When he says "there's zero objects" and the camera zooms in on the blank paper, it really shows the capacity for the human mind to conceptualize things which aren't material. I love this.
ME (NOT A MATHEMATICIAN): "But I don't get it, how do you arrange nothing?" JAMES: "Would you like to see it?" ME: "Yes." JAMES: "There it is." ME: "...oh." JAMES: "Would you like to see it again?" ME: "...maybe." JAMES: "There it is." ME: "...okay."
"How do you arrange nothing" in maths is the same thing as asking "How many different ways can you arrange 0 objects?" The answer is then simple to see. There's only one way to arrange 0 objects, i.e. the one you're looking at: no arrangement. Therefore, you can arrange 0 objects 1 way.
This is the most satisfying explanation I have come across about the zero factorial. On top of the technical, division proof, you made it very easy to understand in practical terms.
Zenovarse -1!=-1 because -5!=-120 since -5 times -4 times -3 times -2 times -1=-120. You would think -5 times -4 times -3 times -2=-120 but it equals 120 since -1 in multiplication makes the answer negative.
@@leetfukk It also means factorial. And 0!=1. I've done a lot of programming, and ! checks to see if two operands are equal or not as in "(A!=B) true" and if they are not equal the function runs. My only point was that as a codemonkey in binary 0≠1, yet in pure mathematics 0!=1.
The cover of this video says "0!=1" In computer programming "!=" is a logic operator that returns true when the two numbers are different. In simplier words, it means "is different from". I read "0 != 1" and said "No shit"
Actually != is written as a = With a dash in it (on paper). I program a lot as a hobby and I still understood what the title ment. Basically because "!=" is only ever used in programming and not in writing.
Rew Rose there is only a small usable range of exponential growth in Cartesian space. Here is an example of exponential growth: 2,4,8,16,32,64,128,256,512,1024,2048 What do you think is the best range to understand and use that data?
In 6:04 min the Gamma function had a little bit of mistake. The gamma function is written as: Gamma fn(n)= Integral of t^(n-1)*exp(-t) dt NOT Gamma fn(n)= Integral of t^(n-1)*exp(-n) dn
@@shoutitallloud t is just a variable here, without deeper meaning behind it. It can take values between 0 and infinity. The function is then integrated over t, and the solution of that happens to be (n-1)! It's quite interesting, and you can show that this relation is true, but that requires knowledge of integration, and is quite complicated.
My favourite part of he video is when James takes away the last coin and Brady zooms in on an empty spot on a table (yes I know it's to show the 0 objects but out of context it's hilarious)
The very first programming language that I was introduced to in school used the symbols "" for "not equal to", which makes more sense because it means "greater than or less than". It's unfortunate that not all languages use "".
i could watch videos of James all day, whether i understand what he's talking about or not his enthusiasm is so infectious he makes everything interesting
For all the people unsatisfied with the explanations offered by the video, how about we just rely on the base definition of the factorial? The factorial of a non-negative integer n (n!) is defined as the product of all positive integers less than or equal to n. There are no positive integers less than or equal to 0, so 0! is the empty product, or the product of no numbers. The empty product is defined to be the multiplicative identity, which is 1. (Just like the empty sum is the additive identity, which is 0).
or we could rely on another definition of factorial, which is n!=(something)*n. You multiply all the numbers less than, and equal to, n. Now following this logic, 0! = (something)*0. Anything multiplied by 0 is 0 so 0!=0.
Arkalius80 0 is where the graphs start. Its not on either side of the graphs, its not negative, but its also not positive SO you can't have 0! as 0 is not "non-negative" the same way he decided to stop completing the pattern at -1 since it's wrong. Its equally wrong to do it with 0 !!!!
You can also just use another statement of the definition of a factorial: For whole number n, n! = n(n-1)! Plugging in 1, you end up with 1! = 1(1-1)! = 1(0)! From the very leftmost term, we know 1!= 1, through transitivity, we now know 1=1*0!=0! Therefor, 0! must be 1
My lightbulb really went on when you said that n factorial represents the number of ways you can organize n objects. Because I was just wondering about applications for n factorial. Thanks for that remark!
It will not be e(to the power)(-n)×dn in the gamma expression, there should be a variable (usually denoted by 't') instead of 'n' , because 'n', as previously defined, is a constant.
