yes, !! is also a math symbol

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We introduce the double factorial !!, also called the alternate factorial. This cute little bit of mathematical notation gives us a way to achieve an effect similar to the standard factorial but while skipping every other number. The double exclamation point will respect the parity of its number, and proceed accordingly. We'll see examples of the factorial and double factorial, and then discuss perfect matchings in graph theory to see an example of where the double factorial is the solution to a problem. #maths #graphtheory #mathematics
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Опубликовано:

17 окт 2024

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Комментарии : 155

@WrathofMath День назад

Join Wrath of Math to get exclusive videos, lecture notes, music, and more: ru-vid.com/show-UCyEKvaxi8mt9FMc62MHcliwjoin More math chats: ru-vid.com/group/PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO

@TheHazbean-ks3gi День назад

At some point mathematicians are just going to be writing equations that look like perfectly functioning English.

@sans1331 20 часов назад

LMAO

@dylanm.3692 16 часов назад

4nd 0n 7he 1ntern3t, 3nglish 15 1ooking m0r3 lik3 m47h3m471cs.

@user14707 16 часов назад

Don’t give the category theorists an idea…

@GerryCatLord 13 часов назад

@@dylanm.36921nf3$+@+ 0n by dm dokoru

@ShadowPhoenix4798 10 часов назад

the volume of a pizza with height a and radius z is pi•z•z•a

@TavoLL1511 2 дня назад

Or in Spanish notation: n! -> ¡n! n!! -> ¡¡n!!

@WrathofMath День назад

yes 😂

@Alice_Sweicrowe День назад

😂😂😂

@sergiocm2911 День назад

n! -> ¡ñ! n!! -> ¡¡ñ!!

@Alice_Sweicrowe День назад

@@sergiocm2911 That's only in specific cases.

@terminusfinity009 День назад

so then (n!)! = ¡(¡n!)!

@danielbranscombe6662 День назад

reminds of this reddit bot I used to see that would reply to any comment where they had an ! after a number and would reply by computing the factorial. LOL.

@tenebrae711 День назад

It's so in "aksselly" style of Reddit, like you say "it was over 100!" and some dumbass "mis"interprets it as 100 factorial, completely neglecting the semantical context of the aforementioned. those people are the worst (I'm a math nerd)

@jaydentplays7485 День назад

@@tenebrae711So it was over 100*99*98*97*96*95*94*93*92*91*90*89*88*87*86*85*84*83*82*81*80*79…

@pillowfromtpotreal День назад

@@jaydentplays7485i wanted to comment the actual factorial but there's no way I'm writing that whole number without finding a way to copy it from somewhere

@Drayson_scalie День назад

93,326,215,443,944,152,681,699,238,856,266,700,490,715,968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,000,000,000,000,000,000,000

@lastchance1036 22 часа назад

@@pillowfromtpotrealhere you go 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

@th1v5 День назад

Subfactorials are so underrated. I love to use them, especially for those problems where you have to take digits and make an expression that equals a given number. There are actually an infinite family of subfactorials, which are longer chains of exclamation points, such as n!!!! for the 5th subfactorial of n, or n * (n-5) * (n-10) etc.

@isavenewspapers8890 День назад

@@th1v5 Shouldn't that be 5 exclamation marks?

@Fperm-t6g День назад

what the fuck!?!? actually 5th subfactorial is not a thing, 5th factorial is

@erdmannelchen8829 День назад

Is there use cases for n!!! , n!!!! etc? And Would a way to abbreviate the amount of ! you write be that you replace the dot with a number? So that way you could have any number of !, even a pi amount of !. Which I think would work like: 9! = 9 * (9-pi) * (9-2pi)

@xshwei День назад

I think you can just use a general product sign for this, denoted as Π. Π(k=0, n/3) (n-3k) which would give you n(n-3)(n-6)...(n-n). I think I screwed the upper bound of this multiplication, but you get the point

@nnoxie.a День назад

n!!! = n(n-3)(n-6)(n-9)... n!!!! = n(n-4)(n-8)(n-12)...

@h20dynamoisdawae37 День назад

Literally found an infinite sum for arcsin that utilized the double factorial and got mad when desmos didn’t support it lol

@lukandrate9866 День назад

it can be expressed using regular factorials. Something like (2n)!/(4ⁿ(n!)²)•x^(2n+1)/(2n+1). then you sum it up from n=0 to infinity

@h20dynamoisdawae37 День назад

@@lukandrate9866⁠i did end up doing that but i was still annoyed anyways lol

@isavenewspapers8890 3 часа назад

I'd like to see them add that, but if they just did it with the regular notation with multiple exclamation marks, it could break some older projects. Instead, they could do it with a subscript on the exclamation mark: subscript 2 for double factorial, 3 for triple, and so on.

