What's special about 288? - Numberphile
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- Опубліковано 23 жов 2023
- Featuring Sophie Maclean and superfactorials. Try the Halfsies challenge at brilliant.org/challenge/numbe... (and tell us your score!) or go to Brilliant's courses and start for free at brilliant.org/numberphile/ (episode sponsor)
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For the notation, we could go in a spanish direction and do ¡4! for superfactorial of 4.
you have my vote
Great idea
¡Bravo! 😅
As a Spanish person, I will sign this petition
Very cool
It's even cooler when you realise sf(4) can be written as 4^1 x 3^2 x 2^3 x 1^4.
So sf(4) = 4^1 x 3^2 x 2^3 x 1^4 = 288 = 4^4 + 3^3 + 2^2 + 1^1
Cool!
So, 288 is multiplicatively boustrophedonic. That's cool.
Ah that's cool
isn't factorial and 'to the power of' the same thing?
@@johnjeffreys6440 Factorials and exponentiation are very different. n! is the product of every positive integer up to n, and x^n is x multiplied by itself n times
Notation suggestion: Using an exclamation point after but instead as a superscript. This makes sense (to me, at least) since superfactorials are repeated factorialization in a similar way that exponentiation is repeated multiplication. And "superscript" uses the word super! e.g. 4^! = 288
I was going to suggest this too, for pretty much the same reason.
That's what I was going to suggest. Either that or putting the exclamation point directly above the number. Like this:
!
4 = 288
(Hard to depict in this format. We need new symbols, like the interrobang, but they're not on the keyboard. Yes, I know we can put in some kind of code to call up these "special" characters, but that's a PITA. People who make keyboards need to get with the program and update them for new characters!)
The exclamation mark as superscript is really easy to confuse with just a regular exclamation mark.
@@stephenbeck7222 Right. Which is why I suggest putting it above the number.
@@PhilBagels That's not how typography works.
You're doing a Numberphile video about 288? I think it's two gross.
💀
Ever since the English speaking world started to move over to metric, that statement will be lost on most native speakers nowadays.
@@nottmjas don't worry it's obvious what they mean
@@nottmjasAh, yes, the metric dozen is ruining everything.
@@joshmyer9demolition, not ruin
As a long-time viewer of Numberphile (about 8 years and counting!) it's quite surreal seeing someone I used to see in uni lectures make it famous on here. Thank you for continuing to spread the love of mathematics to a new generation.
Mix the dollar sign with an exclamation point! So it has an "s" for "super" and a "!" for factorials.
Came here to say exactly the same $!
I feel like the favorite number videos is closest to the spirit of Numberphile. You get some much passion from their explanation
girls are not known for their brain, that's a fact. i wonder why so many girls in this channel. someone explain to me.
This was good, more of Sophie please
Does a mathematician ever hear a new number fact and not make it their new favorite number
Calling it now. This woman is the next Hannah Fry. Wonderful exposition - anticipates the reader, has fun quips and keeps folks engaged.
Most of the video was just muttering 1st grade math, and then an one-liner fact. Just like those 'keep watching' shorts.
@@sicapanjesis3987She’s clearly in number theory, which is all about first grade math and making it super complicated.
All that AND an affinity for the interrobang? Definitely the next Hannah Fry
Exactly my thought!
This Numberphile video in particular has a playfulness to it that I think is absolutely awesome, one of my favourites already!
Hi
Thank you, very cool! Fun fact: 288 is twice 144 which is 12 squared and is the 12th Fibonacci number.
If maths doesn't work out for Sophie then there's always scriptwriting - there were so many twists in this video before the crescendo! 😆in all seriousness, I loved the enthusiasm!
Also love that we've got what I like to think as a "classic numberphile" video. A fairly run of the mill, bog standard base ten number that we never think about but - when unpicked - turns into something beautiful!
Simply numberphile from latin phileo = loving. Numberphile = loving numbers.
That's pretty cool! And I think the sf notation is perfectly fine, since in music sf means "sforzando", which tells you to play a note with a sudden emphasis!