The way I would say it is: when presented with these situations that sort of break the assumptions of a naive formula (like raising something to the power of zero, or finding zero factorial, or figuring whether or not 1 is prime), it's convenient for mathematicians to just decide on a consistent *convention*. They can leave the answer undefined, or they can assign a value, and what generally happens is that the value is assigned in such a way as to make general calculations easier, without having to add some extra specifications about special cases all the time. In this case, saying 0! = 1 is the least awkward choice for general formulae. It means that n! = n *(n-1)! is true even for n=1. It means that you can write the Taylor series expansion for a function using factorials in a consistent way, without having a special case for n=0. And there are a lot of other areas where this is the most convenient choice.
***** It's like asking the question, "How many times can you divide by zero?" I could see the answer being 0, 1, or ∞ but, as it is, the answer remains "undefined."
3:14 "who says Mathematician don't make a lot of money? See I got 50p here" I feel so sorry for them already :( p.s.... nice timing when you says that tho. lol pi
im sorry but that´s not even close to pi. In fact that is infinitly far away from pi since pi has an infinite amount of decimals. It is the three first number of pi though..
***** To my knowledge, no coding language has pi as a native constant value...... So you would have to manually type in 3.14 each time, or define it in the program for repeated use, Variable=(Value of pi, to however many decimals you are willing to type). Of course, I have yet to actually study programming enough , so this comment could be completely wrong.... it is merely just a speculation.
aiklarung "Of course, I have yet to actually study programming enough , so this comment could be completely wrong.... it is merely just a speculation."
Oh my god I can't believe it. I love these videos but for the most part they are real pop-math stuff. But he just explained the gamma function! Thank you! I have encountered that stupid thing for YEARS and I've seen the definition of it a thousand times but no one has ever explained WHAT it is. "Factorial for real numbers" how hard was that to say! Thank you!
Well , i have a question , if you pick a random number in between 0 and 1, on the paper i can see the "y" value equal to 1, but i cannot see it with the integral that you gave us . Nice proof.
Does that function curve indicate that (1/2)! is the lowest number of any factorial, and that negative factorials are on the same curve as positive factorials but with one added? In other words, "(-n)! = (n+1)!" ?
You should be given an award for proving that we do not understand 0 at all, I think.. You are a super cool and twisted person for even thinking of this. Hats off. You made math interesting, and I think also proved we have more to learn about our understanding of it.
Adam -亚当- Factorials are permutations. An empty set has 1 permutation (only one way to arrange) regardless of it being an empty set or a set with only 1 element...which is why 0! = 1 and 1! = 1.
Adam -亚当- You can still have permutation of an empty set, it is basic probability theory. You have to remember that permutations are a bijection of set S to itself as f:S → S. If there are no elements in set S then you can still map an empty set {Ø} to itself in a bijection or one to one correspondence.
You can't use the original definition of the factorial for 0 (or any non-natural number e.g. 1/2). It only holds for natural numbers (1, 2, 3, ...). For everything else, the gamma function is the definition of the factorial. So 0! is defined as gamma(1), (1/2)! is defined as gamma(3/2), etc.
The "An empty product equals 1" bit is the most important here. An empty sum is zero, which is straightforward: Let's assume I have a set of zero numbers and a set of n numbers, and that the sum of the latter is X. The sum of ALL numbers in these two sets is expected to be X, too, since the first set is empty doesn't contribute anything. Therefore, total sum = X and also X = (sum of 0 numbers) + X. The sum of zero numbers should obviously be zero to make that equation work out. Now, let's say we want to multiply all numbers, and we already know that the product of all numbers in the 2nd set is Y. Again, the empty set shouldn't change that, since we're again multiplying the same n numbers. Therefore, total product = Y and also Y = (product of 0 numbers) * Y. (Edited: typo correction) To make that work out for any value of Y, the product of zero numbers must be 1, the "neutral element" of multiplication. I think the main issue with the products is that one might think that we're getting "something" (1 in this case) from "nothing" (no numbers at all). The key to that paradox is that 1 _isn't_ "something" when it comes to multiplication; multiply by 1 and nothing changes, and therefore the "product of zero numbers equals 1" convention is mathematically sound.
