yes, !! is also a math symbol

[@WrathofMath]

Join Wrath of Math to get exclusive videos, lecture notes, music, and more: kzbin.info/door/yEKvaxi8mt9FMc62MHcliwjoin More math chats: kzbin.info/aero/PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO

[@KenFullman]

You have the word "math" twice in the thumbnail alone and still refuse to admit it's plural. 🤣

[@TheHazbean-ks3gi]

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

[@sans1331]

LMAO

[@dylanm.3692]

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

[@user14707]

Don’t give the category theorists an idea…

[@GerryCatLord]

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

[@Oricorio-Pom-Pom]

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

[@NetRolller3D]

0!=1 is true both in math and in programming.

[@WrathofMath]

A classic!

[@Jacobconnor525]

False is not equal to true

[@AJJ0904]

Because programming is math

[@ShawnGamerTV]

@@Jacobconnor525 let me break it down for you: if 0 != 1: print("True") else: print("False") output: True

[@HexagonPankake]

@@WrathofMathoh god what’s classic factorial equal to

[@cragasm]

! = excellent move !! = brilliant move

[@vectorboom1982]

And he sacrifices THE FACTORIAL!!!

[@cragasm]

@@vectorboom1982 is that a !!! magnificent move???

[@ytozylea8869]

#ChessRelated

[@trphoenix_.]

AND HE SACRIFICES THE QUEEEEENNNN!! **he sac'd the queen dance**

[@Axcyantol]

what

[@TavoLL1511]

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

[@WrathofMath]

yes 😂

[@Alice_Sweicrowe]

😂😂😂

[@sergiocm2911]

n! -> ¡ñ! n!! -> ¡¡ñ!!

[@Alice_Sweicrowe]

@@sergiocm2911 That's only in specific cases.

[@terminusfinity009]

so then (n!)! = ¡(¡n!)!

[@reminderIknows]

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

[@AbdAlHakamJunaid]

!!! - that's crazy!!!

[@DaviUndertale]

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

[@TheInstrumentMass]

I see whatcha did there hhhehehehehehehahahhaha 🧌

[@DefenseScavenger]

AND HE SACRIFICES THE ROOK!!

[@V1c._.]

e4 or e5?

[@studiogervassi]

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

[@isavenewspapers8890]

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

[@briaplhalore]

34560

[@danielbranscombe6662]

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]

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]

@@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…

[@cloudyfromtpotreal]

​@@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]

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]

​@@cloudyfromtpotrealhere you go 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

[@h20dynamoisdawae37]

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

[@lukandrate9866]

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]

@@lukandrate9866​​⁠i did end up doing that but i was still annoyed anyways lol

[@isavenewspapers8890]

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]

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

[@spaceguy20_12]

I wonder what 5!! is, would you just be screaming “FIVE” Anyways I wonder what 5!!! is, would you just be screaming 10 or something (because uh, I think 5!!! would be 5x2 and that’s 10 for all you who haven’t done 1st grade)

[@WrathofMath]

turns out, you'd be screaming fifteen

[@ÞeSheep1]

@@WrathofMathFIFTEEN

[@spaceguy20_12]

@@WrathofMathFIFTEEN

[@XDBot_GOATED]

@@WrathofMathFIFTEEN

[@JasmineLi]

@@WrathofMathFIFTEEN

[@undoubtedly7157]

simply the difference between something great and something brilliant

[@daneclarke186]

I knew I wasn’t the only one…

[@isavenewspapers8890]

Holy heck.

[@someaccount5200]

Did they do a rook sacrifice

[@isavenewspapers8890]

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.

[@Speed001]

What is an n-ball?

