0! we could also write e as 0.111111...

@MathVisualProofs 21 күн бұрын

@@Dalroc for sure. I was thinking about the 1.11111... version but hadn't thought about 0.1111... :)

@narfharder 20 күн бұрын

I'm not sure how you could consistently extend this to fractions, since the n!-th place "digit" d represents the product of (d >= 0, d < n) and (n!), where n is negative. Does the undefined nature of these two factors somehow "cancel out"? Also note: even in that case, the 0!-th place must be undefined, since only one of the above factors is undefined.

@alvargd6771 19 күн бұрын

fun algebraic proof: n(n!)=(n+1-1)n!=(n+1)n!-n!=(n+1)!-n!, and hence you get a telescoping sum for 77 just count as always, 24 is in there thrice so u have 4, then u have a 3 left u can do with 2! and 1! so u get 3021 and e is just 10+1+1/10+1/100+1/1000+1/10000... (which in number systems with a constant base is just 11.111111111...)

@fdileo 18 күн бұрын

I didn't know this theorem. It's amazing

@MathVisualProofs 18 күн бұрын

👍😀

@Smartas599 18 күн бұрын

Amazing.Thanks

@MathVisualProofs 18 күн бұрын

Thanks for checking it out!

@vanshsingh7270 21 күн бұрын

Damn! Math is beautiful

@MathVisualProofs 21 күн бұрын

No disagreement here :)

@LeftGuard 21 күн бұрын

Great video and two nice bits of homework. 👍🏻

@MathVisualProofs 20 күн бұрын

👍😀

@alanthayer8797 21 күн бұрын

Thanks for da VISUALS Visuals visuals! Life w/o them Ain’t da Same!

@MathVisualProofs 20 күн бұрын

😀

@oida10000 19 күн бұрын

The first 5 digit factorial number is 120 as 5!=120 and the digit representation of 77 in base factorial is 3021 (3*4!+0*3!+2*2!+1*1!) as 2*4!=48

@mr.bastolas.7478 20 күн бұрын

Can you do a visual representation of sir ramanujan's sum of integers from 1 to infinity. ?

@yusufdenli9363 20 күн бұрын

Factorial number system is very cool 💥

@MathVisualProofs 20 күн бұрын

for sure!

@konstantindrumev8036 18 күн бұрын

Hi, can you please recommend any books you found useful in your journey of math? Ty❤

@MathVisualProofs 18 күн бұрын

I will see about making a video describing some of the books I have found helpful, especially in this particular endeavor. The short answer, though, is to check out books by Roger B. Nelsen :)

@nickm3694 13 күн бұрын

Fun fact: this numbering system only becomes more efficient than the decimal system after 25 digits (assuming you don't factor in that you need as many symbols for the value of a digit as the number of digits in the largest number you'll use) Also xkcd 2835

@SumanYadav-wr3cn 18 күн бұрын

Please make videos on sieve theory

@David-bh7hs 21 күн бұрын

3:20 - does this mean each n must be equal to or less than its index i?

@MathVisualProofs 20 күн бұрын

Yes. Each digit is less than or equal to its position.

@stevehines7520 21 күн бұрын

"by two" limitless from be-ginning!

@Izzythemaker127 17 күн бұрын

Oh I see, so instead of digit x being kˣ with k being the place value constant, its x! instead. Does a system like that use an infinite amount of digits? If not what is the minimum?

@MathVisualProofs 17 күн бұрын

yes. this is a problem with this system. you need more and more digits for each position.... so the number of digits you need is unbounded.

@eonasjohn 21 күн бұрын

Goes beyond my comprehension.

@RenanMoreira1728 12 күн бұрын

(m+1)!-1=n.(n-1)!

@KrasBadan 19 күн бұрын

5:19 I think that the entended answer was either 10.011111... or 1.11111... This is because e is both 1•2!+0•1!+0•0!+1/1!+1/2!+... and 0•2!+1•1!+1•0!+1/1!+1/2!+... But this answer is wrong in my opinion. You see, in our number system the base function is 10ⁿ, and it makes sense that for fractions we just put negative n. But for factorial number system the base function is n!, and there is no reason to assign (-n)! to 1/(n!). In fact, the factorial function is undefined for negatives (and even gamma function is undefined for negative integers). So the actual answer is there does not exist a representation of number e in this system. Also, there is actually no ambiguity with the fact that 0!=1!. One may think that in this system 1=0.1 or 10=1.1, but actually we can't put any digit in place with the position less than this digit, so just like we can't have a number 2.0 because 2 is greater than 1, we also can't have a number 0.1 because 1 is greater than 0.

@RohitKulan 19 күн бұрын

I'm guessing e would be 10.1111111...

@YT-AleX-1337 8 күн бұрын

77 = (3, 1, 0, 1)

@MathVisualProofs 8 күн бұрын

Might be just off. This looks like 79…

@SeanSkyhawk 20 күн бұрын

wait.... you can do that?

@MathVisualProofs 20 күн бұрын

I hope so, 'cause I just did :) What part are you referring to?

@SeanSkyhawk 20 күн бұрын

@@MathVisualProofs the part where one basically represents all natural numbers solely on the basis of linear combinations of factorials

@MathVisualProofs 20 күн бұрын

@@SeanSkyhawkyep. That’s one possible positional system. There’s a link in description to the wikipedia entry on it. Nice connections to enumerating permutations.