marbanak what do you mean? It’s because you’re saying it with excitement, so you use explanation points; making 0! = 1. Screaming 0 loud enough makes 1.
@@marbanak are you *actually* implying that zeroes can't be equal to one? that is literally numbers-ist. reported for numerical supremacy and hate speech
Honestly my favorite out of these is reason #2 because of how philosophical it is. How many ways can you arrange nothing? Exactly one way: no arrangement at all. The complete absence of an arrangement is itself a valid arrangement when you have nothing to arrange. :D
Calyo Delphi Yeah, but you could add that nothing to all positive factorials, in which case they should be their current value + 1: 2! should then be 2 + 1 = 3, for example. Yet no one uses that fact to conclude that, but it is not logically sound in that case.
J.J The Great: I don't think so: You cannot NOT arrange any number of items when you have one or more items. As long as the items are present, they are arranged. So the additional +1 is NOT an option for n! as long as n>=1. But when you have NO items, then the option NOT to arrange them is the ONLY arrangement. It's an option that appears ONLY when there are zero items.
In reason 4, what I really like is that, if you use one of the other reasons to accept 0!=1, than, you can turn the other way around to prove the «convention» 0^0=1.
@@lolme2646... This guy keeps assuming open 0 in the limit as x ---> 0 for y = x^x being y @(0, delta) = 1 for very small delta. His #4 solution shows this. But L'Hopital's Rule showed f(x) = 1 as x approaches 0 at [0, delta)! y= x^x has a limit of 1 at x=0 not only for series expansion reasons of defining e^x = a series limit but for other mathematics where 0^0 appearance problems.
Zero is not a number; it is the empty set and acts as a placeholder nothing more. Numbers that are greater than 0 are numbers. All of those numbers with a negative sign in front are the same exact number with the same exact magnitude just that they span out in the opposite direction as they are pointing 180 degrees or PI radians in the other direction, vector computation proves this. Also the negatives are reflected about the perpendicular bisector that is located at zero on the number line or x axis and the point at zero acts as the point of rotation and the point of symmetry. Zero has no quantitative value therefor it is not a number! Yes operations can be performed on it just as any other number, but the outcomes can vary from one operation to another that doesn't hold to the operations of all other numbers either if they are the set of all real numbers or the set of all imaginary and complex numbers. For example we were all taught that you can not divide by 0 as it is undefined. Let's not look at this as a numerical expression but as a conceptual expression instead. I'll use some regular fractions as examples first to illustrate the point being made. If we take 1 and divide it into 4 such as the fraction (1/4) we are saying that we want to take a slice out of the whole in 4 equal parts and this gives us a ratio proportion of 0.25 or 25% of the whole. These all mean the same thing. If I was to take it's reciprocal and say take 4 and divide it into 1 such as the improper fraction (4/1) we are saying we want 4 equal parts of the original which in turn gives us 4 similar objects of the original. Let's try to do the same thing this time with 0 and 1. I'll start with 0 in the numerator. Let's divide 0 into 4 such as the fraction (0/4) we are saying that we want 0 parts out of the whole which in turn gives us a value of 0. We can conclude that (0/n) = 0 as long as n does not equal 0. Let's now take 4 and divide it into 0 objects this is saying that we want 4 exact objects but there are no objects to take it from. This could yield two valid results. It could either be 4 as we have 4 equal empty sets or it could be 0 as we have no objects; both interpretations are valid assessments. So we could conclude that (n/0) = n or 0 as long as n does not equal 0. We have one case left and that is when we have (0/0) and with this situation we could have a value of either 0 or 1. When we divide anything by 0 (numerator) we end up with 0 which is true. Also when we divide anything by itself we end up with 1 (identity) which is also true. So when we have the indeterminate form (0/0) the answer can be 0 or 1. A little more complex than this but there are other contexts where 0/0 could also possibly equal +/- infinity but that is beyond this discussion. The main reason we consider it to be undefined because we always assume that operations on operands must act like a function with a single input and a single output but we also know that this doesn't always hold in nature, there are many times you can input a single value and get multiple results and when this happens if fails the one to one rule as it then becomes a one to many rule. And since we do most of our computations on electronic or digital devices such as calculators, computers, etc. These