Suggestion for notation. Since it's supposed to be super, it should be a buffed up exclamation mark. That could be a triangle (inverted delta) with a bold dot at the bottom. Like a comics BAM exclamation mark
Classic numberphile! Love it!!
Try the Halfsies challenge at brilliant.org/challenge/numberphile (and tell us your score!) - Brilliant sponsored this episode.
FYI this link is giving me a 404 page not found right now
The real link seems to be /halfsies instead of /numberphile
broken link
@@1vader the numberphile link would work if it had challenge and not challenges (btw @numberphile i solved it)
Thanks @@1vader. I got a 94% with 4 perfect scores (including the "cover US with Australia" one!)
It's been a while since we've had an OG Numberphile video about an actual number! Love it!
288 has some other nice properties also. It is a refactorable number, which is a number whose total number of positive divisors is itself a divisor of the number. (A smaller example is 12 which has 6 divisors one of which is 6).
288 also is one of the rare numbers which are both twice perfect square and one less than a perfect square. There are infinitely many of these and they connect to what is known as Pell's Equation.
I was aware of the second property but not the first. Thanks!
Not sure how "rare numbers" and "infinitely many" can both be true. I'll have to think about this.
@@backwashjoe7864 It's about how frequent a type of number is, rather than the sheer number of them. Take powers of ten, for example. There are two in the first ten positive integers (1 and 10), so at that point it looks like 20% of positive integers are powers of ten. If you sample the first hundred, though, you only get three powers of ten, so adjust that estimate to 3%. Sample the first thousand, and we've only got four powers of ten, so adjust again to 0.4%. This trend toward zero continues rather rapidly. If you generate a number at random, the probability of it being a power of ten is 0%.
On the other hand, if we do this same repeated sampling of positive integers but count even numbers, we get a trend toward 50%, and if we generate a number at random, it's got a 50% chance of being even. So powers of ten are rarer than even numbers, even though there's a countably infinite number of each.
I can always remember that second property coming up in a question I faced when I was a kid, which was along the lines of "If you add the numbers from 1 to 8, you get a square number. What's the next EVEN number for which that's true?" The answer was 288. If you think in terms of the formula for triangle numbers you can see why that property is important.
Of course, there is an ODD number that works before 288, but that has slightly different requirements.
@@scudlee Neat. Let our even number be 2n. T(2n)=n(2n+1) must be a square. n and 2n+1 are coprime (i.e. have no prime factor in common) and so must separately be squares: n=y^2, 2n+1=x^2, therefore x^2=2y^2+1, a Pell equation.
Sophie is very nice! Need more videos with her
Simp
@@wernergamper6200 agree
@@wernergamper6200Why not?
@@ForAnAngel I don't know. Maybe it's the uninteresting topic
In a bikini would be interesting ! 🎉
Is it not confusing to use n!! notation as given in the video as opposed to taking a factorial twice (as in, factorial of n!)? The second one seems like a more obvious way to interpret the double exclamation.
Maybe, but if you're gonna use two exclamation marks, this choice is obvious, since taking factorials multiple times can be expressed with parentheses:
(n!)!
It is confusing, but it's also standard 🙃
@@KeimoKissaExcept (n!)! would be taking the factorial of n!. For example (3!)! would be 6! not the superfactorial of 3.
I would say n!! makes more sense for superfactorial, but that notation is already taken.
The fact that 5!! and (5!)! are different numbers is kinda confusing
@@Henrix1998 it's pretty common that you need parenthesis to make a distinction in maths. For example:
4^3^2 =/= (4^3)^2
I propose as the notation for super factorial an exclamation point overlayed over an S. So similar to a dollar sign, but instead of a simple straight vertical line, it's an exclamation point!
Sophie is beautiful in so many ways.
I used this to generalize further extensions of superfactorials. By using my modular reduction technique i`ve developed this pattern holds for factorials, superfactorials, and my generalized extension, yielding the same 6 numbers of cyclic permutation group
My favorite mathematician changes a lot, but for now it is Sophie Maclean.