[@isavenewspapers8890]

@@Speed001 To start, imagine a circle. This circle lives in an infinite flat 2D space, known as a plane. Note that we're not counting the inside region as a part of the circle. In math terms, that inside region is called a disk. Anyway, the radius of this circle is the distance from the center of the circle to a point on the circle. This distance is the same in all directions. No matter which point on the circle you choose, it'll always be the same distance from the center. So we define the circle like this: the circle is the set of all points in the plane that are a certain distance, the radius, away from a certain point, the center. With a sphere, it's a similar story. Every point on the sphere is the same distance from its center. Therefore, the sphere is defined as the set of all points in 3D space that are a given distance, the radius, away from a certain point, the center. The inside of a sphere is another shape, known as a ball. Now, let's talk about dimension. For example, a plane is a 2-dimensional object. But how do we know? Well, we need 2 coordinates so that we can label every point in the plane. There are different types of coordinate systems. The most popular is (x, y) coordinates, known as Cartesian coordinates. You can use a different kind of coordinate system, like polar coordinates, but you will always need at least 2 coordinates in a given system to span the entire plane. So, what is the dimension of a sphere? Let's look at a real-world example. Earth is a ball, and its surface is a sphere-well, approximately. We have a system for identifying points on Earth's surface: the geographical coordinate system (GCS). This system has two coordinates: latitude and longitude. Latitude tells you how far north or south you are, and longitude tells you how far east or west. Using two coordinates, you can label any point on the surface of Earth. For example, the Great Pyramid of Giza has a latitude of about 28° N and a longitude of about 31° E. So, a sphere is 2-dimensional. What about a circle? Let's take the equator of Earth, for example. If a point is constrained to fall on the equator, then we just need its longitude to know its location. That's only one coordinate, so the circle is 1-dimensional. A circle is called a 1-sphere, and a sphere is called a 2-sphere. This concept can be generalized to other dimensions as well. The n-dimensional version of a sphere is called an n-sphere. Now, remember how a disk is the region of a plane enclosed by a circle? You need 2 coordinates to label a point on the disk, so the disk is 2-dimensional. Meanwhile, a ball, the region of 3D space enclosed by a sphere, is 3-dimensional. A disk is called a 2-ball, and a ball is called a 3-ball. This concept can be generalized to other dimensions as well. The n-dimensional version of a ball is called an n-ball.

[@Speed001]

@isavenewspapers8890 i see, thank you. You meant n-dimensional versions.

[@_TheNoobPlayer]

​@@isavenewspapers8890Thank you for the explanation! Do other formulas exist for other forms too?

[@isavenewspapers8890]

@ You mean, like, other generalizations of shapes to n-dimensional space? Well, I know of a simple one. An n-cube is the n-dimensional version of a cube-think squares and cubes, including the inside regions. A square with side length s has an area of s^2, and a cube with side length s has a volume of s^3. In general, the volume of an n-cube with side length s is given by this formula: V_n(s) = s^n So yeah, that one's not complicated. Edit: I meant the cube has a volume, not an area. Oops!

[@th1v5]

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]

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

[@Fperm-t6g]

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

[@th1v5]

@@isavenewspapers8890 yea woops

[@dalemonshateu6948]

Had a math assignment where I could use (n+1)!!! No hesitation, best way to solve the problem

[@th1v5]

@@dalemonshateu6948 the correct kind of solution

[@pncka]

It indicates a brilliant sacrifice of parts of equations in order to solve them.

[@erdmannelchen8829]

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]

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

[@zachshorts]

Triple and quadruple factorials work in the way doubles do (which skip every 2nd number) Triples skip every 3rd number, and quadruples skip every 4th number. This can go on infinitely. Example: 10!!! = 10x7x4x1 2nd Example: 10!!!! = 10x6x2 Absurd Example: 10!!!!! = 10x5

[@TechnoTurbo]

a use case could be for some infinite series if you feel lazy in the notation, maybe in some taylor series

[@word6344]

"Why are the numbers yelling?" -- high school philosophy teacher when my group presented something about math to the class

[@WrathofMath]

Great time to share my funny philosophy teacher story. Class in college called "Thinking for Yourself". Took a test, got a problem wrong, argued with the professor that my answer was in fact correct. He agreed, then said "But the text says otherwise, so your answer is wrong".

