0! = 1 Explained in 5 Levels from Counting to Math Major

Подписаться 17 тыс.

Просмотров 423 тыс.

50% 1

The factorial of 4 is 4! = 4 * 3 * 2 * 1 = 24. But what is 0 factorial? At first, we might guess it should be 0, but we actually define 0! = 1. Let's explore why 0! = 1 in five levels, ranging from the meaning of factorials when counting, through explanations from Calculus and beyond.
00:00 Introduction
00:22 Level 1: Counting
01:12 Level 2: Algebra
02:44 Level 3: Combinations
04:07 Level 4: Calculus
04:54 Level 5: Math Major

Опубликовано:

1 авг 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 481

@FlowMichael 4 месяца назад

Wow thanks so much! I tried to ask my Calc teacher about why this worked and he basically used the "1 way to arrange 0 objects" explanation, but this is way clearer! I hope your channel keeps growing; you do great stuff!

@mattholsten7491 4 месяца назад

Agreed! That is a classic explanation, but to me it’s also a bit lazy and not satisfying since I could just as easily imagine a universe where we say “there’s ‘undefined’ ways to arrange 0 things.”

@JJ_TheGreat 4 месяца назад

@@mattholsten7491Agreed! You could argue that there is an infinite number of ways to arrange 0 objects, since they do not exist!

@sbok9481 4 месяца назад

You all got lost along the way. Start with the original definition of factorial. I'm gonna ask one simple question: is zero (0) a positive or negative integer? And what positive integer below it did you multiply with? It would make more sense to work with fractions as factorial between 1 and 0 because 0 itself doesn't have a factorial.

@asddsaasdfg2846 4 месяца назад

@@mattholsten7491there’s actually 1 way to do it. Bc you can’t do anything

@frimi8593 4 месяца назад

@@JJ_TheGreatthat argument would be much harder to make. You either can’t arrange nothing or there’s only one way to arrange nothing. I can’t imagine an argument for there being infinite ways to arrange nothing that doesn’t also imply there’s infinite ways to arrange, for instance, 1 thing

@stevenjones8575 4 месяца назад

I like how you emphasize that we define it to be 1 because it's useful to be defined that way, rather than just saying it equals 1. I think it's important to acknowledge what is convention, and why the community has largely chosen that as convention.

@paulfoss5385 4 месяца назад

I think the gamma function pretty clearly shows that nature agrees it equals one. If we had a different convention, then we'd simply be wrong.

@lerarosalene 4 месяца назад

@@paulfoss5385 it isn't about "nature". There is no "natural" about math, it's just our own invention. Gamma functions was invented to extend factorial, there is no hidden truth there.

@Nartymer 4 месяца назад

⁠⁠@@paulfoss5385it doesn't "show" as in "proves". Mathematics in itself is a convention we created to study our world. The caveman had an easier time saying "me have 3 rocks" than saying "me have a rock and a rock and a rock". It's why 1+1 = 2 for all objects and we can be sure of it. Same goes with the factorial. We created it as a convention for how you can arrange n objects where the order matters. Now how we arrange 0 objects has no meaning to us, but it needs a certain accepted convention, which in our case, making 0! = 1 simplifies our formulas of any possible exceptions we'd need to create around it. Something somewhat similar usually goes for 0^0. Some calculators just define it as 1 as convention because the limit of x^x as x approaches 0 is 1. But there ARE limits with case 0^0 that converge to 0, so it can be disproven when needed

@Muhahahahaz 4 месяца назад

Agreed, for most of his examples. However, I would say that the empty product directly obviates the need for any “convention” in this case If n! is the product of the first n positive integers, then 0! is the product of no integers, which is automatically 1

@mdcoulson 4 месяца назад

@@paulfoss5385It wouldn't mean we're wrong, it would mean we would constantly need to handle separate cases. It is a convention made of convenience. Note that empty product being 1 and multiplicative identity being 1 are technically different things.

@MrCocktaiI 4 месяца назад

Actual Level 5: "Yeah we just define it like that" - "understandable have a nice day"

@mateherbay2289 4 месяца назад

Yeah, but you cannot define it other ways because analytic continuation doesn't work then.

@yuezie 4 месяца назад

@@mateherbay2289You could, but you'd just have to work with the consequences of a function being non-continuous right?