Every Numberphile's video comment section: X is so great!! Love watchin them. We need more videos with X.
6:20 It's square because it's an even power, not because it's a power of two.
I like her way/manner of explaining things.
"My favourite number changes a lot." I like her already 😄
Yay! Another reason to use the interrobang! I heckin love that bit of punctuation.
Thank you for introducing me to the 'Interrobang' ; ‽ I'm nearly 70 years old ; and I still learn something new every day.
She's brilliant - more of her please!!
288 is special because it's my room number today.
What a coincidence 🤯
For sure 288 is an absolute beautiful number, it's the number of blocks available in Trackmania Nations Forever
For Super Factorial, as an ease of use, I would write 5^! (in computer form) or just add a small "!" as an exponent on the number.
That could suggest that superfactorial is defined: n^!= n^(n-1)^(n-2)^...^2^1. At least I thought that was going to be what superfactorial is, but it's probably something totally diferent
@@filipsperl well, in some specialized field, to put an exponent it is often referred as "sup", so putting the exponent "!" would quite literally translate to "sup factorial"
And I used the notation x^! because "^" is often use in computing to express a "sup exponent"
Superfactorial notation should be a fat exclamation point, so instead of a line and a dot, it's a rectangle and a circle. Kind of like the blackboard notation for number systems - integers (Z), rationals (Q), complex (C), etc.
I dislike the notation "n $" specifically 𝘣𝘦𝘤𝘢𝘶𝘴𝘦 I write monetary values with the monetary unit symbol (such as "$") exclusively after the numerical part. That notation is consistent with other units, allows for prefixes, and does not interfere with negative signs.
On the other hand, univariate functions should typically prefix their inputs (which should be in parentheses), the negative sign being the major exception. So, I would use "!(n)" for the factorial of n and "$(n)" for the superfactorial of n. Also, I prefer to put a dot under the single vertical stroke of "$", so that it looks like an "S" superimposed on "!", for the superfactorial. My monetary unit symbol "$" has two vertical strokes.
I like her teaching style. It is very lively!
Thanks for all the hard work ;)
Her excitement is infectious 😀
A lovely video. Got me hooked halfway through. Fun to see the mathematicians of the future show their passion for maths and creative spirit. Good luck girl!
I don't know that this has been done before and I'm just now noticing it, but recreating the on-screen graphics using the same writing from the actual brown papers is a nice touch!
I've lived and worked on both sides of the Atlantic, and I don't think I've ever heard anyone say "n take 2" for n-2, for example. I have heard people say "n take k" to refer to C(n, k), though it's more common to say "n choose k". In elementary/primary school, some teachers might say "n take away 2" but "n take 2" on its own sounds odd to me.
Yeah, it feels like a case of someone wanting to sound like they are talking about something more complicated than they really are. I mean, how hard is it to refer to a basic function that already has a name (minus) by that name?
0:00 🧮 Introduction to the fascination with the number 288.
0:22 🌀 Factorials: Definition and variations like double and super factorials.
1:26 🔍 Super factorial definition: Product of n! * n-1! * ... * 2! * 1!.
1:50 🔢 Examples of factorials: 1!, 2!, 3!, 4!, 5! and corresponding super factorials.
2:41 🤔 Proposals for a symbol for super factorial: Interabang or 'super' text.
4:07 🧩 Interesting property: 4k super factorial divided by 2k factorial yields a square number.
6:35 💡 Fascination with 288 due to its unique expression as 4! * 3! * 2! * 1! and through powers sum.
8:04 🧠 Advertisement for the Halfsies game by Brilliant, showcasing a puzzle involving visual estimation skills.
8:52 🔍 Mention of a recent discovery of a shorter super permutation than previously known.
I love how Mathematicians often have multiple favorite numbers
Love her enthousiasm, instant part of the numberphile furniture 😅
Great new collaborator. Looking forward to see more of her. She reminds me of Matt with her enthusiasm and new favorite number :)
So much enthusiasm. I'd say this is a SUPER video.
I think the interrobang works really well because it also represents how I felt to find out about the superfactorial.