[@tcatking9761]

PROTOGEN!!

[@Axcyantol]

@@tcatking9761i'm starting to get sick of this exact thing happenening every time i see one

[@FireyDeath4]

@@WrathofMath Seems like the text is right by definition but incorrect. This is why you do not trust what is "right". In reality, only functional logic and mathematics are right, which will eventually give people who've declared contrary statements as "right" an unpleasant wake-up call

[@WizDaPenguin]

bro summoned the chess players

[@rathercurly]

Hehe

[@Drxcture]

I'm here 😃

[@plokenv]

​@@Drxcture Hehe same!! ❤

[@Drxcture]

@@plokenv Yay! :D

[@FootballisLife11993]

I'm here

[@brknfs]

!! Brilliant Move

[@zachshorts]

Triple and quadruple factorials work in the way doubles do (which skip every 2nd number) Triples skip every 3rd number, and quadruples skip every 4th number. This can go on infinitely. Example: 10!!! = 10x7x4x1 2nd Example: 10!!!! = 10x6x2 Absurd Example: 10!!!!! = 10x5 Then, there's primorials. Primorials are much more complex, with it being subtracted by every prime number. Example: 10# = 10x7x5x3x2 2nd Example: 30# = 30x29x23x19x17x13x11x7x5x3x2 I'd assume a double primorial (I don't think this exists), would act the same way as a double factorial. Example: 10# = 10x7x3

[@daveanderson2107]

So do triple etc factorials have similar uses as the double factorial, just in slightly more complex problems? Or would such uses be vanishingly rare?

[@Scrumboiling]

"!!" Is the sum of a brilliant move

[@davidchedester8181]

But the derivative of a brilliant is a blunder 💀

[@doughnutplayz]

and he sacrificed, PEMDAS!

[@davidchedester8181]

@@doughnutplayz and he sacrificed, Algebra!!

[@Totally_Not_Greek]

that's a brilliant move!

[@yungdkay1008]

Your teaching is very articulate and wow

[@WrathofMath]

Thank you!

[@UntitledVideoChannel]

“It’s over 9,000!!” Just got a new meaning 💀

[@spaceguy20_12]

better name it “it’s over 2601218943565795100204903227081043611191 5218750169457857275418378508356311569473 8224067857795813045708261992057589224725 9536641565162052015873791984587740832529 1052446903888118841237643411919510455053 4665861624327194019711390984553672727853 7099345629855586719369774070003700430783 7589974206767840169672078462806292290321 0716166986726054898844551425719398549944 8939594496064045132362140265986193073249 3697704776060676806701764916694030348199 6188145562519559256691883082551494294759 6537274845624628824234526597789737740896 4665539924359287862125159674832209760295 0569669992728467056374713753301924831358 7076125412683415860129447566011455420749 5899525635430682886346310849656506827715 5299625679084523570255218622235813001670 0834523443236821935793184701956510729781 8043541738905607274280485839959197290217 2661229129842051606757903623233769945396 4191475175567557695392233803056825308599 9774416757843528159134613403946049012695 4202883834710136373382448450666009334848 4440711931292537694657354337375724772230 1815340326471775319845373414786743270484 5798378661870325740593892421570969599463 0557521063203263493209220738320923356309 9232675044017017605720260108292880423356 0664308988871029738079757801305604957634 2838683057190662205291174822510536697756 6030295740433879834715185526028053338663 5713910104633641976909739743228599421983 7046979109956303389604675889865795711176 5666700391567481531159439800436253993997 3120306649060132531130471902889849185620 3766669164468791125249193754425845895000 3115616829743046411425380748972817233759 5538066171980140467793561479363526626568 3339509760000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 000000000000000000000000090!” (factorial of (6!)!, it should be 6x4x2! and that is 720! and that is 2601218943565795100204903227081043611191 5218750169457857275418378508356311569473 8224067857795813045708261992057589224725 9536641565162052015873791984587740832529 1052446903888118841237643411919510455053 4665861624327194019711390984553672727853 7099345629855586719369774070003700430783 7589974206767840169672078462806292290321 0716166986726054898844551425719398549944 8939594496064045132362140265986193073249 3697704776060676806701764916694030348199 6188145562519559256691883082551494294759 6537274845624628824234526597789737740896 4665539924359287862125159674832209760295 0569669992728467056374713753301924831358 7076125412683415860129447566011455420749 5899525635430682886346310849656506827715 5299625679084523570255218622235813001670 0834523443236821935793184701956510729781 8043541738905607274280485839959197290217 2661229129842051606757903623233769945396 4191475175567557695392233803056825308599 9774416757843528159134613403946049012695 4202883834710136373382448450666009334848 4440711931292537694657354337375724772230 1815340326471775319845373414786743270484 5798378661870325740593892421570969599463 0557521063203263493209220738320923356309 9232675044017017605720260108292880423356 0664308988871029738079757801305604957634 (I removed a lot to not absolutely decimate your device)