@Benjamin-xv9le 4 месяца назад

@@yuezienah, for any value you assign to 0! there would be continuos functions that go through n! for all n

@vladimirkhazinski3725 4 месяца назад

Yeah this video is pretty bad and doesn't explain it well at all. The reason its defined like that is because the gamma function is the only continuous function that satisfies that f(x)=xf(x-1) at all points and that is the property that defines factorials

@JACabLab 4 месяца назад

@@vladimirkhazinski3725not really, there are other functions that extends the factorial, the property that makes the gamma function special is that its log is convex

@parzh 4 месяца назад

As a software engineer, I can confirm that 0 is indeed not equal to 1.

@paulgoogol2652 4 месяца назад

ya mean not 0 equals 1

@deckerfranks7098 4 месяца назад

Ha, I'ma steal that pun for my students

@lifeai1889 4 месяца назад

This is what I thought after seeing the thumbnail

@mheermance 4 месяца назад

I LOLed so take my upvote.

@eddyr1041 4 месяца назад

Maybe not in quantum qubit😅😂

@DrZaius3141 3 месяца назад

When I teach 10-year-olds (not necessarily about factorials, but with them in mind), I put it this way: If you want to add something without a change, you add 0. If you want to multiply something without a change, you multiply by 1. If you don't add any numbers together, you get 0. If you don't multiply any numbers with each other, you get 1. Essentially, it's about neutral elements but without any terminology needed.

@tille4376 2 месяца назад

This ist actully a perfect explanation for a 10-year-old. I have 2 questions. Is he a Genius? Where do you live?

@user-li5lx3vi8o 2 месяца назад

Horrible

@Marconius6 4 месяца назад

As a programmer, I just need to keep in mind that 1 is the identity for multiplication. So if 0! wasn't that, you'd need to constantly make special cases for it in formulas and calculations.

@pedazodeboludo 4 месяца назад

True, so the special case is still there, just implemented in fact(0) 😅

@hlebysq4840 4 месяца назад

As a programmer, in the preview just true: 0 not equals to 1

@pedazodeboludo 4 месяца назад

Base case

@feandil666 4 месяца назад

never heard of the "zero product must be 1" before but it makes total sense when you think about it in group algebra.

@viktorsmets29 4 месяца назад

Yeah, in group theory it makes sense to have a⁰=1 for every a in the group. It's the same reason we have that x⁰ = 1 for all x, except 0 itself. Looking at the group of (C, *, 1) the property in groups and the property of exponents we are used to is the exact same.

@viktorsmets29 4 месяца назад

The reason why the Gamma function is often used as a way to generalize the factorial is because it has the property z*gamma(z) = gamma(z+1), which is just saying that z * (z-1)! = z!. And that property is also the one used when using reasoning from algebra.

@MuffinsAPlenty 4 месяца назад

Thank you! I so often see people on RU-vid disparaging the "algebraic" method in the video and then turning around and saying that the Gamma function is the _"real"_ reason that 0! = 1. But as you say, the reason the Gamma function gives 0! = 1 is because the function is forced to satisfy the recursive property of factorials, not because of being a meromorphic function or because of log convexity or anything like that.

@joshuamackaman4070 4 месяца назад

Please keep making videos like this. This makes getting back into math easier and more enjoyable.

@hcoguitar 4 месяца назад

Fantastic delivery and clear explanation. I wish I had you as my calc professor in college!

@Nick-dj5bx 4 месяца назад

Awesome video, keep it up! I like that the same mathematical fact can be explained in several ways to connect it to all those different settings where this question would arise.

@haasjeoverkonijn6961 4 месяца назад

Wow. So crisp and clear. Thanks!

@atrus3823 4 месяца назад

The way I think of it is not so much that 1 is left, but that we start from one. Any number, let's say 5, can be thought of as starting from 1 and scaling it up 5 times its size (it's often useful to think like this). So multiplying, let's say 2*3, can be thought of as 1*2*3, meaning, start from a whole of something, scale it to 2 times its size, then scale the result to 3 times its size. Factorial can be thought of this same way. Think of it algorithmically: f

@tonioene2262 4 месяца назад

I like this explanation. Where can I find more information on it?

@PROtoss987 4 месяца назад

same reason 0^0 = 1, because 1 is the multiplicative identity

@bartgertsen6181 3 месяца назад

@@PROtoss987that doesnt make sense to me. How does 0^0 relate to the multiplicative identity. That just means that A*1=A.

@PROtoss987 3 месяца назад

@@bartgertsen6181 What else would you get when you multiply 1 by 0, but for zero iterations? If I have a nonzero A which I add to 0 only zero times, my sum is still zero.

@JJ_TheGreat 4 месяца назад

Thank you for taking my suggestion and doing a video about it! I see you incorporated the nC0 into the video! It makes more sense now!