devices have physical limitations and we don't know how to represent single operations that yield multiple values due to the fact that the transistors or switches that makes up the logic gates are defined as two state boolean devices either on or off, high or low, open or closed 1 or 0, etc. Yet the evaluation of 0/n, n/0 and 0/0 is well defined and not undefined, but since it has this complexity unlike numbers the empty set or null set or zero vector, zero point or the unit digit place holder it is easier for us to say it is undefined and not have to deal with its complexity. The same situation arises when dealing with the slopes of lines. We know from algebra that a slope with two points p1 and p2 has a slope of m = ( (y2-y2)/(x2-x1)) which is also dy/dx which is also sin(t)/cos(t) with respect to the the angle from the origin above the horizontal or x axis which is also tan(t). We we have 0 slope we have horizontal lines and this holds true because the numerator part the sin(t) or dy = 0. We we have vertical lines we say the slope is undefined because of "division by 0" and in this case the cos(t) or dx = 0 which is exactly where the vertical asymptotes show up in the tangent function. Now intuitively we say that when the ground is level or flat and there is no change in one's height as you move across the xz plane (y axis vertical) that you have 0 or no slope and this makes sense. However when you have no change in the xz plane but your height is constantly moving up or down in discrete intervals I'd don't like the idea the slope is "undefined" I tend to think of the slope as full slope not partial as (m/n) where m and n are not 0, but as in infinite slope. If you are going up you are approaching + infinity and if you are going down you are approaching - infinity. The reason I state this is because if we closely evaluate the slope of a line when m is in the form of the trig functions sin(t)/cos(t) both functions on their own have a domain that accepts all possible values and their range is between [-1,1] as long as they are standard form (meaning there were no transformations applied to them such as scaling, skewing, or rotations). So when we look at the limit as cos(t) approaches 0 we need to look to see what is happening to sin(t) at the same time, it is approaching either +1 or - 1 which implies +/- infinity just as you can see from the vertical asymptote in the tangent function. This is just pure reasoning and logic and basic computations to prove these assessments. Also you can apply the dot product using these points to find the actual value needed to calculate the angle in which they from each other (0,0),(1,0), (0,1),(-1,0),(0,-1) and these are the 5 crucial points that make up the unit circle. You can take any 2 of the points and use the dot product in terms of the cosine function and you would get values of 0, +/- 90, +/-180, +/-270, +360 depending on which direction you chose your points to be in but of course you would have to apply the arccos to get the actual angle after using the dot product. Just something to think about how complex the empty set truly is. Depending on the operation being applied and the context in which it is being used it can yield 0,1, +/- infinity and sometimes it could yield more than one in a single evaluation depending on which direction you are taking the limit from.
The fourth reason doesn't really count. The reason 0! works in the power series is BECAUSE it is defined to be 1. So that’s circular reasoning. If it wasnt 1, the sum would be expressed as 1 + Sum(n = 1)(infinity) x^n/n!
Also, I’d asume that for the definiton of exp(x) as a power series you could not define exp(0), but you should do instead lim(x->0) exp(x)... and in this case you are not using the convention of setting 0^0=1 For me this is an interesting topic, yet I am an engineer so maybe I dont know at all what I am talking about :)
I'm a big fan of the empty product. And you can use the empty product to explain 0! = 1. For nonnegative integers n, you can say that n! is the product of all positive integers less than or equal to n. For 0!, you are then taking the product of all positive integers less than or equal to 0. But there are no such numbers. Therefore, under this definition, 0! is a product with no factors, i.e., the empty product, which is 1. :)
it's 0!/0 which is +infinity. (-2!) then is (-1)!/(-1) which is +infinity/(-1) = -infinity. (-3!) is (-2)!/(-2) which is (-infinity)/(-2) = +infinity and so forth.
DerToasti not exactly lim(x to 0+) ((x-1)!) Would be lim (x to 0+) (x)!/x = +inf But lim x to 0- gives -inf. If you look at x! For x in (-1,0), (-2,-1), (-3,-2) etc (where (a,b) is the set of numbers between a and b exclusively) You can see that the sign changes every time you move to the next set, so all negative integars factorials are undefined
Recall that the definition of a "factorial" is only defined for a positive integer(including zero). Otherwise, is undefined. -1 is a negative number(integer) thus -1! is undefined.
