Why is 0! = 1?

Why 0! equals to 1? Okay. Because it does seem to defy the idea of you multiply down until you get down to whatever one, and then you stop. I'm going to argue for this the same way I would argue a very sort of reverse of.. Oh no, not that. How would i do this? How do you argue that? Because you'll take it to be true, right? But it's the same problem when you say on 'a' square, but we define powers as you multiply this number by itself that number of times and That makes sense to you, but the definition breaks down The U7 definition breaks down when you go to cases like this So how do you argue it, or do you just accept it and don't think about it? *Laughs* Don't think 'bout it All right people let's play a game, okay Someone at the audience: Mario! *laughs* No I still with a number, shall we? Okay. Let's go. To power of one? Everyone: 2. 2 squared? Everyone: 4. 2 cubed? Everyone: 8. Okay. You can start to see what's going on, right? We don't need to… *Laughs* People who spent time on their phones recently know this numbers VERY WELL.

Now, ah… Now, you're used to do it forward aren't you? You do it forward and the numbers get bigger, okay, but just as equally You can go backwards and the numbers get smaller, do they not? As you power decreases you divide by two And then you divide by two and you divide by two and when you get to here We have no reason to think that the pattern changes right? so this pattern becomes this and And so we divide by two and you get this. And you keep on going and it keeps making sense Okay, now therefore Can you argue for me? Why is your factorial is what? Tell me what 1! is.

Everyone: 1. 2 factorial? Everyone: 2. 3 factorial? Everyone: 6. 4 factorial? *Laughs* Okay. So, this time, how do I go backwards? I divided by 4 and I divided by 3 then I divided by 2 Everyone: Oh my god Can i just say, can i just say… the reason why I bet a big deal that this is because See this is what makes maths It's what makes math interesting to me. You see math is, uhm, it's an imagined world That's the point of it, right? Like I know maths can be used to do stuff, and that's, that's nice, okay but um The point is that oh actually I'll come back to that in a second The point is that it doesn't matter if it can be used for something or not the the point is that it has a consistent system of rules. That's meaningful, okay? Here's the amazing thing right There was a guy. [there's] a guy, a french guy, and his day was for you, okay? And he was the guy who we earned he discovered this Have a look at this for a second.

What is this ah? Yeah, language for this. This is a wave function [now]. You don't know what wave function is. I just drew it okay, but Amazing thing about this. This is what, I mean, this is what fourier proved, okay? is that this wave function can be made up of can be Composed of just take a whole bunch of sine functions, right? The whole variety of them some of them will be big, right? Some of them will be small. So you have to change… let's say you have to change this coefficient on the front and you change the frequency, okay, and if you add enough of them together Okay, and so on You can make you can make anything, anything! You can even make weird looking things like this You like, I can make a function like that. It's a wave function. I know it doesn't look like it It's got straight lines, [and] what's with that, okay? If you add up enough wave functions Different ones like this, okay like this one.

You can give us a else now. Here's the interesting thing What was fourier after? He just thought this was kind of cool. It was like. This is interesting, right? I wonder if you can do that, and he gave it to go and it came, okay? Uh he was not at all thinking about practical applications, right? He was not thinking about, you know, a world of electric communications which uses this routinely in order to, you know, How does this thing know to convert weird stuff, which is in the air? into sounds Sounds, right? How does it do that? Answer: using these, but fourier wasn't thinking of those, here's just like: Cool! Let's just see what happens. And the application came later, almost every field of mass Complex numbers, like, why are we doing that for? They come up all the time for engineers, but that's not what the people who were thinking about them were thinking.

They're just like it's cool. Zero factorial equals one. It does make sense just like Something power of zero equals one.
Subtitles by: Instagram: @franckcid | Twitter: @franck_cid.

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