Factorial Fact Frenzy (!)
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- Опубліковано 24 лис 2022
- Let me show you why "factorials" are so important and share a bunch of fun(damental) fact(orial)s, such as: factorials hiding in clocks/calendars, why 0 factorial = 1, mutant variations like "hyperfactorials", and more !! As usual in the Combo Classroom, things gets a bit crazy by the end ! Thanks for watching ! Leave a comment ending with an exclamation point so it'll be ambiguous if you're yelling or writing a factorial symbol !
To everybody who asked if I have a Patreon: I'll be starting one December 1, so stay tuned for that if you want to support the channel and get behind-the-scenes videos and other rewards ! Thanks !
Disclaimer: Do not copy any actions you see in this video, apart from the mathematical knowledge. This is for educational purposes.
Combo Class links:
Bonus Channel with more videos: / @domotro
Discord: / discord
Reddit: / comboclass
Patreon: (coming in December)
Some topics in this video include: factorials, permutations, music, chairs, decks of cards, clocks, calendars, 0 factorial, primorials, alternating factorials, hyperfactorials, exponential factorials, double factorials, multifactorials, prime numbers, the gamma function, Stirling's approximation formula, candles, and more... !
Combo Class, taught by Domotro, is an unconventional learning experience where anybody (whether they're a fan of normal school or not) can become excited to learn rare things about math, science, language, and more. Also check out the shorter videos on the Combo Class Bonus channel. Thanks for coming to Combo Class !
Ok here's the Patreon that people requested, which I just launched (with various cool rewards for different tiers) www.patreon.com/comboclass
fun fact: factorials don't actually exist, your math teacher was just very excited about the number 6
Lmao
We math teachers are excited about every number. Every number factorial!
@@NoActuallyGo-KCUF-Yourself but dislikes 1 and 2 more than any other numbers bigger than it, or 0
@@PowerOf47 wdym
6! It’s a six!
Deserves a like just for "welcome to the factorial factory for a factorial fact tutorial"
Your channel is highly addictive, great maths facts and problems spinning from easy to mind-bending. You clearly know a lot of maths but your modest and chaotic back yard approach is refreshingly enjoyable. One of the best maths channels I’ve seen. Keep the videos coming! 👍❤️😎
I love this guy so much.
I missed out on maths having quit school in year 9 (because reasons). When I turned 30 I wanted to try and learn something before I was dead and enrolled in a "get-your-dumbarse-into-uni"-course for adults. A year later I got into computer science.
After so. many. late nights after work doing the very basics of algebra, then trig, then calculus (god bless Khan Academy), I kept my head above water enough to graduate.
Halfway through I realised that maths is not a calculation thing like I always thought, it's a creative thing! A deeply imaginative thing and it's chock full of fascination.
This guy embodies that realisation so perfectly. "And look! There's pi and e lurking in there!! *wide-eyed stare*
I wish I had a teacher like him when I was a boy.
damn man, a lot of people learn this stuff just because they were told to, but you have made a conscious decision and followed it through which I think is way harder. That's impressive, you should be proud of yourself
@@ker0356 Hey thanks man. To be honest, my missus deserves a medal because she picked up my slack due to lost time. Stuff like cooking, shopping, and forcing exercise breaks. God daaaaamn it's important who you marry, and I got mad-lucky 🍀. We have a kid now and "her time" is still yet to come, and I fear it never will, revealing an unfortunate statistic for many women. I am now sloggin' it out, trying to earn a motza in an attempt to provide some freedom for her, but so far it hasn't paid off 😄🔫. The work/husband/father balance is bloody tricky.
Thanks for watching ! Leave a comment that’s ambiguous whether you’re yelling or using factorial symbols ! Also, to everyone who asked if I have a Patreon: I’ll be launching one at the start of December, with cool behind-the-scenes videos and other rewards, which I’ll link in next week’s “snack break” episode here ! Also check out my Bonus Channel for more videos: www.youtube.com/@Domotro
How many eyes do most people have? 2!
I really love ur videos but I cannot understand why these arent more popular, but I have suspicion why, thing is u keep telling same thing over and over again as if viewers were 8 years old. I want u to be more popular but I think u should "tighten up" a bit, so many videos u essentially saying same thing over and over again as if people have difficulty understand what u are saying. This is just my personal opinion and I think u deserve to be more popular than u are. (youtube algorithm is bitch) mr beast is best source of how to make videos "viral". There is science behind it :) btw keep the stupid comedy, its great :D
In the letter video, you said you would remove c, but what about the ch noise?