@Hassan Akhtar Yes! The video was not rigorous in saying if n \in N (or W, depends on your def of N), or not, and that is why some people tried to put 0 in places where it shouldn't go and find "issues"
@Jatin Parmar That is incorrect. (-1+1)! = (0)(-1)! You've assumed that (-1)! is finite, which, as explained in the video, is a false assumption. Using the result of 1/0 from the video (which tends toward infinity), you could say (0)! = 0*infinity The product of zero and infinity is undefined by definition. This does not conflict with 0! = 1
Yea lol, but they fixed it, see "show more" t isn't even defined, nor is he integrating over it. Bit of a derp there. Replace t with n and... EDIT: WAIT, no. This still makes no sense lol. Just look up the actual gamma function haha.
If anybody is asking "Why, isn't there infinite ways to arrange zero objects?" then you forgot the fact that not arranging something is a way to arrange it, you can't arrange zero objects and that is the only thing you can do about it.
Sounds weird, yes! It would get much more so if you were to think about equivalence classes. I could very well say that 2 is equivalent to 0, and that 0 + 0 = 4, if we are working with a mod 2 equivalence relation, haha.
Yeah, I loved that he at least mentioned continuous factorials. Although I was looking for a video that much better explained that. I would like to know how I could calculate the exact value of (1/2)! by hand.
![Lp. Сердце Вселенной #60 РОЖДЕНИЕ ЛОЛОЛОШКИ [Финал] • Майнкрафт](http://i.ytimg.com/vi/YoR0pAV9FVQ/mqdefault.jpg)
"You've broken maths, Brady. Stop that."
This is why I now love math.
Yeah I cracked up at that part! Lol
I love the little smirk Dr. Grime gives just before showing the example!
I love this guy so much. James is my favourite.
It's not math it is maths this is a British channel
I keep finding you on UA-cam lol
Aw, I was really wanting to see him try to arrange -1 coins.
That's 1920s German money
Money made out of anti matter.
There are 1/0 ways to arrange -1 objects.
Acutally, I've been arranging -n dollars every week.
I...I...uhm...
**dies**
When he says "there's zero objects" and the camera zooms in on the blank paper, it really shows the capacity for the human mind to conceptualize things which aren't material. I love this.
I laughed so hard when he was like "here watch me do it again"! 🤣
Exactly. Lol.
Computer Engineers looking at the thumbnail - "Ah! yes. That's true."
😂😂😂
Lol, I got that joke mainly due to messing around with Minecraft commands.
For the peoples who are dont understand the joke: != means in programming lang. "Not equal" so 0 is not equal to 1
😭😂😂😂
yep yep
"There's zero objects..." *Zooms in on blank paper*
That paper is an object... XDDDDDDDD
Is it a bird, is it a plane? A plane.
Sarah Boes i laugh when he said it
XD
It is a birdplane!
"See who says mathematicians don't make a lot of money, there's literally 50p here"
Your sense of humour satisfies me
What is 'p'? is it pound?
no, it's "pence" (one pence = 1p = £0.01)
Ethan Goldsmith you uncultured swine!! -(joke)
Eternia Dr. James has an awesome sense of humour. "How many way of arranging 0 objects. There it is. Wanna see me do it again? There it is!"
@@SpyridonJohn1633 that me made smile :)
"there it is, wanna see me do it again? there it is!" xD
that made my day
I like how his extra British accent came out during that 😄
@Lester Meza That's the reference
Same :)
griffin tucker no there’s only one.
@griffin tucker you don't get it. There is only one way to arrange it. By having it empty.
The moment starting at 4:14 looks like it was taken straight out of some kind of math version of The Office.
I thought the exact same thing 😂.
I can't see it but don't do anything rash and stay healthy.
Omg you are spot on! Never struck me the first time but now it's even funnier.
I couldn't see that.
But maybe that is because I have never watched "The Office".
I like your profile name
I asked my math teacher why 0!=1 he said "because it's like that"
Teachers are goons
@Mr. H We get free education from youtube but teachers be like: That's the law. :/
Mr. H 🖕
Mr. H because you are implying that private education is superior and that normal people people get nothing because they didn’t “pay”
True by definition isn't as satisfying, but it works.
I liked the cheeky little zoom to get a more detailed shot of the nothing.
I love Dr. Grimes he has such an enthusiasm to him that i absolutely adore
Fr hes not even tryin to teach hes just having a blast
Big facks
Love him too
@@-enzyme
Bbbb
Bbfff
Isn't it like a common thing on this channel?
4:21 when my teacher asks where my homework is
"You wanna see me do it again?" XD
😂
Lol
🤣😭
🤣
If you shout 0 loudly enough, it becomes 1.