[@tavii79]

Now we can do maths with EVEN MORE enthusiasm!!

[@unnevenn]

5:04 tower of paradise

[@unnevenn]

mf liked my comment as if bro knows😭

[@DuBCraft21]

I wish I had learned about the double factorial 6 months ago when I was learning about taylor and maclorian series... This would have been so damn helpful for simplifying things instead of writing it with only single factorials and exponentials

[@Deficard]

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

[@WrathofMath]

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

[@MyOneFiftiethOfADollar]

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

[@Deficard]

@@MyOneFiftiethOfADollar yeah, that's how i interpret it before i knew it. even my android calculator do this. but just a week ago, i know multifactorial. never has something, fooled everyone. 💀💀💀

[@Tommy_1nn1t]

And it is called a brilliant move

[@D.U.D.E.-LSTRMS]

They added brilliant moves to math

[@ST3LLAR.mp3]

TL;DW: !! is a factorial, but skip the numbers that aren't the type of _n._ So 7!! = 7 x 5 x 3 x 1 (skip evens, only odds and vice versa for even numbers)

[@dieuwer5370]

So, 7! = 7!! x 6!! Also 6! = 6!! x 5!! etc. Another fun one: 10! = 6! x 7!. Therefore, 10! = 7!! x (6!!)^2 x 5!!.

[@rgbatom5145]

brilliant move !!

[@erenerbay]

♙♙♙

[@TheInstrumentMass]

🧙🏿‍♀️

[@prenstopper]

??

[@amazingSoupaDoopa]

⁠@@prenstopper that’s a blunder

[@FF_1488]

!! is briliant move

[@lmao-uw7vp]

What a *brilliant* symbol.

[@booferguy]

I did 6!! on a calculator and it counted it as a number so awfully big, it’s considered infinity

[@someaccount5200]

It probably read (6 factorial) factorial then

[@booferguy]

@ yeah it did

[@spaceguy20_12]