Another reason, closely related to #2, is a general combination/permutation problem. If the number of ways to choose k elements out of a set of size n is equal to n!/(k!(n-k)!), then when n=k (that is, you choose all the elements of the set, of which there is only one way to do so), it makes sense to say (n-k)!=0!=1.
I really like number 2 because it helps illuminate a purpose driven use case of factorial. I think ultimately what is useful for factorial depends on what you hope to happen when you hit that "hole" in the function. The use case of what you're actually trying to describe matters.
There is a 5th reason. If we define n! for n >= 1 to be the product of the integers 1 ... n, then 0! is the empty product which by default in mathematics is always 1, just like that the empty sum is always 0.
Right now Matt Parker is trying to solve a problem really hard. I don't wanna explain it but it's a probability problem about coins with a mild twist. I'd love to see your take on the problem. I know it's really different from what you normally do, but it'd still be cool to see a video on it.
Andrew Criswell well he stated that 0^0=1, and from there algebra dictates 0!=1. Really it's a limit problem that you could work through, but I think you'd have to substitute in the pi or gamma function for 0!, and then you may as well just state 0!=1.
Thank you for saying "should be." This is an axiom, which I am fine with. It grinds my gears when people say this is true in the sense that it can be proven.
It depends on your definitions, though. If you set up your definitions of multiplication and factorial in a certain way, then 0! = 1 _can_ be proven. It's also the "right" value in the sense that it makes _every_ formula where 0! shows up give the correct value in those situations, and the certain-way-of-setting-up-your-definitions I gave above explains how we know it will _always_ work.
I prefer "5 choose 5" as an example, 5!=(0! 5!) It's a longer formula than permutations of the empty set (0!), but it's a lot more obvious that there is one way to do it without having to philosophize about the empty set to get there.
I still like to present the first one as just your requirements for the Pi and Gamma functions: x! = x(x-1)! 1! = 1 Simply plug in 1 for x, and you get 0! 1! = 1(1-1)! 1 = 1 * 0! 1 = 0!
Yes, but your "contradiction" relies on -1! having a defined and finite value. [0! = 0*(0-1)! = 0*(-1!) which is not zero, but undefined since -1! is undefined in the reals]
No we are saying that -1! is undefined because it is... differently from 0! which is DEFINED to be 1 (and this can be justified since no matter what definition of "!" one uses there is no pole or division by zero involved in the value of 0! or Gamma(1))
Sorry - we will disagree deeply on the statement "We know some fundamental things from life and we do math with these thing." Mathematical objects/concepts have no obligation to have a connection to "life" (or the physical world), and very often they do not. What's the physical meaning of the pi-th derivative of e^i? Yet it has a mathematical meaning and a value. We can write n! = Gamma (n+1). It's the absolute truth for n in N, and if you try to find the value of Gamma for integer negative values of n you will find it's undefined. So Gamma (0) is undefined (in R), as is Gamma (-1), Gamma (-2) and so on. Which is why above I was saying that -1! is undefined because it is: no matter what (commonly used) definition of factorial you try to use, you simply cannot define a value for it that makes sense given other simple mathematical concepts (other "non log-convex" Gamma functions excepted... which aren't necessarily all that simple)
Before anything I would like to say that I love all your videos and the way you explain. Although it is clear that the video is not a proof of why zero factorial is 1 but why we can define it as 1 there are some observations. The reason #1 uses the definition of factorial and this definition finishes at 1. So using the logic of (n-1)! = n! / n loses meaning if n = 1 because the the factorial is defined until 1. This is a logical argument that MAKES SENSE but it not something that is supposed to find shelter on the factorial definition. The reason #2 is almost philosophical because there are those who could say that if you have no objects to be arranged than there is no way to arrange what does not exists while there are those who could state that if there is nothing to be arranged than this nothingness is arranged itself in only one possible way. The reason #3 is elegant, truly beautiful, but it is an extension of factorial and in order to be accepted as extension of factorial it must matches 2 conditions (and it does). - Working with Gamma Function we need to make sure Gamma (1) matches the 1! and it does AND n! = n x (n-1)!. There is a video of yours (beautiful) profing it. So it matches the conditions. - The PI function fits also but with a parameter shifted by 1. Thus it is an extension of factorial and when we set n equals to zero it returns 1. So it is fine but what it says is that it matches a convention established when the factorial function was defined. It is not a proof of zero factorial equals 1. The reason #4 is also elegant but it is based on o power to 0 which is in some situations an indetermination while in other situations it is defined as 1. Once we overcome this it is very beautiful approach. However there is one approach that is a reasoning alternative for approach #2 that it is not explored. The combination of a set of n elements where we pick all the n elements. It means (n k) as a column matrix like notation. It is solved as (n! / [k! (n-k)!]). If n and k are equal we get (n! / [n! (n-n)!]) = (n! / [n! (0)!]) We know that there is only one possible way to arrange a set of elements if we pick all the elements of that set. Therefore (n! / [n! (0)!]) = 1. We cancel the n! and we get 1 / 0! = 1, so we finally get the definition but it is logical however I think is less philosophical and closer to a math proof but I am not confident that we can call it that way.