*0!*
Nice one
i dont get it :(
Its an exclamation mark
George Washington +Pyrrhic Victory it's 0 factorial which is 1.
Pyrrhic Victory oh but if george watched the video he should've known anyway
We need t-shirts saying "You've broken maths brady!"
-stop that
Shut up and take my money!
I'd buy one. At least consider it.
I bid $100
I second that
"Hi, -1 factorial?"
"Sorry, mathematics broke"
"Understandable, have a nice day"
ME (NOT A MATHEMATICIAN): "But I don't get it, how do you arrange nothing?"
JAMES: "Would you like to see it?"
ME: "Yes."
JAMES: "There it is."
ME: "...oh."
JAMES: "Would you like to see it again?"
ME: "...maybe."
JAMES: "There it is."
ME: "...okay."
"How do you arrange nothing" in maths is the same thing as asking "How many different ways can you arrange 0 objects?" The answer is then simple to see. There's only one way to arrange 0 objects, i.e. the one you're looking at: no arrangement. Therefore, you can arrange 0 objects 1 way.
I hate it when people flex their overflowing wealth like that.
@Lo Po I think you missed the joke my friend
@Lo Po r/woosh
@Lo Po james ain't rich I'm guessing
I was more upset with the idea of coins in a wallet, do many people do that?
manw3bttcks where else would you put them
The zoom in on "nothing" just made my day
Haha I noticed that too.
2:55 1÷0
2:56 'You've broken maths Brady, STOP THAT!!!'
This is the most satisfying explanation I have come across about the zero factorial. On top of the technical, division proof, you made it very easy to understand in practical terms.
All of us programmers read "0!=1" as: zero is not equal to one, which also happens to be true.
"You've broken maths, stop it"
I love this channel
I wish he was my math teacher ..
Joshua Rage well today he was!
Numberphile I'm sure this wasn't the last time! :)
-1!=?
Zenovarse -1!= -1Remember, you always do factorials first then apply the unary operator for the negation.
Zenovarse -1!=-1 because -5!=-120 since -5 times -4 times -3 times -2 times -1=-120. You would think -5 times -4 times -3 times -2=-120 but it equals 120 since -1 in multiplication makes the answer negative.
4:24 you wanna see me arrange these objects?
You wanna see me do it again?
😆
You've broken maths Brady stop that
thats the funniest thing I've ever heard on this channel
that was amazing
+diabolicallink Ikr I told my dad that
That was the best
+diabolicallink I went to the comments just after hearing that sentence convinced that it had not gone unnoticed, and I wasn't dissapointed!
"there it is, wanna see me do it again? there it is!" xD
+Dave Galindez hahahaa that killed m
Damn AHAHAHAHAEHEAHEHEAHEIHIHIHEHIEHII
haha, made my day
read this at the exact moment he said it, and its my first time watching this vid
Loved that actually...
I really like the way you arranged the zero objects. You are a true artist.
Programmers: "Well yeah obviously 0 isn't equal to 1..."
"quantum computer say hello"
Code monkeys: 0≠1, but 0!=1.
@@joelschama1735 In a lot of common programming languages, != means "does not equal"
@@leetfukk It also means factorial. And 0!=1.
I've done a lot of programming, and ! checks to see if two operands are equal or not as in "(A!=B) true" and if they are not equal the function runs.
My only point was that as a codemonkey in binary 0≠1, yet in pure mathematics 0!=1.
Well, most programming languages count from 0, so the 1st object in a list is 0
The cover of this video says "0!=1"
In computer programming "!=" is a logic operator that returns true when the two numbers are different.
In simplier words, it means "is different from".
I read "0 != 1" and said "No shit"
Find 'N' Frag LOL...so you read it as "zero is not equal to one" that's hilarious
Programmer jokes. I love you.
Sapphire Shard Was it a joke? I am a programmer too, and I did the same thing. More of an observation really :P
Actually != is written as a = With a dash in it (on paper).
I program a lot as a hobby and I still understood what the title ment. Basically because "!=" is only ever used in programming and not in writing.
salmjak "As a hobby". If you did it for a living, you would be reading more code, than nearly anything else :P
He doesn't do drugs. He does maths. It gets him so high he can graph an exponential function in Cartesian coordinates.
?
Rew Rose Exponential growth quickly does not fit on a Cartesian coordinate system. We use logarithmic graphing for exponential growth curves.