@@Pr1sk1Your talking about him repeating himself?? Have you even read your own comment🙄 Seriously, I bet he's thankful for your help telling him what he's doing wrong with his channel. Especially coming from someone with such a huge, and successful Utube channel like the one you have...... RIGHT??🙄🙄🙄
The simplest way to distinguish a mathematician from a programmer is to ask them the meaning of!
The absolute insanity and chaotic nature of these videos just gets me hooked every time, along with the interesting random number facts! Awesome work, keep it up!
Double factorial is actually part of an infinite family of factorials, as things like a triple, Quaduple, and Pentuple factorials also exist, each giving even smaller results than the last
Yeah that family is the "multifactorials" I mentioned :)
Didn't see you mention that
Also curious to see if something like a "Double Primorial" can exist
My coworker once reminded us that we needed some parts: "we need 170!"
You’re killing it, bro 👍🏼
Bro are you a real teacher? You're one of the most passionate educators I've ever seen, definitely the kind of person a kid would remember for the rest of their life.
I was so confused on why 0! is 1, but you explained it really well. you're a great teacher
Another, more rigorous idea, is to write factorial in terms of its successor: n! = (n+1)!/(n+1). Plug in 0 and you get 1.
@@faming1144 It's another way to motivate 0! = 1, but I wouldn't consider it "more rigorous". You either define factorials so that 0! = 1 is true by definition from the beginning, or you motivate it by showing it has a property we want and then define 0! = 1 separately. For example, you can prove n! = (n+1)!/(n+1) for all _positive_ integers n. But 0 isn't positive, so plugging in 0 isn't more rigorous. It's just another case of "0! = 1 is what we need in order for this pattern to hold", just like the "permutations of nothing" example.
@@MuffinsAPlenty we can rearrange the formula into "n!=(n-1)!*n" and plug in 1. Would that be more satisfying to you?
@@cheekibreeki904 It's not a matter of being satisfying; it's a matter of rigor. I find using formulas like n! = n*(n-1)! or (n+1)!/(n+1) = n! to be quite satisfying ways to _motivate_ 0! = 1. And they are absolutely lovely things to mention to people! I think it does a disservice not to mention something like this. But unless 0! is already defined, you can't _prove_ that any formula will hold when plugging in something to get 0! = 1. Doing so only shows that _if you want 0! to have a value which makes this pattern continue,_ then 0! must be 1.
You could define factorial in such a way that 0! = 1 pops out immediately.
For example, some people define factorial recursively as 0! = 1 and then (n+1)! = (n+1)*n! for all nonnegative integers n. Here, 0! = 1 is automatic.
My personal favorite way is to define n! is that, for nonnegative integers n, n! is the product of all positive integers less than or equal to n. Here, 0! = 1 can be proven provided that one accepts the empty product convention. In this case, 0! is the product of all positive integers less than or equal to 0. Since there are no positive integers less than or equal to 0, 0! is the empty product, which has a value of 1.
There are other things one could do like defining factorial in terms of the gamma function, which also implies 0! = 1.
Or you can _define_ n! as the number of permutations of a set of cardinality n. From this definition, 0! = 1 is provable.
But if your definition of factorial doesn't imply 0! = 1, then plugging into a formula or continuing a pattern isn't a rigorous proof of 0! = 1. It's just a motivation for defining 0! = 1 separately from your already specified definition.
@@MuffinsAPlenty Maybe rigorous is not the best choice of words, although I called it an idea not a proof.
But factorial as n! = n*(n-1)*(n-2)*...*1 and seen as the number of ways you can arrange n items, both only apply to positive integers. 0! as an extension does not fit above formula, and hence calling it 1 way to arrange zero items seems a bit filosophical to me. The derived formula n! = (n+1)!/(n+1) gives a more mathematical way to define 0! as 1.
Moreover 0! is just one instance of an extension of factorial to all real numbers. Such an extension has been defined by the "Pi" function: integral[0->inf]e^-t•t^ndt, which indeed has Pi(0) = 1. And n! = (n+1)!/(n+1) not only holds for positive integers but is consistent with Pi for all real numbers.
E.g. (1/2)! = Pi(1/2) = √π/2. So (-1/2)! = (1/2)!/(1/2) = √π. And indeed Pi(-1/2) = √π. (And no filosophising about arranging -1/2 items in √π ways.)