I look at it intuitively. Suppose we're adding some terms. Before we begin, the sum is zero, the additive identity. Then we add in each term to get our sum. Now suppose we're multiplying some factors. Before we begin, the product is the multiplicative identity, one; then we multiply by each factor. If there are no terms at all, the sum is zero. Likewise, if there are no factors, the product is one. So zero factorial and zero to the zeroth power should both be defined as one.
You can also look at the recursive definition of n!. If you define 0! to be anything else than 1, you won't be able to compute n! recursively in a way that makes sense.
No. If 0 wasn't defined, then we would just start at 1 instead of zero which is just as arbitrary of a place to start. If you wanted you could be explicit and specify it's domain as positive integers.
THIS! Yes thank you for pointing that out. I think everyone is forgetting the basis of factorials is to multiply all natural numbers before a number no 0 included- there is a limit already to what factors there are for positive numbers, though a pattern is made factorials themselves are not the basis of other factorials. 0 is a neutral number and can have it's own domain for its factorial.
I like the reason that if you have a list of consecutive numbers raised to the power n, the nth difference is constant and equal to n! (so 1,4,9,16,25,36,... have differences 3,5,7,9,11,... and second difference 2,2,2,2,... for an annoying example, as 2=2!) When you raise the numbers to the power of 0, you get 1,1,1,1,1,... which has a constant 0th difference (you take the difference no times) and it's equal to 1. Also works as a reason to say x^0 must be 1
I disagree with the 4th reason. 0! can be defined whereas 0^0 is undefined. Therefore, it's more of a reason why we sometimes define 0^0=1. Here's a better reason to define 0!=1. 0! is by definition an empty product and the result of multiplying 0 factors is the multiplicative identity 1, like the empty sum - the result of adding 0 numbers - is the additive identity 0.
archimidis I see what you mean. To really be rigorous, he should have said the limit as x approaches zero from the positive direction. Since that limit on x^x is equal to 1, I think the proof still stands.
"whereas 0^0 is undefined." I mean, that's why he said it's "conventionally accepted," for convenience's sake. He was very careful during part 4 if you'd notice to avoid saying "equals," even when he was saying 0! should be 1, not 0! equals 1.
A^x/A^x = A^x/A^x When dividing variables with powers, subtracting the exponents gives the correct answer. So, subtract the exponents on the left. x-x=0 Now we have, A^0=A^x/A^x Anything over itself is 1 Apply this to A^x/A^x A^0=1 Therefore, any number to the 0th power is 1. This also applies to 0^0, making it 1.
Dear blackpenredpen, Hi! I am a subscriber from Hong Kong, I'd like to ask you three questions. 1. What is i! ? ( a Hindu-Arabic number, cool.) 2. What is i^π^e ? 3. What is log(base i)π^e ? Hope these questions can help you have more theme on the questions about i.
With the third argument you just say "this doesn't matter because it's just 1" (referring to t^0). But on the lower bound it will be 0^0, which is undefined.
If you can cancel something in the integral out before performing the integration, it's a completely valid mathematical step. t^0 is equal to 1, so it can be removed from the expression entirely. Additionally, not removing that t^n from the integral while n=0 would mean requiring that you perform integration by parts.
I prefer the gamma function explanation as the most important thing is that N! Is produced by the gamma function. The first one is good as well as it's very natural and simple to understand.
