0! = 1
Dr James Grime tries to explain why this is the case - follow James on Twitter at / jamesgrime
More links & stuff in full description below ↓↓↓
Regarding the equation at the end - James says it should be e^-t dt NOT e^-n dn ... sorry for the mix-up!
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"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!
+Jay Kay I checked the comments to see if anyone said that.
I love this guy so much. James is my favourite.
Asriel Dreemurr My favorite is Matt Parker. Except, all the people on Numberphile are great!
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
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
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"! 🤣
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**
"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!
4:21 when my teacher asks where my homework is
"You wanna see me do it again?" XD
😂
Lol
🤣😭
🤣
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
"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.
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
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.
4:24 you wanna see me arrange these objects? You wanna see me do it again?
😆
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.
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
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 "".
"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 :)
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?
“We have zero objects” *zooms in on blank paper*
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
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!
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
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.
I really like the way you arranged the zero objects. You are a true artist.
2:56 'You've broken maths Brady, STOP THAT!!!'
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.
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_
3:59 I love the dramatic zoom in
Also see 4:13
The zoom in on "nothing" just made my day
Haha I noticed that too.
2:55 1÷0
"Hi, -1 factorial?" "Sorry, mathematics broke" "Understandable, have a nice day"
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.
Love this channel! Opens up my mind to new mathematical ideas and refreshes me on ones I already knew.
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
"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...
So contagious and engaging the way he carries his argument. ❤️
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!
All of us programmers read "0!=1" as: zero is not equal to one, which also happens to be true.
2:55 *math.exe has stopped working*
Lol tumbhh Error exception 000xxxxx0x00x
Rather my brain.exe has stopped working
l'hopital's rule
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
Brilliant and concise explanations for O! and the so useful Gamma function. Thanks a lot Dr James!!
You've broken math Brady! Stop that.
"You've broken maths, stop it" I love this channel
I love this channel, and 2:54 is such a perfect moment to encapsulate the whole thing.
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.
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
I hate when math breaks
The camera work during the arranging of one coin is amazing. Loving the zoom - made me laugh.
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.
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
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?
2:23 I love how you can see the “oh shit” in his eyes when he knows he can’t answer the question
when he says “wanna see me do it again, there it is” gesturing to the empty piece of paper, my day was made
2:53 is the funniest part of any video I've seen
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
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.
tremendous explanation. I owe to you for providing us with such a crucial lesson
You've broken maths, stop that! lol
xD
"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?
"There is zero objects" *Zooms in on the paper*
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.
"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*
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)
I literally know almost nothing about math beyond multiplication and division but this kind of stuff makes me want to know math.
You seriously need to understand algebra if you want to write computer programs or do spreadsheets. But I figure you need to know some serious advanced mathematics, if you want to balance the field equations to levitate the flying cars of the future.
The moment you say "there isn't" is so insightful and comical, thanks!
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
so -1! breaks math and 1! is 1 well that's discrimination!
Thank you for correcting the formula for gamma(n) in the description.
Brilliant. Great teacher. 0! = 1! = 1. Very tempting drop off the factorial notation and we get 0 = 1.
You've broken the maths!
3:12 *pulls out his entire life savings*
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
*zero objects* _INTENSE ZOOMING_ *Nothingness intensifies*
"Who says mathematicians don't make any money? There's literally ***50 p*** here!" I've never laughed so hard in a numberphile video
You've broken maths, Brady, stop that...
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
I think it's worth stressing here that there's a certain pragmatism to extending definitions beyond their initial domain. Factorial is well defined for the natural numbers. It's a useful function there, so maybe it's worth finding a natural extension of that in other domains. When we add 0 to the natural numbers, setting f(0)=1 preserves a bunch of properties that we like about factorial.
Thank you ❤️
Mathematics is a spirit and I’ve been impacted very much. I’m really appreciating your goodness
Took such a short time to make me understand this. Now I can use it boldly without having to memorize anything.
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."