Michael Winter
ok . . . ( why though? )
Michael Winter
Rew Rose there is only a small usable range of exponential growth in Cartesian space. Here is an example of exponential growth:
2,4,8,16,32,64,128,256,512,1024,2048
What do you think is the best range to understand and use that data?
In 6:04 min the Gamma function had a little bit of mistake. The gamma function is written as:
Gamma fn(n)= Integral of t^(n-1)*exp(-t) dt
NOT
Gamma fn(n)= Integral of t^(n-1)*exp(-n) dn
I was looking for this cause n is a number where as t was the variable...thanx
@@sunandinighosh6037 i was thinking "how can you differentiate with respect to a constant?" I am learning calc and thought i missed something
I don't uderstand quite what is "t" here. Could you explain please?
@@shoutitallloud t is just a variable here, without deeper meaning behind it. It can take values between 0 and infinity. The function is then integrated over t, and the solution of that happens to be (n-1)! It's quite interesting, and you can show that this relation is true, but that requires knowledge of integration, and is quite complicated.
@@shoutitallloud the variable of integration
“We have zero objects”
*zooms in on blank paper*
My favourite part of he video is when James takes away the last coin and Brady zooms in on an empty spot on a table (yes I know it's to show the 0 objects but out of context it's hilarious)
Mathematicians: Does 0 !=1?
Programmers: Well yes, but actually yes
Well, the only thing in the factorials programmers have a problem with:
1!=1
@@JonathanMandrake 2!=2
and that's why you use spaces...
@@geryz7549 programming languages that use white space are considered bad
The very first programming language that I was introduced to in school used the symbols "" for "not equal to", which makes more sense because it means "greater than or less than". It's unfortunate that not all languages use "".
2:53 is the funniest part of any video I've seen
i could watch videos of James all day, whether i understand what he's talking about or not his enthusiasm is so infectious he makes everything interesting
2:55
*math.exe has stopped working*
Lol tumbhh
Error exception 000xxxxx0x00x
Rather my brain.exe has stopped working
l'hopital's rule
I was sort of expecting the paper to burst into flames when they tried (-1)!
Technically you can do negative factorials. It's undefined at negative integers, though.
*****
The factorial function can be extended to be Γ(x+1), so.
Well, I just typed "This sentence is false" and my keyboard didn't explode, so...
dt!
0!/0=-1!
0!=1
1/0=-1!
-1! is undefined
Took such a short time to make me understand this. Now I can use it boldly without having to memorize anything.
I've been on a binge watching OLD numberphile videos. You guys were (and still are!) legit.
Thanks for all the wonderful content over the years. 😢😊
"0!=1" well that's quite obvious for Programmers!!!
!0 == 1
but, !0 !== 1
@@bowel_movement 0!=1
@@bowel_movement you forgot the semicolons. That's going to explode
@@TheMegaxPlus that's only if it's c#
!0.
Not 0 = 1
*BINARY QUCK MAFFS*
For all the people unsatisfied with the explanations offered by the video, how about we just rely on the base definition of the factorial? The factorial of a non-negative integer n (n!) is defined as the product of all positive integers less than or equal to n. There are no positive integers less than or equal to 0, so 0! is the empty product, or the product of no numbers. The empty product is defined to be the multiplicative identity, which is 1. (Just like the empty sum is the additive identity, which is 0).
I wasn't satisfied with the "complete the pattern"-explanation but the coin method makes sense, since that's what factorial is really meant for.
Pravat Kiran Timsina or psychological minors? xD
or we could rely on another definition of factorial, which is n!=(something)*n. You multiply all the numbers less than, and equal to, n.
Now following this logic, 0! = (something)*0. Anything multiplied by 0 is 0 so 0!=0.
Arkalius80 0 is where the graphs start. Its not on either side of the graphs, its not negative, but its also not positive SO you can't have 0! as 0 is not "non-negative" the same way he decided to stop completing the pattern at -1 since it's wrong. Its equally wrong to do it with 0 !!!!
You can also just use another statement of the definition of a factorial: For whole number n, n! = n(n-1)!
Plugging in 1, you end up with 1! = 1(1-1)! = 1(0)!
From the very leftmost term, we know 1!= 1, through transitivity, we now know
1=1*0!=0!
Therefor, 0! must be 1
"You broken maths Braidy. Stop that!" 😂 I love numberphile.