Of those I like 2 the most. another one I like is by defining the factOREO as a productoria (or however is called), defined from 1 to n, therefore when n=0 we get an empty range which give us an empty product which give us the identity of the operator which just 1, much like with summations with empty sum which give us 0
I love this very much especially since I came over lot of situations where 0! Must be 1 in order to fulfill something that I was calculating. For example Cauchy Integral theorem That states : Cn= f^(n) in function of z0/n! = (1/2pi*i)* integral of f(z)/(z-z0)n+1 dz Which in terms where derivative is " 0 " has n! down which would be 0! And Cauchy placed that without derivative f(z0)=1/(2pi*i) * integral of f(z) / (z-z0) dz (which they also call theorem of average value) Great video as always.
Yes there is a complex factorial (extended) because GAMMA function is meromorphic in the whole complex plane. Beware the negative real part though, you will need the Euler's mirror to compute it (the functional equation) The integral shown only converges for Re(z) >0 where z = a+b*i Also, (-1)! = GAMMA(0) = undefined (a simple pole with residue 1)
Like an empty sum is set to the neutral element of addition (zero), an empty product should be set to the neutral element of multiplication (one). Just define 0! as the product over n from 1 to 0 (hence an as an empty product) …
reason 1: ok yeah i've heard of this before cooI reason 2: IoI thats nice way to put it I wonder what other ways there ar- *REASoN 3:* oh dear god... Reason 4: *dies of a stroke*
I prefer 3, then 1, then 4, and lastly 2. 3 doesn't rely the 0^0 convention, and it doesn't rely on keeping a pattern going, unlike 1. With the second method, I think that you could also conclude that 0! is undefined; it doesn't make sense to talk about arranging non-existent things, at least in classical logic.
I like the simple explanation of #2: it can be explained without any numbers. It also explains why 0! can still have a useful 'real' meaning, but (-1)! doesn't.
The second reason (in my opinion) is wrong since if you wanted to arrange nothing, you wouldn’t arrange it and the ways of arranging it wouldn’t exist meaning 0!=0
I'd like to know why by convention zero to the zero power equals 1, in infinite series. Why other things to the 0 power equals 1 is not as big an issue
I personally like to think of it in the 2nd way, of combination, you add 2 values, alpha & omega, alpha must be the first term, and omega must be the last term, you add 0 terms, you get alpha beta, and there is no other way to combine them, i.e. you get 1 solution, same with 1, you get alpha, 1, omega, no other way to combine then, etc.
The factorial function is defined on the IN U {0} but it has a natural extension by the Gamma function defined by Bernoulli which is defined on IR/Z-. Z- = {-1, -2, -3,.... }. So it is a natural thinking to find the 0! by this way.
AndDiracisHisProphet I'm not sure why stage 4 needed x=0. From the Taylor expansion we get e^x = 1 + x/1 + x²/(1*2) + ... + x^k/k! +... So e^x = 1+ sum(n from 1 to inf) (x^n/n!) But this isn't very neat, we would want 1 in the sum as well. So would want 1 = x^0/0! For all x. Hence we want 1 = 1/0! (for x not 0 for now), so 0! Should be 1. Can then deal with 0^0 afterwards, lim (x->0+) x^0 = 1 and lim (x->0-) x^0 = 1, so in this case our 0^0 = 1.
i admit i´m not into thje last reason. that 0^0 business is kind of hard to swallow. yes by convention it is set to one but it just is an undefined headache in most scenarios. So it is not the best way to make a point. the first one definitely is the best though: plain and simple. no need for fancy maths magic. Just intuitive observation of patterns. However if you say it should be defined as one. are there people who think differently? maybe if there are could you also share their perspective?
Good point. Although a slight tweak to the last reason would have also worked. If we calculate e instead (e^1), then we have as the first term 1^0/0!, so we can avoid the 0^0 ambiguity. So e=1/0! + 1/1! + 1/2! + 1/3! + …. Rearranging, we can get 1/0! = 1/1! + 1/2! + 1/3! + … - e. Taking the reciprocal of both sides, we get 0! = 1/(1/1! + 1/2! + 1/3! + … - e). Just doing the first several terms, we will see that the answer converges to 0! = 1.
An alternative to reason 1) is to notice that factorial by definition follows (n+1)! = (n+1)*n! which can be rewritten as n! = (n+1)! / (n+1). Therefore substituting in n=0 you get that 0! = (0+1)! / (0 +1) = 1! / 1 = 1/1 = 1 . Similarly this property of factorial shows why -1! is undefined, since -1! = 0! / 0 and dividing by zero is undefined. Really though the main moral of the video isn’t so much that we should define 0! to be 1 but rather that we should define factorial to be either the pi function at n or the gamma function at n+1. Redefining factorial using those functions includes defining factorial at many numbers that aren’t included in the classic definition including zero.