Also note that in this extension Pi of negative integers is not defined. (n+1)!/(n+1) is still consistent as (-1)! = 0!/0 which is also not defined and hence all smaller negative integers are not defined by it.
Is anyone else loving the length time of this video? The sum of two cubes, twice!
Please explain
@@smbs471729 famous Ramanujan taxicab number, is the smallest number that can be expressed as sum of two cubes in two different ways
I've never seen so many clocks in a person's back yard but I love it
Your videos continue to blow me away! Absolutely love the content
This is like Breaking Bad but they cook math instead.
That's one way to keep students attentive : at any moment, either I will learn something mind-blowing or chaos will explode on screen.
you honestly deserve more subs than 17.4k with this type of content in my opinion, 2 things i love are math and (occasionally) watching things collapse and crash and stuff, so mixing those together makes the perfect video for me. keep up the good work
Can't wait for comment bots to attack your comment sections though because that's definitely good and wont cause any harm whatsoever /j
2:00
The clock on the far right is 4:24
4! = 24 clever
So there’s this kid in my math class that always sleeps
One day the teacher decides to call him out by asking him “Hey! What’s 5!”
Pretty simple.
The guy woke up and said “uhh 120”
And then fell back asleep
He is like the Explosion & Fire dude but with math
Ha I said that a few videos ago.
These episodes are obviously planned, but don't think they're scripted. Such great awareness of presence of presentation. Just enough repetition to reinforce the concept without boring the viewer. Just enough silliness to entertain, but honestly accurate about communicating the topic concept. Best wishes and appreciate what you're doing.
By the way, if 0! = 1, and 1! = 1, does that mean 0! = 1! ?
Thanks and yes 0!=1!
Is also works if you are using ? As the termial function (like factorial, but instead of multiplying, you add. Example: 3?=1+2+3=6. Fun fact, 3? = 3!)
So, 0! =1! ? Because 0! = 1 and (1!) ? =1? = 1
@andrebenites9919 isn't the ? operator you mention simply triangular numbers (n? = T_n = n(n+1)/2)?
a pyromaniac obsessed with clocks teaching weird math on his backyard? never knew anything better.
Old geezers like me will remember the old Texas Instruments calculators from the 70s would do factorials, and since the biggest number it could display was 10^100-1, the biggest factorial it could do was 69. And it took several seconds to calculate that, which was a novelty, so it was fun to do 69!
I think the most practical use of knowing factorials is about the lotteries, knowing what the chance of our bet's winning to know if it's worth it.
These are the official names of the following factorial variants.
Subfactorial:
The number of arrangements of n elements where all n elements are out of order.
Superfactorial:
Πₖ₌₁ⁿ(k!)
Hyperfactorial:
Πₖ₌₁ⁿ(kᵏ)
Ultrafactorial:
Πₖ₌₁ⁿ((k!)^(k!))
You can even count the number of trailing zeros with these factorials. There's a REALLY CUTE way that uses the floor function.
This entire channel is phenomenal.
HAHAHAHA at 15:49 I think domotro and I were making the EXACT same face at each other AAAAAAAHHHHHH what
My three favourite topics in maths are Permutations Combinations, Probability, and Statistics. So let's just say I'm a huge fan and leave it at that.
Factorials grow really quickly, and some computer systems have a hard time calculating big ones, but if you're only interested in knowing how many digits you'd need to write a factorial, there is a much less computationally intensive way to do it. If you remember how to use logarithms you can maybe figure it out.
Hint:
log(A) + log(B) = log(A*B)
I remember messing around on a calculator and noticing 2^2^2 is only 16 but just 3^3^3
is already in the trillions which was kind of wild to me
@@monhi64 I was about to say that doesn't seem right, but then I remembered you can associate both left and right. (3^3)^3 is only moderately large, but 3^(3^3) is, as you say, in the trillions.
"Pi and e hanging out in there !" made me laugh 😄Gotta admit that's pretty crazy !
Good video. I like the ending!
9:35 Although, 432 is a cool first number for that pattern to break given that it's 4! * 3! * 3. More factorials...
Glad you brought up playing cards, one of my most favorite math-in-action examples.
52! is a number so huge, it's like holding a small universe in your hands.
Goddam! Mister Domotro, you had me totally bamboozled there! I really thought i was getting myself into a factorial factory for a factorial fact tutorial but luckily it was Combo Class afterall...
"Welcome to the factorial factory for factorial fact tutorial, unfractured edition!"