@DrSeanGroathouse 4 месяца назад

Glad you liked it, and thanks for the suggestion!

@baronch-workaccount-gr9fx 5 дней назад

really great math video. it definitely deserves more.

@tylerbakeman 4 месяца назад

Im really happy that I knew all of this already. I was expecting something new / crazy. Great video btw! 0! being synonymous with a “multiplicative identity” is a cool idea, with potential useful insights in Unital-magmoid Theory: There is probably a lot of research on those very specific Groups anyhow.

@FlatEarthMath 4 месяца назад

So excellent! All this time I'd only known the combinations reason, but I always assumed 0! was just arbitrarily defined as 1 so the combination formula would always work.

@ChatGPTGamingReal 4 месяца назад

this came at a perfect time! i was trying to derive some formula i learned in stat today and this one trick I didn't know was the bridge I needed to finally understand it.

@popaye.8968 4 месяца назад

You look so young yet you've achieved so much. Very impressive sir.

@Tribalchief69690 3 месяца назад

The new math channel in the market ❤

@placeholderblankspace 4 месяца назад

This is… kinda crazy. I’ve always just accepted this without understanding and the way you explained it so simply at first and kept building up is such a nice way to learn the reason behind this! not a math major myself and yet while this is my first time hearing about Gamma and stuff the explanation still made sense.

@LeonardoRamos01 3 месяца назад

Very nice video, congrats! I find it concise and entertaining.

@Trenz0 4 месяца назад

My calculus instructor at community college was amazing and she really did a fantastic job explaining a bunch of concepts. I owe her a LOT in terms of my math prowess, especially in comparison to my peers here in university. However, one thing I'll never forget was her explaining that 0! = 1 was best understood by just taking her word for it lol. She kind of went into the "1 way to arrange" explanation but she made it clear, it's best not to get too hung up on it. Meanwhile, she spent a ton of time explaining famous proofs and other important concepts

@eggchipsnbeans 4 месяца назад

There are time when your teacher doesn't really understand fully everything he or she teaches. For example, in Statistics I can explain how to use the tools pretty effectively and often give explanations about why the ideas work but mathematical statistics is a dark and lonely road with creepy trees and best avoided.

@JohnDiGrizUkraine Месяц назад

Honestly the answer is essentially: "we decided to define it like that cause it behaves more consistently and allows as to avoid having special cases for it". The rest of the explanations is essentially different ways in which 0!=1 is more consistent and useful than any alternative

@artursruseckis4242 3 месяца назад

Nice. I was always fighting against this concept ( I wanted 0! to be 0), but this video finally convinced me. As soon as you went into group theory and identity element of multiplication, it was a major eye opener. I don't even remember when was the last time when a single sentence suddenly crushed my world view with "oh, that's why! it's so simple!"

@jazzabighits4473 2 месяца назад

It is 0, don't fall for this. There are 0 ways to arrange 0 things.

@alexdotdash7731 4 месяца назад

Love these videos!

@potato9832 4 месяца назад

Thank you, Dr Sean!

@Ubeogesh 4 месяца назад

"multiplicative identify" really hit the spot for me

@disafear3674 Месяц назад

Currently working on my chemistry PhD thesis and focusing on anything but that. Interesting video!

@AndreyEfimov Месяц назад

"Level 2" is my favourite. Explains the reasoning very logically.

@amirmehr151 Месяц назад

Great video and explanation

@johnmiller6696 4 месяца назад

Absolutely brilliant! Subbed

@MGmirkin 4 месяца назад

Brilliant explanation of a **nonsensical** and **technically wrong** assertion. Convenient, maybe. Correct? No. The null set / empty set does not have a value of 1, it is **at best** "undefined."

@ExtraTrstl 4 месяца назад

This is great - thank you! WONDERFUL presentation; subscribed!

@DrSeanGroathouse 4 месяца назад

Glad you liked it!

@markd8004 4 месяца назад

Very interesting, thanks!

@enderyu 4 месяца назад

1:55 - 2:44 That is a whole level of explanation by itself, and I also think it is the most convincing one as well

@ww_artemis 4 месяца назад

Saying that 1 is the neutral element to a product is just the most simple way to explain it, and the most logical way too, in my opinion.

@pikminlord343 4 месяца назад

What a great video!

@zachrodan7543 3 месяца назад

As someone who has sadly only been able to get up to the calculus level in my math education, or slightly beyond (i'm aiming for math major, but it takes time), i had never really thought about 0! Being 1 due to 1 being the muliplicative identity... but i like that

@seany2751 4 месяца назад

love it dr sean

@markgraham2312 3 месяца назад

Great job!