Thank you for all the great content, Brady! You rock almost as hard as the Mighty Black Stump! And Sophie was all kinds of awesome! I hope you have her back again sometime
I'd love to see how many favourite numbers Sophie and Matt Parker end up going through because I feel like it's a lot :D
Now this is some classic numberphile stuff! More numbers!
17/12 is a pretty good approximation of the square root of two. 17 squared is 289. 12 squared, multiplied by 2 is 288.
To denote superfactorial, follow it with the squirrel (technically chipmunk, but chipmunks are squirrels) emoji 🐿, e.g. 4🐿==288.
Very very interesting! It reminds me of the pythagorean tetractis. 4+3+2+1=10. 288 is part of the 432A "natural" scale. That factorial and n^n factorial relationship also reminds me of the perfect shapes wich have = values for perimeter and area... Oorr squaring the circle. Its like 4321 is a "perfect number" number in aloooottt of ways and i never heard of that one! Sweet! Thanks!
Sophie reminds me of Tom Scott.
That last fact about 288 is nifty, I'll have to remember to show it to my students.
For notation I suggest !n!. You can stack the exclamation marks on each side.
- So if a super-factorial is a the product of the factorials (S5=5!×4!×3!×2!×1!), then would a hyper-factorial be the toootally useful factorial of the factorial of a number (H5=120!)? 🤔
- 2:58 "You never put a dollar-sign after a number, so it feels so wrong" - Just making it clear that Sophie has never been to France or Québec or various other places.
I personally feel like the double factorial notation is better suited for super factorial. You're factorial-ing the factorials, so it would make sense to have two exclamation marks for it. Double factorial could be something different, maybe using a sub-script 2 if that's not already used. Could be expanded for triple or quadruple factorials if those are/can be things
for the super factorial I suggest !_2 (where _2 is sub 2) because the normal factorial is !, i.e. of order 1, and this is of order 2, and of course u can extend it to 3, 4, etc
Same.
I was about to suggest the same (but with a superscript instead, same thing).
A superscript could be confused with a power. Is 3!^2, 3!*3! or sf(3)? You would need a additional symbol, like writing the level into a square or something.
@@adrien5568 Yeah, I was thinking of writing 3!^(2) for example, i.e. the superscript in round brackets, like you see sometimes for other things like derivatives in Lagrange notation, to avoid having to add new notation. But the subscript might be better. Anyway, the important point is that it should be a number representing the "order" of the factorial as Amir said, for generalization. This was the point I also wanted to make.
From the definition of sf() it follows that sf(n) is equal to the product for k=1 to n of k^(n-k+1) [because n occurs in just one factorial, n-1 in 2 of them, n-2 in 3, etc. until 1 which occurs in all n factorials]. Isn't this a much simpler expression @numberphile ?
288 is also the number of valid Sudoku arrangements for a 4 by 4 grid.
My notation for superfactorial is the number with a bowl of soup poured over it.
I've been watching these Numberphile videos for years. Even though the math is almost always way over my head, I am still drawn to the videos and I think I finally realized why: The presenters are always cheerful, often to the point of humorous, in their demeanor. Whether they are young old, male, female, native- or non-native English speakers, their unfailingly bright and upbeat attitudes, just lift my spirits a little. Am I being nutty here?
Fascinating.
Sophie is a great presenter!
288 has another interesting property, connected to the Erdos-Straus conjecture. The conjecture is that for any integer n>1, there are positive integers x, y, z which represent n like this: 4/n = 1/x + 1/y + 1/z. Suppose a value is chosen for n. If there are positive integers a, b, d where 4ab=dn+a+b and d divides ab, then n is represented like this: x is the positive integer where ab=dx; y=an; z=bn. It is known that there are no such a, b, d if n is a square. Only three non-square n are known where there are no such a, b, d. The lowest of these is 288.
I liked the video, but I loved that dollar sign! I had to pause it to make sure she hadn't printed it or used a stencil.
Is this a version of Matt Parker from a parallel universe? This feels like a video Matt would have made, down to every detail, even the everchanging favorite number.