Thanks a million! Your channel is one of my favorites. I love it. Fantastic job! Keep up the great work. God bless you.
Easier way to explain why 0! = 1: Notice that 5!/5 = 4! and 4!/4 = 3, etc. In general, n!/n = (n-1)! So 1! / 1 should be 0!. Thus 0! = 1
Alright this is a fun channel. Keep it up, I can't wait to hear more!
Your channel keeps coming up in my recommended. i finally officially subscribed. Nice work!
Been watching a lot of your videos, great stuff. Fascinated by the primes, and I never realized the importance of the number 4!
Oh man, I love it when pi and e hang out together!
That intro hit my funny bone out of left field man; keep making videos, your personality is simply splendid and your content thoroughly interesting ❤️
Love it, that was so fun and clearly explained!
Such an underrated gem of a channel!!
When this reached the part about how 432! does something special, my initial thought was along the lines of "if a large number is going to introduce something special to a pattern, of course it'd be something like 432, that's twice the cube of 6, it has a lot of factors and comes up often". And then it turns out that the special thing it does is _breaking_ a divisibility pattern? Bizarre.
I like your use of set pieces to demonstrate the universe's tendency toward entropy.
Funny thing about that last curve for fractional factorials ... it actually dips down between 0 and 1 before taking off again, meaning there is a minimum value less than one. Yes, a value less than 1 given that both 0! and 1! equal 1.
Once again, you never fail to amaze with your unique interpretations. Factorials being highly divisible relating to common intervals of time being made from multiple small prime numbers? 8! minutes in February? That's absolutely brilliant! I'm absolutely stealing that for future use. Keep up the great work m8!
Hey dimitri and carlo, I really enjoy your videos about math. I'm always getting my mind blown by how complex but beautiful math is (even though I'm in grade 11 and still haven't mastered what I'm being taught lol). Keep up the good work and I'm 100% sure you'll make it big one day and get the attention you deserve. Love you, from an avid combo class student ❤️🫶
I love this channel so much
Man I loved this video, I'll watch it a weird number of times, like 7! Maybe more!
my favorite factorial fact is that the product of (n) consecutive integers is always divisible by (n!). for example, 58*59*60*61*62 (5 consecutive integers) is divisible by 120 (5!)
I almost never make it through a whole video because I get so many ideas along the way! A really great channel! 🐢
Alright, this is my third or so video of yours I've watched, and the fact you used the actual definition of factorial instead of leaving it at "1x2x...xn" means that you're worth your salt. Glad to be a new subscriber, and I can't wait to watch your channel grow!
I actually love that he keeps dropping shit.
Your channel is gold, hope it'll grow soon
great class professor!
You are absolutely 💯 awesome! Interesting! Knowledgeable, Diffrent! And More!
You kinda feel like the 'beakmans world' of mathematics. Just need a guy In giant rat costume :) Love it
God I love this channel I'm so sad it's not any bigger
Those cards: one deal per second, every billion years, one 1cm step along the equator, each time you pass the pacific empty out a medicine spoon (5mL) of water, every time it empties add a sheet of A4 paper to the pile, every time you reach the moon start actually counting, when you get to the millionth visit, realise you are not even 90% of the way to having completed the exercise (Oh, the universe ended aeons ago)!
In complex analytics, we define it with just 2 rules:
* factorial(0) = 1
* factorial(z) = z * factorial(z - 1)
* factorial is continuous and it's 1st derivative is continuous too
* or is the rule that all derivates must be?
* some rules on complex derivatitves, based on the real derivatives of functions defined by the 1st 3 rules
I see Stirling's Approximation scores another point for tau against pi.
My favorite number is 5!
This man may be the greatest teacher of all time he’s so passionate and charismatic
this is so underrated!!!!
Great video! wasn't the stunt at the end dangerous though?
also I've mean meaning to talk to you about some areas in math I've explored, but haven't gotten around to it, I'll do it eventually though
Those clocks in your yard are really great!
It makes you realize, it is time for some fun :)
Or time for math, depending on your perception.
fun and maths are not mutually exclusive.
@@TigruArdavi neither is "or" :)
It is helping my daughter and brushing me up.🎉 I enjoy it.
This one is a lot of fun and very funny 😂
Excellent video as hinted on the Discord channel!
What this video made me curious was about Permutations / Combinations: what if the things I'm ordering have some kind of characteristics to them that would sub-group them? Like in a deck of cards some numbers are printed in different colors (like Spades, Cups, etc), or when something have more than one sub-grouping characristic in uneven quantities? How to model this on mathematical terms?
