Zero Factorial - Numberphile 

"You've broken maths, Brady. Stop that." This is why I now love math.

If you shout 0 loudly enough, it becomes 1. *0!*

Computer Engineers looking at the thumbnail - "Ah! yes. That's true."

4:21 when my teacher asks where my homework is

Aw, I was really wanting to see him try to arrange -1 coins.

"There's zero objects..." *Zooms in on blank paper*

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.

The moment starting at 4:14 looks like it was taken straight out of some kind of math version of The Office.

I asked my math teacher why 0!=1 he said "because it's like that"

"there it is, wanna see me do it again? there it is!" xD that made my 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."

4:24 you wanna see me arrange these objects? You wanna see me do it again?

We need t-shirts saying "You've broken maths brady!"

"See who says mathematicians don't make a lot of money, there's literally 50p here" Your sense of humour satisfies me

Programmers: "Well yeah obviously 0 isn't equal to 1..."

Mathematicians: Does 0 !=1? Programmers: Well yes, but actually yes

I liked the cheeky little zoom to get a more detailed shot of the nothing.

I wish he was my math teacher ..

“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

You've broken maths Brady stop that

I hate it when people flex their overflowing wealth like 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.

I really like the way you arranged the zero objects. You are a true artist.

I love Dr. Grimes he has such an enthusiasm to him that i absolutely adore

2:56 'You've broken maths Brady, STOP THAT!!!'

Me seeing the thumbnail as a programmer: *hmm yes the floor is made out of floor*

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.

"Hi, -1 factorial?" "Sorry, mathematics broke" "Understandable, have a nice day"

"there it is, wanna see me do it again? there it is!" xD

Love this channel! Opens up my mind to new mathematical ideas and refreshes me on ones I already knew.

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

The zoom in on "nothing" just made my day

All of us programmers read "0!=1" as: zero is not equal to one, which also happens to be true.

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!

so -1! breaks math and 1! is 1 well that's discrimination!

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"

when he says “wanna see me do it again, there it is” gesturing to the empty piece of paper, my day was made

I love this channel, and 2:54 is such a perfect moment to encapsulate the whole thing.

"You've broken maths, stop it" I love this channel

He doesn't do drugs. He does maths. It gets him so high he can graph an exponential function in Cartesian coordinates.

2:23 I love how you can see the “oh shit” in his eyes when he knows he can’t answer the question

Brilliant and concise explanations for O! and the so useful Gamma function. Thanks a lot Dr James!!

2:55 *math.exe has stopped working*

I hate when math breaks

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.

The camera work during the arranging of one coin is amazing. Loving the zoom - made me laugh.

2:53 is the funniest part of any video I've seen

I was sort of expecting the paper to burst into flames when they tried (-1)!

"There is zero objects" *Zooms in on the paper*

The moment you say "there isn't" is so insightful and comical, thanks!

You've broken math Brady! Stop that.

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)

Has a huge blackboard Wastes sharpie and cardboard

6:04 No one noticed, that it should be the integral of t^(n-1) e^(-t) dt instead?

"youve broken math! stop that"

I read the thumbnail as "zero is not equal to one". lol

tremendous explanation. I owe to you for providing us with such a crucial lesson

It was cool to see him go into continuous factorials at the end, as a math major, I haven’t learned that yet

2:56 I don't know why but I laughed so hard

"0!=1" well that's quite obvious for Programmers!!!

*zero objects* _INTENSE ZOOMING_ *Nothingness intensifies*

Mathematics is a spirit and I’ve been impacted very much. I’m really appreciating your goodness

3:12 *pulls out his entire life savings*

philosophically i think there are infinite ways to order 0 objects.

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.

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).

You've broken maths, stop that! lol

"Who says mathematicians don't make any money? There's literally ***50 p*** here!" I've never laughed so hard in a numberphile video

Thank you for what brings you joy

You've broken maths, Brady, stop that...

Everything was completely fine until the Gamma thing kicked in.

This is a great lesson in the difference between what a mathematical operation is and how to compute it.

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

"You broken maths Braidy. Stop that!" 😂 I love numberphile.

I love the way the camera zooms in on the zero objects.

4:12 : “We have zero objects” Camera : *zooms in for a closer look*

so basically the factorial function is defined such that it works how we want it to work.

Took such a short time to make me understand this. Now I can use it boldly without having to memorize anything.

3:59 I love the dramatic zoom in

Everything was going fine. Then came 5:38

I love this guys' passion for math

You are much better than my math teacher

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

Very enlightening video ❤️

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.

"You've broken maths, Brady stop that"

i saw thumbnail 0!=1 and i thought he explains why zero isn't equal to one... oh programming.

This reminds me of my favourite quote " If you change nothing, nothing will change"

Thanks really interesting. Never heard of non integer factorials before. Nice integral

0 != 1. Zero is not one. Programmers..?

You broken math Brady! stop that! :D :D :D :D :D

This and the proof of m^0 = 1 are some of my favorites. They are seemingly inexplicable, yet the proofs are surprisingly simple.

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.

"You've broken Maths, Brady, stop that!"

0! = 1 0 != 1 There's a difference

I love things like this. Because it forces you to have a complete understanding of the process. Like exponent zero.

that thumbnail expression is PERFECTION