Me seeing the thumbnail as a programmer:
*hmm yes the floor is made out of floor*
Junior roblox
I'm not a programmer what do you mean
!= is the difference operator
0 != 1 means "zero is different from one"
@@KieranHelix heh? What
more like: _the floor isn't made out of doors_
I love the way the camera zooms in on the zero objects.
You've broken math Brady! Stop that.
I read the thumbnail as "zero is not equal to one". lol
the struggles R real!
the struggles R real!
+Ryan G-P
return true;
+JusesCrustes Unless the struggles are integer or complex. :)
>>>print 0!=1
True
So contagious and engaging the way he carries his argument. ❤️
i saw thumbnail 0!=1 and i thought he explains why zero isn't equal to one... oh programming.
Feel u m8
At least you get a true statement regardless.
Everything was completely fine until the Gamma thing kicked in.
thekenmatax at least I have a half baked idea of what γ is... Sort of
The lowercase gamma is used for Euler-Mascheroni constant. The gamma function is always denoted by the uppercase gamma.
All I know is that it ended WW2, haha
he drew/wrote it in his own way of calligraphy... Gamma upercase is that - Γ - you may know it that way.
same here
"youve broken math! stop that"
Maths***
+Szymon Gorczynski It's an UK and US thing: US uses math and UK uses maths
Remavas Yes, I know the Americans can't spell.
+Szymon Gorczynski h!=hs
thats why 'stop that' wasnt exclaimed
flawlessgenius And where did I question that?
My lightbulb really went on when you said that n factorial represents the number of ways you can organize n objects. Because I was just wondering about applications for n factorial. Thanks for that remark!
I like the multitude of different ways that it was explained. Great work, numerophile!
2:56 I don't know why but I laughed so hard
i know why
I can't believe you've done this.
its like hes bullying math
I laughed to at that XD
Hahahahahah, sameeee
I hate when math breaks
"You've broken maths, Brady. Stop that."
This is the quote that keeps bringing me back to this video
so -1! breaks math and 1! is 1 well that's discrimination!
What's pi factorial?
Decimal factorial makes no sense because there's an infinite amount of space between any number so what numbers are you so posed to choose?
0
***** O
Leon the third no.. seriously i got it now. Pi = 3.14
so, the comment actually ask what's 3.14 factorial. the answer is 7.17
7.1880827...
3:59 I love the dramatic zoom in
Also see 4:13
Love this channel! Opens up my mind to new mathematical ideas and refreshes me on ones I already knew.
You are much better than my math teacher
He's also much better than you. And so is your math teacher.
Bloodbath and Beyond Well that was rude. no need to put people down because you had a bad day.
I had a great day, made better by your reaction :)
Peter Xenopoulos
I usually just Flag those people nowadays, this at least seems to hide their comments so I don't need to see them. xD
Laurelindo yeah
You've broken maths, stop that!
lol
xD
as a programmer I couldn't see the problem with "0!=1"
If you're well-versed in mathematics, then you _shouldn't._
0==1 would have been worse :)
that's a JS thing. ( *Usually* ) not found in other languages
That's only because you use a really bad language for programming
Compilation warning : Statement always true
6:04 No one noticed, that it should be the integral of t^(n-1) e^(-t) dt instead?
I was going through the comments to check that too
yeah really bothers me
It was a human error
THANK YOU! I had to scroll down A LOT to find your comment, it was driving me crazy!
He said that in the description.
It will not be e(to the power)(-n)×dn in the gamma expression, there should be a variable (usually denoted by 't') instead of 'n' , because 'n', as previously defined, is a constant.
if you go by the logic he uses then 0! should be 0. there's no "one" way to arrange something that's not there. there's no way to arrange them.
Put 0! In a calculator, nurd.
The gamma function looks wrong. It should be integral from 0 to infty of exp[-t]*t^(n-1)*dt
Yeah, they goofed; but they caught themselves -
Click on "SHOW MORE" under the video windowpane, and you'll see that correction given there.
Yup u r ryt...👍
The way I would say it is: when presented with these situations that sort of break the assumptions of a naive formula (like raising something to the power of zero, or finding zero factorial, or figuring whether or not 1 is prime), it's convenient for mathematicians to just decide on a consistent *convention*. They can leave the answer undefined, or they can assign a value, and what generally happens is that the value is assigned in such a way as to make general calculations easier, without having to add some extra specifications about special cases all the time.