@isavenewspapers8890 3 часа назад

Note that Desmos does support product notation, using which the double factorial can be written.

@undoubtedly7157 День назад

simply the difference between something great and something brilliant

@daneclarke186 День назад

I knew I wasn’t the only one…

@isavenewspapers8890 3 часа назад

Holy heck.

@HoSza1 23 часа назад

someone should also opt for !!! triple factorial

@spaceguy20_12 19 часов назад

I wonder what 5!! is, would you just be screaming “FIVE”

@WrathofMath 18 часов назад

turns out, you'd be screaming fifteen

@TheSheep1 14 часов назад

@@WrathofMathFIFTEEN

@spaceguy20_12 14 часов назад

@@WrathofMathFIFTEEN

@reminderIknows День назад

! - that's great! !! - that's brilliant!!

@AbdAlHakamJunaid 16 часов назад

!!! - that's crazy!!!

@DaviUndertale 7 часов назад

@@AbdAlHakamJunaid !!!! - I'm a psychopath

@rgbatom5145 15 часов назад

brilliant move !!

@erenerbay 13 часов назад

♙♙♙

@WizDaPenguin День назад

bro summoned the chess players

@rathercurly Час назад

Hehe

@Deficard День назад

Haven't knew double factorial and other multifactorial until now. i made the same mistake before i knew. super helpful!

@WrathofMath День назад

Thanks for watching! There are a lot of weird factorial operations, will definitely talk about more in the future

@MyOneFiftiethOfADollar Час назад

Not a mistake, but a possible interpretation of 3!! = (1x2x3)! =6!

@isavenewspapers8890 8 часов назад

The volume of an n-ball with radius r is given by this formula: V_n(r) = 2^n ((π/2)^⌊n/2⌋ / n!!) r^n, or, if you prefer, V_n(r) = 2^n ((τ/4)^⌊n/2⌋ / n!!) r^n. This formula, which is quite beautiful to me, features the double factorial function. Due to this, the double factorial will always hold a special place in my heart.

@mjohnson5030 20 часов назад

04:30 on even numbered double factorial... "You could include 1 if you want to..." Can you? You said parity, "even or odd numbers only",was the definition of double factorial. 1 is an odd number. Wouldn't every even "!!" end in zero? 8!!=(8*6*4*2*0) ?

@WrathofMath 18 часов назад

no, you only continue the multiplication down through positive integers. I only said you could include 1 as a passing remark, where people might wonder what the stopping point of the multiplication should be. Of course, typically it's 1; but I should have addressed 0 rather than flippantly throwing out the 1 remark.

@MyOneFiftiethOfADollar Час назад

I think I heard this called a double exclam post operator in grad school study groups once?!(interrobang) Like you said, a natural interpretation is n!! = (n!)! Your excellent graph theory exposition clearly justified the parity based definition n!! = n(n-2)(n-4)....1 What happened to the coffee pot dude?!?!?

@The_fox_furry_lover День назад

Me as a speedcuber that hears the word “parity” (aka a state of a rubiks cube that cant be solved with only the outer layer turns or has to be solved with a really long algorithm) (aka a pain in the ass)

@Bober26681 День назад

same

@geopediashorts День назад

same

@MyOneFiftiethOfADollar Час назад

Are there prize paying speed cubing tournaments?

@lkern6238 2 дня назад

Very clear presentation!

@WrathofMath День назад

Thank you!

@isavenewspapers8890 День назад

Excellent video!

@WrathofMath День назад

Thank you very much!

@RunstarHomer 2 дня назад

Super helpful, thank you :)

@WrathofMath День назад

It is my pleasure!

@osakablinladen 23 часа назад

i feel so represented!!

@barretthoven 4 часа назад

3:09 what is this gorgeous music????

@ernstboyd8745 10 часов назад

using named functions like sin(x) make possibilities nearly infinite but sin and cos and exp are so much more commonly used than x!! it seems like they should have special 1 character symbols emojis anyone?

@BritishBeachcomber 9 часов назад

Mathematicians do not steal symbols. They just repurpose them for convenience.