@ThiagoVasconcelos46 4 месяца назад

I liked this channel. Pretty nice.

@Patricula 4 месяца назад

Interesting video! Constructive tip: Either keep your hands in frame all the time or maybe gesture higher so your whole hand enters the frams instead of just your fingers periodically popping up. :)

@aporifera 4 месяца назад

I think it's helpful to think of factorial as in the same tier as powers, logs, and roots etc., the tier above multiplication and division, on which they are all based.

@tcmxiyw 4 месяца назад

In a computer program, if you want to compute a product p by computing each factor and incorporating it into the running multiplication leading to the final product p, you will initialize p to 1. I.e. the empty product is 1. Similarly, the empty sum is zero. To calculate 4!, start with the 1empty product 1, and include the factors 4,3,2,1. To calculate 0!, start with 1, and note there are no additional factors.

@pawel_maslanka 4 месяца назад

cool video, I'd like to see 0^0=1 next

@CapAnson12345 4 месяца назад

I thought about this for a while. And I think the best way to think of it and not be confused is instead of thinking about it from the top down.. so n! = n x (n-1) x (n-2)...x 1. Instead start with 0! and simply define it as 1 by convention. Then 1! REALLY becomes 1 x 1, 2! Becomes 2 x 1 x 1, etc. But there's always that extra times 1 that could be thought of as "and also, multiply it by the empty set identity"

@BucketCapacity 4 месяца назад

I think it's worth mentioning that you could tie Level 2 with Level 5: observe that (-1)! is undefined in the same way that dividing by zero is undefined, diverging to positive infinity approaching from the right, and diverging to negative infinity approaching from the left. This is the same as what we would expect from using the algebraic explanation, 0!/0 = 1/0.

@MGmirkin 4 месяца назад

0! should ALSO be "undefined," *is* technically undefined/illegal [since **actual** factorials stop at multiplying by 1 and actually multiplying by zero itself would just always yield zero], if people were actually being honest about it. Zero is **essentially** an illegal integer in factorials. As, presumably, are negative numbers... Yeah? Wolfram|Alpha says this of factorials: "n! is a sequence with integer values for nonnegative n." I would, personally, amend that to "n! is a sequence with [non-zero] integer values for nonnegative n." ----- "Because we say so" and "because it's convenient" are not good enough, and if push comes to shove, there is no actual legitimate proof for the identity 0!=1 because 0!=/=1 they are not, in fact equal. They are only asserted to be so by fiat not because anything actually proves them to be so. ----- 0! is basically the null set, not 1 or any other value. It is undefined. One should think of factorials as being a multiplicative series of sequential integers with a number of positions equal to the integer being factorialized. 1! has 1 position: (_) or (1) 2! has 2 positions: (_x_) or (1x2) 3! has 3 positions: (_x_x_) or (1x2x3) 3! has 4 positions: (_x_x_x_) or (1x2x3x4) 3! has 5 positions: (_x_x_x_x_) or (1x2x3x4x5) How many positions does 0! have? None: () it is the "empty/null set." There are **no numbers** being multiplied together. Its value is not 1, its value is "undefined." It is a non-expression. There is literally **nothing**. ----- Also, multiplication and division are basically interchangeable by using the inverse fraction. So, 1*2=(1/1)/(1/2)=2 1*2*3=((1/1)/(1/2))/(1/3)=6 1*2*3*4=(((1/1)/(1/2))/(1/3))/(1/4)=24 But this is a problem for 0, since: 1) any number divided by zero is ... undefined or basically an illegal operation. 2) Since there are no "positions" for any numbers or fractions in 0! [0 positions], technically one can't even really **do** a division by said "undefined" fraction. Heck, there's not even a "position" **just** for the 0 itself **in order** to invert it let alone enough positions to put it in an expression relating it **with** another number. Again, it's just basically the "null set." No positions, no "value," etc.