In this case, saying 0! = 1 is the least awkward choice for general formulae. It means that n! = n *(n-1)! is true even for n=1. It means that you can write the Taylor series expansion for a function using factorials in a consistent way, without having a special case for n=0. And there are a lot of other areas where this is the most convenient choice.
I appreciate this explanation more than the one in the video. Thank you.
philosophically i think there are infinite ways to order 0 objects.
+sharon f Okey, present 2 different ways.
+sharon f
Or alternatively, there are zero ways to order 0 objects.
Fr. Victor Feltes But there is at least one way, namelyin no order.
***** It's like asking the question, "How many times can you divide by zero?" I could see the answer being 0, 1, or ∞ but, as it is, the answer remains "undefined."
Except that question is malformed as it assumes an inverse of 0 exists which it doesn't
0! = 1
0 != 1
There's a difference
yeah one is used in programming and the other one is not.
The difference is a space
n-1!=n!÷n .take n=1gives0!=1!=1÷1=1
Kalokal 0!=1
0 !≠1
because: 0(0)!=0(1)=0
(Nothing means 0, and also multiplication)
Both are true
You've broken maths, Brady, stop that...
I love this channel, and 2:54 is such a perfect moment to encapsulate the whole thing.
3:14 "who says Mathematician don't make a lot of money? See I got 50p here"
I feel so sorry for them already :(
p.s.... nice timing when you says that tho. lol pi
"3:14, lol pi" - You need some fresh air....
im sorry but that´s not even close to pi. In fact that is infinitly far away from pi since pi has an infinite amount of decimals. It is the three first number of pi though..
Zain Burney I agree with you.
***** To my knowledge, no coding language has pi as a native constant value...... So you would have to manually type in 3.14 each time, or define it in the program for repeated use, Variable=(Value of pi, to however many decimals you are willing to type). Of course, I have yet to actually study programming enough , so this comment could be completely wrong.... it is merely just a speculation.
aiklarung "Of course, I have yet to actually study programming enough , so this comment could be completely wrong.... it is merely just a speculation."
"You've broken Maths, Brady, stop that!"
Oh my god I can't believe it. I love these videos but for the most part they are real pop-math stuff. But he just explained the gamma function! Thank you! I have encountered that stupid thing for YEARS and I've seen the definition of it a thousand times but no one has ever explained WHAT it is. "Factorial for real numbers" how hard was that to say! Thank you!
when he says “wanna see me do it again, there it is” gesturing to the empty piece of paper, my day was made
You broken math Brady! stop that! :D :D :D :D :D
Well , i have a question , if you pick a random number in between 0 and 1, on the paper i can see the "y" value equal to 1, but i cannot see it with the integral that you gave us . Nice proof.
Does that function curve indicate that (1/2)! is the lowest number of any factorial, and that negative factorials are on the same curve as positive factorials but with one added? In other words, "(-n)! = (n+1)!" ?
You should be given an award for proving that we do not understand 0 at all, I think.. You are a super cool and twisted person for even thinking of this. Hats off. You made math interesting, and I think also proved we have more to learn about our understanding of it.
Brady, the math DESTROYER!
There must be like 10000 people screaming at the lap tops saying, if you don't have anything then there are no ways to arrange it...
I think that is true, there is 1 way to "show" 0 objects but there are 0 ways to "arrange" it since you can't arrange nothing
Adam -亚当- Factorials are permutations. An empty set has 1 permutation (only one way to arrange) regardless of it being an empty set or a set with only 1 element...which is why 0! = 1 and 1! = 1.
Steve McRae permutations of something yes, of nothing....???
Adam -亚当- You can still have permutation of an empty set, it is basic probability theory. You have to remember that permutations are a bijection of set S to itself as f:S → S. If there are no elements in set S then you can still map an empty set {Ø} to itself in a bijection or one to one correspondence.
Steve McRae That sounds reasonable, although I am not sure what you mean by it is a bijection of set S to itself. I think i need to read up more.
By definition, 0! is the product of all the integers greater than 0 up to and including 0. So 0! is the empty product = 1?
yeah, we don't have any factors to add, so the logical value would be the unity corresponding to multiplication, which is 1.
Sup This is the explanation I like best.
less than or equal to?
By definition 0! is the product of all the integers *lesser* than or equal to 0 greater than or equal to 1.