@WrathofMath 9 часов назад

True, if I said steal I take it back!!!

@MyOneFiftiethOfADollar Час назад

they have been known to steal colleagues' research and "repurpose" it just enough to avoid plagiarism accusations

@RichardPenner 19 часов назад

Assuming we know how to take the factorial of a number halfway between integers, we have: m!/(2^((m - 1)/2) × ((m - 1)/2)!) ≤ m!! ≤ (m/2)! × 2^(m/2) The left comparison is equality when m is odd, the right comparison is equality when m is even. Thus if k is any non-negative integer, (2k + 1)!! = (2k+1)! / (2^k × k!) and (2k)!! = k! × 2^k So for any integer n >= 1, n! = n!! × (n-1)!!

@MuneebMakes 16 часов назад

nerd

@studiogervassi 13 часов назад

At first I thought; 5!! Would intuitively be: 5!*4!*3!*2!*1! How would you write this then?

@isavenewspapers8890 3 часа назад

This is called the superfactorial, denoted sf. Specifically, it's the one used by Neil Sloane. In your case, you'd have sf(5).

@Manisphesto 5 часов назад

If this is a double factorial then this (!!!) is a... ... triple factorial. 15!!! = 15×12×9×6×3 14!!! = 14×11×8×5×2 13!!! = 13×10×7×4×1 12!!! = 12×9×6×3

@glarynth 17 часов назад

We'll steal quotation marks next.

@rewixx69420 День назад

n!! is so cool how about the swinging factorial tho

@WrathofMath День назад

Jeeeez, I was looking up weird factorials the other week and I never heard of that one! Will have to investigate; there are so many!

@Jamal_Almansoub День назад

Look at my comments if you can. Its amazing factorial. I formatted before one year ago and its so useful.

@YunxiaoChu День назад

@@WrathofMathcool

@ichika._ 17 часов назад

i thought it would be come cool stuff like: 4!! = 4^3^2 i was dissapointed but cool video

@isavenewspapers8890 3 часа назад

That's called the exponential factorial. The exponential factorial of 4 is 262,144.

@WilliamLister-wx8uk День назад

Now, i have actually used a factorial symbol once, it had nothing to do with anything i was intended to be doing. I just wanted to find a pattern in a pattern of x^n.

@WilliamLister-wx8uk День назад

Now I need to find a use for !!

@8083music 20 часов назад

This is very well explained, even I can understand it Lol

@thegrandyata День назад

Subscribed because you used super Mario sunshine music

@Alexander-oh8ry 10 часов назад

This video progresses kind of slowly, at least for my taste

@Jamal_Almansoub День назад

I have a new way to do that and its so easy. !!

@Jamal_Almansoub День назад

In my way i write 7₂! = 7 * 5 * 3 * 1 nₘ! = (n-0*m)(n-1m) (n-2*m)...* 1 And that's for any number n,m ∈ ℕ And also can be zero only for n , Not for m. And i will tell you why. I think thats enough for now.

@Jamal_Almansoub День назад

In my way I write 7₂! = 7 * 5 * 3 * 1 nₘ! = (n-0*m) (n-1m) (n-2m)...* 1 and thats for any number ∈ ℕ and also can be Zero. But only for n And I will tell you why. I think that's enough for now.

@lememaker День назад

double factorial is quite odd

@Jahwobbly 12 часов назад

The comma says hello.

@WrathofMath 10 часов назад

I assume you mean this in response to my line about ! being the first punctuation mark adopted since the period? I disagree because, whether it was clear in my words or not - I meant adopted into math with a distinct usage. Like, the period in english is certainly different from the decimal point in math. The comma, on the other hand, I think serves a very similar purpose in math and English. It's not doing anything super unique in the number 123,456,789. The comma is just separating ones, thousands, and millions. It's a list just like in English, 123 millions, 456 thousands, 789 ones.

@isavenewspapers8890 4 часа назад

@@WrathofMath Or, say, separating elements in a set.

@WrathofMath 4 часа назад

@@isavenewspapers8890 right

@rarearyantroops День назад

Excellent 👌

@WrathofMath День назад

Thank you!