@anarchistmathematician 4 месяца назад

Very nice video! Here's one more that is more "math major" in construction but "feels like" level 1. The symmetric group on n objects Sn has as its underlying set all bijective self-maps on a set of cardinality n. This is just what a permutation is, so it should come as no surprise that the cardinality of Sn is just n!. Let's consider S0. Well, this is all bijective self-maps on a set of cardinality 0. Only one such set exists: the empty set. There is indeed an empty function from the empty set to itself: it vacuously satisfies univalence and being left-total. It similarly vacuously satisfies bijectivity. Thus at least one object in S0 exists. It is not hard to see that it is the only object, since any other such function would have to vary in at least one output, but there are no outputs to vary. Therefore S0 has cardinality 1. And, as we know, the cardinality of Sn is n!, so we have that 0! = 1

@lukamtc9188 4 месяца назад

the last method is very common use for me, it's very common to deduce solutions using correlating functions like that in all of science and especially chemistry and biochemistry I truly am good at procrastinating to be watching this content, but the videos are high quality.

@oethe9954 4 месяца назад

I never even learned about factorials until taylor polynomials which really makes no sense, I wish I learned about them earlier they are really cool

@sadmanchild9351 2 месяца назад

Very good video I don't quite get the Combinations at first but it makes sense 😺😺😺 I still do not fully understand the calculus or the math major but I hope to one day

@user-vp9zt3sk7b 2 месяца назад

What about abstract algebra and for multiplication which factorial is based on, the neutral element is 1. that’s why it should be 1 does that make sense?

@eaterofcrayons7991 4 месяца назад

This guy's a doctor?! He looks 20! Amazing video

@Nr_Logic 4 месяца назад

Good explained

@darthTwin6 2 месяца назад

Question: The gamma function is used because it happens to satisfy factorial for the natural numbers and can be evaluated at integer inputs, right? Or is there a deeper reason? Do the non-natural results somehow follow the core behavior of the factorial? Can we make sense of it on an intuitive level?

@brilliance. 4 месяца назад

for the algebraic proof/reasoning; what if you continue and try to get -1 factorial? You cannot divide by 0 soo

@trialbyicecream 4 месяца назад

I know what I’m going to reteach in BC calc tomorrow!

@Figorix 2 месяца назад

Gotta love how wet start with "it makes sense" and gradually go to "if it doesn't, it fucks up other theories"

@antekb1034 4 месяца назад

helped me :)

@Dark_Souls_3 4 месяца назад

Is your thesis available? I’d like to read it. Or do you have a video on it? Or could you make one?

@DrSeanGroathouse 4 месяца назад

It's on ProQuest which requires a university login, but I'll get it added to my website in the next couple days so I can share a link. I don't have a video about it yet, but I think that's a great idea!

@KalijahAnderson 4 месяца назад

What makes this confusing is the rule of anything times 0 = 0. Factorial at its base level is multiplication, so most would think 0! = 0. But the consideration that it means the number of ways to arrange something reveals its not just multiplication.

@MGmirkin 4 месяца назад

Other things make it nonsensical or argue against it, as well. Multiplication being essentially equivalent to division by the inverse fraction: 1*2=(1/1)/(1/2)=2 etc. Problem: any number divided by 0 is undefined. As such, presumably 0! must surely be undefined if it can in some way be rewritten as some kind of "division by the inverse fraction," putting 0 in the denominator? ----- More fundamentally, factorials are a multiplicative series with a number of positions equal to the number being factorialized. 5! has 5 "positions" in the expression 5x4x3x2x1 4! has 4 "positions" in the expression 4x3x2x1 ... 1! has 1 position: "1" 0!, presumably must therefore have 0 positions. It is basically the "empty/null set" (). There are no numbers being multiplied together. The null set does not have a "value," does it? The null set does not contain 1, or equal "1" does it? Surely its value, per se, is either 0 or more likely "undefined"? Yeah? ----- IMO, 0 is basically an "illegal number" to use in factorials, just as are negative numbers, basically by definition, since a factorial stops once it reaches the integer 1. 0 is **NOT** included in a factorial. It is simply never included in a factorial. The definition should *be* "Factorials are the multiplicative sequential series of [non-zero,] non-negative numbers between the number being factorialized and 1. ([Zero and] negative numbers are not allowed.)" ----- Just my opinions, when thinking rationally about it.

@bertsimpsan 4 месяца назад

Can you use a de-ess filter the lisps are making it hard to concentrate

@atatdotdot Месяц назад

In the Taylor polynomial (4:40), wouldn't it make more sense to write 𝑥⁰ and 𝑥¹ as the first two numerators, and then the symmetry would be complete?

@drunkinbda 4 месяца назад

Not sure I can get my head around the Algebra part, so perhaps someone can explain a bit... I get that 4!= 4.3.2.1 can then set as 3!=4!/4.... because it seems you're just divding both by four...(same for 3,2 and 1)... but it seems weird that suddenly 0! shows up after 1!.. The 0 was never in the product to begin with so how does it show up now? Would this be the case that we're always just calculating (n-1)! each time instead quotients/products? Im reading this and even realising its a hella complicated thing to explain...