probably read it as (6!)! and that would be 6x5x4x3x2x1=720 and that makes the (6!) into 720 and it becomes 720!, yeah, not even I could do that first one without a calculator and 720! IS SO LARGE IT TAKES UP MORE THAN HALF OF THIS REPLY: 2601218943565795100204903227081043611191 5218750169457857275418378508356311569473 8224067857795813045708261992057589224725 9536641565162052015873791984587740832529 1052446903888118841237643411919510455053 4665861624327194019711390984553672727853 7099345629855586719369774070003700430783 7589974206767840169672078462806292290321 0716166986726054898844551425719398549944 8939594496064045132362140265986193073249 3697704776060676806701764916694030348199 6188145562519559256691883082551494294759 6537274845624628824234526597789737740896 4665539924359287862125159674832209760295 0569669992728467056374713753301924831358 7076125412683415860129447566011455420749 5899525635430682886346310849656506827715 5299625679084523570255218622235813001670 0834523443236821935793184701956510729781 8043541738905607274280485839959197290217 2661229129842051606757903623233769945396 4191475175567557695392233803056825308599 9774416757843528159134613403946049012695 4202883834710136373382448450666009334848 4440711931292537694657354337375724772230 1815340326471775319845373414786743270484 5798378661870325740593892421570969599463 0557521063203263493209220738320923356309 9232675044017017605720260108292880423356 0664308988871029738079757801305604957634 2838683057190662205291174822510536697756 6030295740433879834715185526028053338663 5713910104633641976909739743228599421983 7046979109956303389604675889865795711176 5666700391567481531159439800436253993997 3120306649060132531130471902889849185620 3766669164468791125249193754425845895000 3115616829743046411425380748972817233759 5538066171980140467793561479363526626568 3339509760000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 000000000000000000000000000

[@kupa121]

6! is already 720 and I don't wanna imagine how big 720! is

[@ILikeCookies-498]

the symbols of ! and !! are also used in chess. the ! means a great move and the !! means a BRILLIANT move. !! > !

[@songnameplscommenter]

Great Move, Brilliant Move.

[@theodriggers549]

BRILLIANT MOVE

[@WaddlerTheDuck]

Math got brilliant moves now? what did I miss 😭😭😭

[@NotSomeJustinWithoutAMoustache]

Brilliant!

[@WrathofMath]

Thanks for watching!

[@Ronxneo]

@@WrathofMathyou don’t get the joke do you?

[@WrathofMath]

@@Ronxneo I didn't when I replied, I do now after seeing a dozen more comments like this

[@ReinCooked]

And he sacrifices... THE FACTORIAL!!

[@ijustlovemath991]

this is brilliant

[@HoSza1]

someone should also opt for !!! triple factorial

[@CCABPSacsach]

Next, the solution to the collatz conjecture will be “Wow^ie that(sUre/was) hard!!”

[@davidchedester8181]

👁️👄👁️

[@PhillipHarris-nr4wt]

brilliant move

[@sheerun]

4:24, watch 2 seconds

[@ruifengguo2019]

i don’t get it

[@sheerun]

@ruifengguo2019 it is all some mathematicians need to know

[@TowerOfChromaticInclination]

At This Point ? Might Be A Symbol Of Math

[@Win090949]

!! is also the symbol in chess for “Brilliant” like this video’s spo-

[@Blutania]

So 9!!! = 9×6×3 = 162 (only the numbers which have the same remainders when divided by 3)

[@Vitrivius]

brilliant move!

[@barretthoven]

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

[@lkern6238]

Very clear presentation!

[@WrathofMath]

Thank you!

[@qsquared8833]

Can tou go over this again more because MAN I loved Golden Eye and I feel like you only scratched the surface here!

[@The_fox_furry_lover]

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]

same

[@geopediashorts]

same

[@MyOneFiftiethOfADollar]

Are there prize paying speed cubing tournaments?

[@geopediashorts]

@@MyOneFiftiethOfADollar Yes, there are!

[@MyOneFiftiethOfADollar]

@@geopediashorts big bucks?

[@dtvjho]

! (Bang) and !! (Bang bang) are both useful in the C language

[@question4984]

Didn't know you can sacrifice odd or even numbers with this brilliant move. Jokes aside, this really was insightful to watch. Thank you for making this!

[@beragis3]

You can also write n! as n!! times (n-1)!!

[@geomniom]

Mathematicians are like the British museum

[@otomatik48]

3!!=720

[@lavinissensonthecommenter4197]

Brilliant move

[@KadGd]

Maths made a brilliant move…

[@milkplaysgames69]

Brilliant!!