You can't use the original definition of the factorial for 0 (or any non-natural number e.g. 1/2). It only holds for natural numbers (1, 2, 3, ...). For everything else, the gamma function is the definition of the factorial. So 0! is defined as gamma(1), (1/2)! is defined as gamma(3/2), etc.
"Who says mathematicians don't make any money? There's literally ***50 p*** here!"
I've never laughed so hard in a numberphile video
Has a huge blackboard
Wastes sharpie and cardboard
They hoard chalk, mate
It is brown paper
@n a they do lines of chalk
This is their brand, man.
Since I'm a programmer, I was confused because everyone knows zero doesn't equal one
0 kan = 1. 8 holes = 2 holes.
To me (a computer) this is also confusing
I read 0!=1 as "zero does not equal one"
That's the java talking! Or is it the C++
It's actually used in many programming languages
even in that way it makes perfect sense lul
Phillip Patakos
Python
Python
2:23 I love how you can see the “oh shit” in his eyes when he knows he can’t answer the question
He talks about completing patterns but he can't put 0! in the n * (n-1) *...*(n-n+1) pattern. I feel cheated.
You can complete that. It's just an empty product which yields 1.
The "An empty product equals 1" bit is the most important here. An empty sum is zero, which is straightforward: Let's assume I have a set of zero numbers and a set of n numbers, and that the sum of the latter is X. The sum of ALL numbers in these two sets is expected to be X, too, since the first set is empty doesn't contribute anything.
Therefore, total sum = X and also
X = (sum of 0 numbers) + X.
The sum of zero numbers should obviously be zero to make that equation work out.
Now, let's say we want to multiply all numbers, and we already know that the product of all numbers in the 2nd set is Y. Again, the empty set shouldn't change that, since we're again multiplying the same n numbers.
Therefore, total product = Y and also
Y = (product of 0 numbers) * Y.
(Edited: typo correction)
To make that work out for any value of Y, the product of zero numbers must be 1, the "neutral element" of multiplication.
I think the main issue with the products is that one might think that we're getting "something" (1 in this case) from "nothing" (no numbers at all). The key to that paradox is that 1 _isn't_ "something" when it comes to multiplication; multiply by 1 and nothing changes, and therefore the "product of zero numbers equals 1" convention is mathematically sound.
@Hassan Akhtar Yes! The video was not rigorous in saying if n \in N (or W, depends on your def of N), or not, and that is why some people tried to put 0 in places where it shouldn't go and find "issues"
@ehud kotegaro Well actually it starts at 0 but yeah
@Jatin Parmar That is incorrect.
(-1+1)! = (0)(-1)!
You've assumed that (-1)! is finite, which, as explained in the video, is a false assumption. Using the result of 1/0 from the video (which tends toward infinity), you could say
(0)! = 0*infinity
The product of zero and infinity is undefined by definition.
This does not conflict with 0! = 1
"You've broken maths, Brady stop that"
Grimey you have that mad look in your eyes again
Everything was going fine.
Then came 5:38
Really yuh
Exactly my experience. I was feeling quite smug up to that point.
Yea lol, but they fixed it, see "show more"
t isn't even defined, nor is he integrating over it. Bit of a derp there. Replace t with n and... EDIT: WAIT, no. This still makes no sense lol. Just look up the actual gamma function haha.
If anybody is asking "Why, isn't there infinite ways to arrange zero objects?" then you forgot the fact that not arranging something is a way to arrange it, you can't arrange zero objects and that is the only thing you can do about it.
3:12 *pulls out his entire life savings*
LOL "You've broken maths, Brady. Stop that! "
The camera work during the arranging of one coin is amazing. Loving the zoom - made me laugh.
So 0! is also equal to 1! ... Weird
Sounds weird, yes!
It would get much more so if you were to think about equivalence classes. I could very well say that 2 is equivalent to 0, and that 0 + 0 = 4, if we are working with a mod 2 equivalence relation, haha.
Maths messing with our heads.
It’s like the same as saying 3-1 = 4-2. Factorial is a function.
Actually, 0! often equals 0. Math is not always accurate.
So neeenennneen
"See, who says mathematicians don't make a lot of money; there's literally 50p here."
It was cool to see him go into continuous factorials at the end, as a math major, I haven’t learned that yet
Yeah, I loved that he at least mentioned continuous factorials. Although I was looking for a video that much better explained that. I would like to know how I could calculate the exact value of (1/2)! by hand.
The moment you say "there isn't" is so insightful and comical, thanks!