@Sonicgott 4 месяца назад

If only our math teachers were more like this. I had such difficulty focusing when I was a kid, but as it turns out, when you're entertained, you tend to pay attention. Teachers need to be as much an entertainer as a teacher, because if you're just saying facts like reading the newspaper in a monotone voice, you'll forget. Aside from that, showing graphically on why the things are the way they are helps us remember. Beautiful demonstration.

@w1111-vs3dd 4 месяца назад

TRUE!! I could very easily understand how 0! was 1 with this video, but by searching all I learned was that 0! is 1 because it just is..

@Thechessrocker1 4 месяца назад

Yeah sure, let's compare someone who had many hours to prepare a 6 minutes video that is only watched by interested people to a teacher who didn't have much time to prepare hours of content and has in front of him 35 immature children that did NOT choose to be here. Seems legit ! And btw : this guy IS saying facts like he's reading the newspaper, almost all the teachers I know are way more entertaining than he is ... What happened there is simple : you got old. And now you realize some stuff was actually more interesting than you thought when you were a teenager ! This "wish our teachers were like this" needs to stop, it's pure nonsense AND insulting.

@w1111-vs3dd 4 месяца назад

@@Thechessrocker1 has a stroke trying to read the first part

@Chakamatics 4 месяца назад

@@Thechessrocker1I agree with this take

@tsotnats 4 месяца назад

Seeing the gamma function gave me flashbacks to statistics class! 😅

@MalcolmRoberts0 4 месяца назад

I think that the cleanest way to define it is, as you alluded to, the empty product: use sets, and, using the property that multiplying all of the elements of a set can be done via sub-sets, extend this to the empty product. x = \Pi_{A} = \PI_{A} \times \Pi_{} = x \times \Pi_{} , so \Pi_{} = 1, and 0! is just the product of the elements of the empty set.

@monoyiosioannis5535 3 месяца назад

This guy’s voice is in my head all these years when I study maths and I am Greek 😂🤷🏼‍♂️

@dastardlyexpressions 4 месяца назад

I would've strengthened the math major argument a bit because it fails to argue why the gamma function is the continuation we want. (1) Yes, there are multiple ways to extrapolate the factorial for nonzero naturals to the reals, but all analytic continuations of the factorial will agree on the integers and indeed predict 0!=1. (2) Though tangential to the original point, if you want to define the gamma function, it's good to point out that it's the only analytic continuation that always obeys the defining equation of the factorial Gamma(x) = (x-1)Gamma(x-1) .

@akademesanctuary1361 2 месяца назад

A product series to allow for negative factorials arrives at an imaginary j=0!, as opposed to i. The j operator commonly appears as ±, but it flips signs on every operation due to making selections. Oliver Heaviside used it to describe the field of selections 1892 solution for impedance. Because it initially selects position (+1), it is easy to show 0!=+1. You have to go out of your way to solve for negatives to see it can also be -1. Because it flips signs on each operation, j^2=-1, j+j=0, and j-j=2j. These qualities streamlined Heaviside's solution. The really surprising thing is that you can see it in the math, but no one bothered to document the seemingly paradoxical qualities. I've been working on an imaginary numbers book if you want mind bending reading to put you to sleep at night. ;)

@jawaduddin1160 4 месяца назад

Plz make a video on fractions snd decimal

@jamiewalker329 3 месяца назад

I was thinking the following. The number of bijections between a set of size n and a set of size n is n! So 0! should be the number of bijections from the empty set and itself (this bijection does exist and is equal to the empty set when we consider functions as subsets of the cartesian product). We also know that the number of functions from a set of size n to a set of size m is n^m. So this would also give 0^0 =.1 as being a sensible definition...

@illinialumni 4 месяца назад

Lol. Blew me away on the first simple explanation. I thought I understood math

@leekyonion 4 месяца назад

If you can explain the concept to a 3rd grader, you're the expert.

@chrisparkin6894 4 месяца назад

It really does feel like an explanation to a third grader...we made it that way...ie because I said so. Unfortunately all of the explanations feel like they are circular arguments. The last one didn't work for negative 1...

@vladkolotnikov249 3 месяца назад

Except the first explanation has the most holes in it. Actually the first explanation would be better suited to proving 0! = 0. “putting no flags on” is not a solution to a permutation. If you have 1 flag there are 1! ways of putting it up (aka 1 way). If not putting up the flag was a solution 1! would be 2 (either you put that 1 flag up, or not)

@purewaterruler 2 месяца назад

@@chrisparkin6894how is the second one circular? It's basically the defining property of the factorial: (x+1)*x!=(x+1)!