TO THE MOOOOOON
You deserve 17k! subs!
fantastic factorial factory for a fun free factorial fact frenzy f-uhh_video 😂 f-thanks
Could you go through the general solution of Navier Stockes next class ?
Need this for fluid dynamics
Awesome. One small tip I have would be maybe always speak towards the mic, because the sound gets distorted.
Who wouldn't have clocks to arrange? That's crazy talk.
Ooh, fire. I'm going to try that at home because I didn't see a warning.
The fire at the end tho
Talking about noninteger factorials reminds me of that video I watched on 1/2 derivatives and antiderivatives
I'd like to know if gaussian primorials are useful
Thank you
I find it so funny that I can hear birds in the background
Combo Class: keeping me excited for the math portion of my CS degree and tempting me to switch one video at a time
If you have the option, do numerical methods module, discrete mathematics (sometimes called algebraic structures) and also automata theory. All are really interesting theories that have applications to both computer science and mathematics as a whole.
5:07 Statistically, 7 shuffles is considered a good shuffle, and a wash shuffle is the most fair. a skilled card player can riffle shuffle in such a way that the cards never actually change position by more than 1, and an even number of these perfect riffle shuffles will result in the deck not having changed its order at all. there are handful of seemingly impossible magic tricks that rely on a perfect riffle shuffle to be done at least twice
Most of that is true but since you split the deck in half when doing shuffles, it takes more than 2 perfect riffle shuffles to reset a deck. Can be done with 8 “out-shuffle” riffle shuffles though. I’ll make an episode about that sometime :)
"I could talk all day about factorials." Um... I think you did .
Great video, I'm a mathematician too, and I was wondering if you would mention non-whole-number factorials and the gamma function. You kind of did at the end. What I like about your channel is that you present things in an interesting way that can be understood by a general audience. I was wondering "how is he going to talk about the gamma function, which requires all this complicated integral calculus?"
Yeah the gamma function will be tricky to explain in an easily understandable way, and didn’t fit in this episode more than that brief mention, but I’ll try making an episode about it at some point :)
@@ComboClass oh by the way, the pi function exists, which isn't offset by 1 and also the integral for it looks even slightly simpler
@@ComboClass not everything has to be simplified, even if you approached it with all its complexities but made it easy to digest I'd still say you were successful.
A way to get a lower bounds on how many digits a certain factorial has is thinking about how many times the number with n digits is multiplied with another up to your factorial, then multiply that result with n to get a number, lets call it Z (for Zeroes) and then just add how many n-1 digit numbers there are and multiply that by n-1, repeat until you reach 2-digit numbers.
I know what I've written is a bit convuluted, but I don't know a better way to describe it. So here's an example.
432! has at least how many digits?
There are 432-100 3 digit numbers. So 332 3-digit numbers. 332*3 is 996.
There are 89 2-digit numbers. So 89*2 is 178. | 996+178 = 1174
So by this example, 432! has at the very minimum 1174 digits.
For tetrationial factorials:
1 = 1
2 ↑↑ 1 = 2
3 ↑↑ 2 ↑↑ 1 = 27
4 ↑↑ 3 ↑↑ 2 ↑↑ 1 = 4 ↑↑ 27 is really big
For context
4 ↑↑ 2 = 256
4 ↑↑ 3 = 4 ↑ 256 = 1.3 * 10^154
4 ↑↑ 4 = 4 ↑ 1.3 * 10^154 has 8.0 * 10^153 digits
...
4 ↑↑ 27 is unimaginably large.
The fact about 432! and the primality for the alternating factorials is very surprising and counter-intuitive.
Is there a symbol for adding up all of the numbers between 1 and a certain number, or exponentiating all of the numbers between 1 and a certain number?
the lengths he goes to make his videos interesting..
i feel the urgent need to play factorio now ..
the factory must !
Nothing beats TREE(3)
Once upon a time, someone loved math so much that the love itself became incarnate, put on a lab coat, and started breaking clocks.
wtf I never knew our time-system is so related to factorials, cool fun fact
I would personally write “multifactorials” as ‘x !_n’ (subscript n), and reserve ‘x!!’ for ‘(x!)!’.
I also generally support the use of superscripts for function composition so that f^2 (x) = f(f(x)), but that’s thrown off by the very strange notation sin^2(x) = (sin(x))^2.
I like swinging factorials the most ~
(You're awesome)!
factorials are one of my favorite parts of math