@leventemolnar6818 День назад

Random proposal for an n-tuple factorial: x!ₙ, where n is the number of factorial symbols. So for a double factorial for example, the notation here would be x!₂ or x!!. This creates a relation between the number of factorials in a term and the gap. Suppose 4!!. We know that 4!! = 4*2 because the “!!” skips every other number. Now for 4!!! (4!₃). This just equals to 4*1, which = 4. This time, there is a 2 number gap. A better example would be 10!₃. 10!₃ = 10 * 7 * 4 * 1. Generalization: x!ₙ = x * (x - (n - 1)) * (x - 2(n - 1)) * … This goes on until the last natural number in the sequence. Special cases for 1! and 0! apply, regardless of how many factorials you put. 1!ₙ = 1, 0!ₙ = 1

@lagomoof День назад

It's sometimes written with a subscript on the exclamation mark, and can't be easily confused for other notation that way. e.g. x!₃ for x!!! and x!ₙ in general (assuming that whatever you're reading this on supports those Unicode subscript characters).

@leventemolnar6818 День назад

@@lagomoof thanks. will convert accordingly.

@michaelcutler6118 День назад

Now how about triple factorials

@321Jarn 13 часов назад

For some reason most calculator websites don't support double factorials.

@isavenewspapers8890 3 часа назад

Probably because it's fairly obscure and not used that much.

@gummyrblx6049 13 часов назад

brilliant move ahh video

@ratamacue0320 14 часов назад

Double factorial? More like half-assed factorial. Haio...

@sheerun День назад

4:24, watch 2 seconds

@leefisher6366 16 часов назад

I'm understanding 'parity', however I'm hearing 'parody'. It actually confused me for a while, wondering what the parody of a value is.

@MihaiNicaMath 19 часов назад

Great video!! I made a video for the Summer of Math Exposition about how you can use perfect matchings and double factorials to prove the Central Limit Theorem from statistics. It's the pinned video on my channel if you want to check it out :)

@jaydentplays7485 День назад

So what would be 4!!!?

@leventemolnar6818 9 часов назад

4!!! = 4 * 1, therefore 4!!! = 4.

@Sawatzel 5 минут назад

2:50 exellent spelling haha

@thatoneactualdude 23 часа назад

so this equation is allowed: (ni-0)!!

@vampire_catgirl День назад

Double factorial isn't a very good name for it if it doesn't even equal a single factorial

@beckett67poop День назад

ikr? but it is double as confusing

@isavenewspapers8890 3 часа назад

Oh, hello again! I've seen some of your comments before. Anyway, perhaps you'd prefer semifactorial, used by Donald Knuth.

@Free.palestine8 2 дня назад

Wonderful 🤍

@WrathofMath День назад

Thank you!

@Miguel_I_guess 17 часов назад

Not a brilliant move, just every other one

@YLCinnasnail 4 часа назад

Cool!!

@MuneebMakes 16 часов назад

(3!)!

@isavenewspapers8890 3 часа назад

720

@ProfessorBorax 17 часов назад

How to take 16mins to explain a concept that could be explained in 1 :p

@isavenewspapers8890 3 часа назад

You mean the double factorial part or the graph theory part?

@z0ru4_ 17 часов назад

Cool

@SingularityG 8 часов назад

Cool!! See what i did there

@terminusfinity009 День назад

the double factorial's symbol makes zero sense

@WrathofMath День назад

what do you mean? double factorial - two exclamation points, seems reasonable!!

@ExzaktVid День назад

Its more of a half factorial ngl

@terminusfinity009 День назад

@@ExzaktVid yeah thats why i call it the semifactorial

@terminusfinity009 День назад

@@WrathofMath two factorials i would think would be (the factorial of the number used) factorial like n!! i would think it would be (n! × (n!) - 1 ...) not (n × (n - 2) × (n - 4)...)

@terminusfinity009 День назад

@@WrathofMathplus, what would three factorials be

@RuleofThehyperbolic День назад

what about 5!³ᐟ⁴

@WrathofMath День назад

What bout it?

@CananaMan День назад

me trying to decide whether that should be 120 to the ¾ power, or 5*4¼*3½*2¾*2*1¼

@RuleofThehyperbolic 21 час назад

@@CananaMan the power is on the ! therfore there is ¾ exclamation points

@isavenewspapers8890 4 часа назад

@@RuleofThehyperbolic I believe I've seen an expression like n!^2 being used for the square of the factorial of n before. You might be going against the grain here.

@RuleofThehyperbolic Час назад

@@isavenewspapers8890 yeah my notation is probably less useful than that but its still makes sense especially considering that n!! doesn't equal (n!)! so why should n!² equal (n!)²