@Neodynium.the_permanent_magnet Месяц назад

Gosper's approx for 0 gives sqrt(pi/3) ~ 1.02, very close to 1.

@epicgamersauce3516 2 месяца назад

A way that i thought of it is that there are an infinite number of trailing multiply by ones at the end of any number and its still equivalent, so by going from 1! = 1 = 1x1x1... to 0! =1x1x1.../1 = 1. Taking one 1 out of the whole doesnt change the end result

@isospectral3537 4 месяца назад

Level 1 can be described formally with the empty set and vacuous truths. This is what I consider the "morally correct" definition. Any function X->Y can be formally defined as a graph, or subset G of the Cartesian product XxY for which there is no x in X and distinct y,y' in Y such that (x,y),(x,y') are both in G ("vertical line test"). The function is a bijection if (i) there is no distinct x,x' in X and y in Y such that (x,y),(x',y) are both in G (one-to-one), and (ii) there is no y in Y for which there is no x in X such that (x,y) is in G (onto). Notice I rephrased "there exists" statements into "there are no counterexamples" statements. Since {}x{}={}, the only function {}->{} is the empty function, i.e. {} itself, which is in fact a bijection. 0! is the number of bijections on a set of size 0, so 0!=1. This also shows why 0^0=1 too. And justifies the Level 2 and Level 3 descriptions, as well as power series (Level 4) interpreted as generating functions (also cf combinatorial species).

@tpfoxCastro 3 месяца назад

0! = 0! "But why" don't question math, that just gives you more work. First ten of the video watched, got all the information I needed, now by

@mr.potatochip2881 2 месяца назад

I like your funny words magic man!

@vihansmit572 4 месяца назад

If you think about it, they all start with 1 times sometimes, why would 0! Start vir 0 times something?

@georglehner407 4 месяца назад

n! is a decategorification of the set of automorphisms of a set. There is exactly one bijection of the empty set - Its identity function. So naturally, 0! = 1. This is in a way related to your combinatorics example and basically just a rewording of your counting example, but slightly more fundamental in spirit. Since the category of finite sets exists, so does the functorial function. (The formula for the binomial coefficient is in a similar way a decategorification)

@ok_kite_kiwi 4 месяца назад

i like to think to factorial integer numbers (not negative) as the multiplication of all natural numbers from 1 to that number, so considering there's no natural number smaller than 1, then 0! should be 1

@Antonio-lt1sp 4 месяца назад

Thanks! Please make a video about why 5th order polynomials are not solvable by radicals. Best regards!

@DrSeanGroathouse 4 месяца назад

Thanks for the video idea! I added it to my list

@spacelem 4 месяца назад

If we only defined the Gamma function to be Gamma(z) = \int_0^\infty t^z e^-t dt, then we could have had Gamma(n) = n! instead of (n-1)!. Why did that -1 have to creep in there?

@redwren4182 4 месяца назад

I found the counting definition to be the most logical and the algebraic definition to be a convenient 'it just is'.

@diobrando7642 2 месяца назад

I think we should change the definition of factorial for natural numbers to: 1! = 1 n! = (n+1)!/(n+1) So that we can derive the same formula we have without explicitly stating 0! = 1 as a base case. It would be equivalent, but it would be easier to accept for people.

@sadmanchild9351 2 месяца назад

Looking into this Ty

@hoperanker8395 4 месяца назад

Thanks for this! I get these pretty much up to the gamma funtion. And I even get why Γ(0)=1. But the video explains that there are other ways besides Γ to extend the positive integer factorial function to the reals. So in some sense, isn't the gamma function explanation a bit circular? If we were instead to start from a presupposition that 0! is undefined, we simply wouldn't use Γ to extend ! to reals, right? So I suppose the argument is less "Well, the gamma function makes it one" and more "If we wanted gamma of zero to be undefined, we'd need something way more complicated than gamma to represent it." Am I on the right track there? I'm not trying to challenge the definition that 0!=1, just trying to clearly understand the reasoning.

@ethanbottomley-mason8447 4 месяца назад

The Gamma function is the most useful extension to the reals. If you additionally ask that the extension be log convex, then the gamma function is the only way to extend factorial to R. f is log convex just means that log f is a convex function.