My favourite reasoning is that it is an _empty product_ and 1 is the multiplicative identity - it _is_ the empty product - the product with no terms. Similarly, a^0 is an empty product. In the same way, an empty sum is 0 - if you use ∑ notation and put the limits as n=0 to n=-1, there are no elements of the sum and it is 0.
Exactly! This is also my favorite reasoning. More precisely, you can think of n! as being defined as the product of (1, 2, ..., n). Hence 3! is the product of (1, 2, 3), 2! is the product of (1, 2), 1! is the product of (1), and 0! is the product of (). Since the product of () is 1, 0! = 1.
It's one for the empty set. "Wanna see that again?" Love that.
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
I love the way the camera zooms in on the zero objects.
"Slightly philosophical" Ok that's the part I've been waiting for.
It’s funny how very simple change of perspective can show totally different solution. Where I come from any factorial is 1x2x3x..etc. I know it sounds silly but putting it ‘from behind’ -> 3x2x1 actually changes my perspective and understanding
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.
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
There's a mistake in the gamma function. It's the integral from 0 to infinity of t^(n-1)e^(-t)dt or in TeX, int_0^\infty t^{n-1}e^{-t}dt.
If you see it philosophically, you could say 1 not divided is 1 and as long as there is one observer, there will always be at least one way of presenting what is there or missing. 1/0 doesn't have to mean 1 divided by 0, it could also mean 1 distance to 0. Which musicaly seen is the same as 1/2 also known as an octave. The string alone is the basic octave.
Yes. They are contiguous. It is logic, so philosophy, anyway. 0! asserts that 0 is equal to the absolute value of 1, since 0 is a real integer neither positive nor negative.
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.
"You broken maths Braidy. Stop that!" 😂 I love numberphile.
This and the proof of m^0 = 1 are some of my favorites. They are seemingly inexplicable, yet the proofs are surprisingly simple.
The proof for 0! = 1 isn't simple: it's facile. Facile means appearing neat and comprehensive only by ignoring the true complexities of an issue; superficial. In this case, I think it's facile and wrong. I am certain that, leaving aside what it actually equals (which I say is 0), 0! does not equal 1 if one applies their mind to the matter in line with the explained definition of what a factorial is.
Thank you for what brings you joy
But if 1! = 1 and 0! = 1, wouldn't that mean according to the graph that 0.5! =1? Woudln't that be true for any number between 0 and 1?
Ma4zu6 x^2 is 1 for x = -1 and x = 1, but that doesn't imply that it's 1 for every x in between.
Well his graph implied that. If those points are connected and we can get values between them, and there is straight line between 1 and 0 then the points on that line must have the same value, no?
Ma4zu6 exactly what i was thinking.
Ma4zu6 I was just going to comment that same thing. Based on the graph, the factorial of any number on the number line which is less than zero is equal to one. Unfortunately, I see you left your comment 3 weeks ago and haven't had an answer. Any actual mathematicians out there able to weigh in on this?
Glitch Xero I actually have answered this weeks ago and the answer is still the same: no, this is not true. Just look it up.
"You've broken maths, Brady stop that"
I find that an immense helpful approach: View it as an action! Here as a non-mathematician I would view factorial as an addition, the aspect I am most familiar with! Under this perspective 0! = 1 is incomprehensible to me, I'm stuck! Viewing it as the action of rearranging opens the door! Offering different pathways to the solution seems to me invaluable!
@@Isaac-eh6uu If you are confronted with a mathematical explanation that you don't understand, the claim that it is "easy", leaves you with the information that that which you don't understand is "easy" and "logical".... If however you are offered another approach, moreover an approach you are able to follow, a door opened. I find it notable that the best achieving countries in the Pisa test are those that teach math with an emphasis on creativity and diversity of solutions.
This reminds me of my favourite quote " If you change nothing, nothing will change"
That's written on my shirt while I am reading this ' If nothing changes, nothing changes'
You broken math Brady! stop that! :D :D :D :D :D
I like the multitude of different ways that it was explained. Great work, numerophile!
Do a video on the gamma function and why it is like is. Maybe also include interesting facts like 1/2!
This is a great lesson in the difference between what a mathematical operation is and how to compute it.