"My current favourite number"... Where have I heard that before?
as a Canadian french speaker, I can confirm that we out the dollar sign after the number! so for me, it is 24$ and not $24.
Cool video, also checked out the Halfsies challenge. Got 95% accuracy, 4 perfect cuts.
If all the standard symbols are used up, clearly we need to go into more exotic symbols. The most obvious oneds are, of cours,e emojis, and it has a perfect candidate for the super factorial: ❕
A numberphile video about a number! Classic return to form.
288 demonstrates the square theorem, too!
SF(4(1))/(2(1))! = 288/2 = 144 = 12^2
that is the most neatly written $ i have seen in my entire life @ 2:52
I would go with !x, because it's really doing both operations.
When someone ends an exclamation with a number, I enjoy reinterpreting that as a factorial, massively inflating the number they were expressing excitement about.
The dollar sign notation could let me do that for people misusing dollar signs.
I do like the interrobang, though.
To put the currency sign after the number is very natural. E.g. five Euro are written as 5,-€.
Bagger 288!
I was gonna say it but checked if someone else did already
With blades covered in gore!
The notation should be !! with a strikeout or double strikeout so it’s effectively a # or H with points at the bottom.
The trouble with the Superman shield is it doesn't imply anything factorial, so how about the pentagon with a ! in the middle?
I would think the bigger problem is that it's not exactly present on most keyboards. 😒
@I.____.....__...__ Just start using emojiis. On Windows, press the WINDOWSKEY + PERIOD. 4😒 = 288
In the former Portuguese currency before we changed to Euros, the Escudo, we used the $ sign after the numbers, not before: 5 escudos was 5$. But actually we always placed the centavos (cents) after the $, so 5 escudos would rather be represented by 5$00, and 2,5 escudos by 2$50.
In Québec, they put the dollar sign after the number. The bus fare is 1,50 $ (un dollar cinquante)
My suggestion for super factorial notation is to put the exclamation mark sideways across the top, like x-bar.
3:29 I suggest moving on to Egyptian Hieroglyphs. For this equation I drew a kitty. This one uses a little bird.
My notation would be n!_2, where the 2 is a subscript. That paves the way for a super-superfactorial, written as n!_3:
n!_3 = n!_2 × (n - 1)!_2 × (n - 2)!_2 × ... × 2!_2 × 1!_2.
In general, you could even define a (super)^k-factorial as n!_k, where
n!_k = n!_(k - 1) × (n - 1)!_(k - 1) × (n - 2)!_(k - 1) × ... × 2!_(k - 1) × 1!_(k - 1).
Maybe it's already been done, but somebody needs to make a playlist of all the "here's my current favorite number" videos 👀
For someone who knows what a factorial is, this video has about 1 min of content. But it’s a cool result!
Double factorial, two exclamation marks
Or put it above like an exponent
I like the interrobang or the Spanish punctuation suggested by @ceegers for the superfactorial notation.
You have to make a video explaining how it is possible to write accurately with such a grip of the marker. I was so puzzled I missed all the maths...
Ditto! Like, WTF?
Yeah, it has always baffled me whenever I've seen one holding a pen or pencil like that, it looks so bizarre and odd that they can even have the dexterity and control to write like that. 🤨
thank you, dröide
Ooh... fresh numberphile video.
À dollar sign after the number feels wrong? You’ll be weirded out if you visit Canada.
2:58 can we just take a moment and talk about that perfect dollar sign she drew..... i mean DAYUM!!!!!
Suggestion: Use an exclamation point with two dots instead of one. Sort of like an upside down ï. The nice thing about that is you can construct supersuperfactorials and n-number superfactorials and the symbol is just n number of dots under the exclamation point.
The video is so on fire, we need to call the video’s runtime.
Glad you like that number, and I respect your choice, but for me it will always be just two gross.
What a great beautiful example for my daughter ! Thank you
I feel like I’ve just been taught advanced maths by a primary school teacher. This is not meant as an insult 😅 I found her (?) very easy to follow!