@albertolando5268 4 месяца назад

So you are suggesting that there are many hints on why, in order to have math as clean and beautiful as possible, we should assume 0!=1, as if we are on one side just choosing it to be so, but under another perspective, discovering some underlying pattern that's already there and choosing convections according to it. That's interesting

@_GhostMiner 4 месяца назад

*Mathematicians and programmers agree that 0!=1*

@francescobottino3892 3 месяца назад

Clever one 😂

@mtaur4113 4 месяца назад

It's also convenient that the same 0! appears when you count how many "words" (real or nonsense) you can spell using exactly 1 A, 2 L's, and 1 B (and 0 C's and 0 D's and...). The alphabet has 26 letters, but the formula can be written without explicitly excluding the unused letters. The "contribute nothing to the multiverse of possibilities" action when multiplying possible outcomes is the multiply or divide by 1 action, as you say.

@Seltyk 2 месяца назад

If the gamma function has this arbitrary off-by-one shift from the factorial function, why not just redefine it simply? The integral from 0 to infinity of (x^k e^-x) dx is equal to k factorial. Plus C, if you must. No shift needed. So why does the gamma function do that?

@willj8205 Месяц назад

The Bausffs is doing side quests being a math professional outside of inting top lane

@tomthaxter161 12 часов назад

Almost all of the confusion I think comes from the notation. Once you frame it as a factorial being simply a function, which we could write generally as f(x), then I think far fewer people would question why f(0) = 1 as lots of functions have this property. I think lots of people jump to 0! feeling like it should be zero is just based on it’s similarity to 0+0 or 0x0

@Oberatous-Udurabas Месяц назад

What is 0.5! Or 1/2 factorial

@ianowens1905 3 месяца назад

Why does he look like he just got out of his emo phase (no hate im in mine 💀)

@poiewhfopiewhf 4 месяца назад

i always imagined it as a 1 going into a machine that would get multiplied by other numbers and for 0 it wouldn't get multiplied by anything

@BigMonkeyKnows 4 месяца назад

I feel like the first example is confusing. For example, with 0!, it’s said there is one way to put zero flags up. But, if that’s true, then 1! would equal 2 because there can be one flag or no flag.

@Abcde7213 4 месяца назад

Think of 2 factorial. There is a flag pole with a green flag and a red flag. 2! Is 2 because you can either arrange it with the green flag higher than the red flag, or the red flag higher than the green flag. There are 2 ways you can arrange 2 flags. The options where you remove one or both flags are not included in the way that 2 flags can be arranged. There must be 2 flags. Likewise “no flags” is not an option in 1 factorial; There must be one flag, and there is only one way to arrange that one flag. And for 0! There is only one way to arrange no flags

@LeoAr37 2 месяца назад

@Abcde7213 I understand, but I gotta admit it's confusing that "no flags" count as a way to organize 0 flags but doesn't count as a way to organize 1 flag.

@SuperCrAzYfLiPpEr 4 месяца назад

Not sure if it's relevant but the definition of derivatives makes h=0 which is in the numerator.

@Tharronis 4 месяца назад

I think #1 and #2 were the only convincing explanations. #3, #4, and #5 all seemed to be confusing correlation and causation. Factorial wasn't defined as "0! = 1" to make those formulas work, those formulas were created and work BECAUSE "0! = 1". Explaining "0! = 1" by showing that there are formulas based on that fact is very unhelpful. #3/#4/#5 kind of come off as trying to explain the value of pi by showing that the area for a circle formula wouldn't work otherwise.

@RoboSlaughter 4 месяца назад

There is no correlation here and that is a very unhelpful comparison to make. Second half of your argument sounds awfully platonic too.

@SaloCh 3 месяца назад

Yeah the 1st two feel like a proof of why 0! = 1 intuitively, whilst the others just leave me feeling like it was defined to be 1 by convention. There's a difference between "This is true and here is why" and "We decided this is true because it was convenient" Edit: and yes, it might just be the case that we defined it like that because it was convenient, but idk, that's not what I was led to believe the video was going to prove

@lynackhilou4865 3 месяца назад

The first two are what you would use as proof , however it is not wrong to say that we made 0!=1 by convention , a lot of the time it is needed to be able to apply certain hypotheses. I think the same can be applied to the number 1 being excluded from being a prime number despite the definition applying it to it

@Protagonix 3 месяца назад

Thats cause (no offense) level 3+ requires more then just a basic highschool understanding of math

@asdfghyter 3 месяца назад

@@SaloCh this is a part of the definition, so it’s not possible to prove. all we can do is show that it’s the most sensible definition that makes as much as possible convenient