[@Benioo6]

From all i know the !! Means that a move was briliant and ! Means that the move was exelent

[@Curlack]

So 8!! * 7!! = 8! 😮 Actually n!! * (n-1)!! = n! provided n > 0

[@Kennyliteofficial]

Brilliant move!!

[@JustAnMathAndSpaceEditor]

Now we need a triple factorial

[@Caysen52TheSecond]

9!! is [00],[945], 10!! is 15,360, 11!! is 10,395, 12!! Equates 46,080, and 13!! |is equal to| 132,935.

[@MichaelShaw-dk5gm]

Yeah, the !! Means brilliant move

[@nomyxbg3206]

Great move and Brilliant move. Take it or leave it.

[@WrathofMath]

leave it

[@pollenspike]

14:23 who is hearing Dire Dire Docks?

[@MyOneFiftiethOfADollar]

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?!?!?

[@cuberman5948]

he got us in the first half, not gonna lie

[@geebanimates]

I never knew Great and Brilliant moves were in math

[@isavenewspapers8890]

Excellent video!

[@WrathofMath]

Thank you very much!

[@DOORS_FROSTBITE]

Explanation: So 1!=1x1=1 And 3!=1x2x3=6 While 3!!=The factorial of 3! or 6!. And there is 3!!! Or 6!!. Hope this enjoyed!

[@mkadenwoods]

Brilliant move

[@lizzie-wizzie]

amazing!!

[@WrathofMath]

Math frequently is!

[@alextenor5378]

brilliant symbol

[@really-silly]

brilliant

[@RichardPenner]

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]

nerd

[@RunstarHomer]

Super helpful, thank you :)

[@WrathofMath]

It is my pleasure!

[@mjohnson5030]

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]

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.

[@pncka]

It indicates a brilliant sacrifice of your computers ram.

[@kupa121]

Funny, I actually guessed the right answer. I was a bit loo lazy to solve it completely, but n!/n!! seemed like it would make sense, but then I realized that's actually the same as n(-1)!!, because all the other numbers cancel out!

[@supr0gg190]

briliant move

[@ThatGuywithNoFacecam]

ah yes. brilliancy from chess exists in math too.

[@ernstboyd8745]

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?

[@zachcapricorn3512]

So 0! = (e^ipi)^2 = 1^2 = sqrt(1)

[@AustriaSolarS]

THE ROOOOOOOK!!

[@reviewerofcomments]

didnt watch the video, but in mathematics, !! is used to indicate that an equation is very cool and brilliant

[@ThxnderXD]

she edge my node till i graph theory

[@WrathofMath]

...bruv

[@ThxnderXD]

@WrathofMath 😈

[@mariafilipowicz2645]

U placed too high. 9:42

[@ReviCat]

! = Factorial Number 7!=5040

[@ふわ魔法]

It's not quite as pretty but: Define 𝑛 ∈ 2ℕ = {2,4,6...} or 𝑛 ∈ 2ℕ - 1 = {1,3,5...} for odd numbers. With that you can do a lot more customization like ℕ³ = {1,8,27,64...}, with the 4th cubic factorial number being 13824. Edit: Also ∏(𝑛³) from 1 to 𝑘 would be the same effect as the above without having to imply that the factorial applies to a certain set.

[@davidchedester8181]

The... What- 💀

[@ItzpoomAV1]

3 + 3 = 3!

[@blacklight683]

Double the factories, less than half the production? We're going to starve!

[@LBRF1_]

It means “brilliant”

[@tr2k500]

imagine screaming 5!! and someone says 15

[@PhilipHaseldine]

I was thinking of an interrobang but that's something else

[@ichika._]

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

[@isavenewspapers8890]

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

[@someaccount5200]

You summoned the chess players

[@WrathofMath]

Yeah I noticed 🤣

[@davidchedester8181]

​@@WrathofMathWrathofMath ❌ WrathofChess ✅