I never liked the Gauss' pi function, Euler's GAMMA function is so much better, you can compute the finite harmonic series with it, try 1+1/2+1/3+...+1/65536 = ?? easy the answer is 11.66757818323578.... (less than one minute with a calculator)
It's similar but it's not exactly the same, my explanation above more explicitly uses the formulaic definition of factorial while the video is kind of just hand-waving it.
One would also give the fifth reason: Consider a sum S=Σ_{k=1}^n(a_k) = a_1+a_2+...+a_n. Then for example for n=1 we have of course S=a_1, since we have only one element of this sum. What is S for n=0? We have 0 elements of the sum, so it's natural to put S=0 -- neutral element of addition. For multiplication it's natural to put Π_{k=1}^0 (a_k)=1 - neutral element of the multiplication. So 0! = Π_{k=1}^0 k=1.
@@aleksapetrovic7088 Infinity. Stop disrespecting our creator by calling it "undefined". Infinities gave us our dimensions; we must respect infinities. If we were living in Minecraft, we would call circles "undefined". Since we are living in a world with polar coordinates, the premium package with the spherical bundle, we are accustomed of seeing circles, so they are not "undefined". Also, infinities are everywhere, we cannot move without them, and the Big Bang couldn't happen without them, without them, we would continue to be lumped together in singularity. There are an infinite number of points in a unit line segment alone, and given the fact that infinities are required to extrude something to the next dimension or travel through time, we should divide by 0, spread our wings, learn how to fly, and do the impossible.
Infinity. Stop disrespecting our creator by calling it "undefined". Infinities gave us our dimensions; we must respect infinities. If we were living in Minecraft, we would call circles "undefined". Since we are living in a world with polar coordinates, the premium package with the spherical bundle, we are accustomed of seeing circles, so they are not "undefined". Also, infinities are everywhere, we cannot move without them, and the Big Bang couldn't happen without them, without them, we would continue to be lumped together in singularity. There are an infinite number of points in a unit line segment alone, and given the fact that infinities are required to extrude something to the next dimension or travel through time, we should divide by 0, spread our wings, learn how to fly, and do the impossible.
The second one always bugged me. I disagree that there's one way to arrange nothing, there's no way to arrange nothing because that doesn't make any sense. I would say 0! is undefined, it's meaningless, like dividing by 0. Also, the third one doesn't make much sense to me, where you said 1/infinity is 0, I don't think that's true. If 1/100 is a hundredth, than 1/infinity is an infinitesimal. If you take something and split it into infinite groups, those groups aren't empty, otherwise they wouldn't be groups at all.
You dont know about the concept of limits if you say for example 1/infinity its like saying 0,000....(many zeros)001 then you can say its technically 0 because even in real life if you cut 1 apple for example this many times you wouldnt be able to see anything remaining.
Not sure I'm satisfied with the "by convention" part on the 4th. Yeah, I can deduce the reasoning, but I also have to convince students who've never seen it before. The 3rd one is particularly solid for new calc students, so I'll bare that in mind. Also, will have to practice the two-pen thing. I use really thin markers, and put the lids of three of them between the knuckles of my left hand to draw free whichever market I need.
4th reason should really be: e^x=1+sum(n from 1 to inf) xⁿ/n! . To make this look nicer, we want to include the 1 within the sum. So we want x^0/0! = 1, which leads to 0! = 1. You can only say e^x=sum(n from 0 to inf) xⁿ/n! After you have defined 0! = 1.
My favorite definition is the #3 through GAMMA function, 0! = GAMMA(1) = 1, I have the proof by Weierstrass representation (starting with Gauss) that 0! =1
Similar to dividing by 0, you cannot factor a system containing nothing into any discrete sets There is only 1 way to factor a 3! system which has 6 discrete sets
The fourth argument is circular because the formula for Exp(x) can be write this way because we already admit that 0! = 1. Anyway, i like very much your videos and also your juvenile enthusiasm. I am over 60 by the way.
Well (i)! = Π(i) = int (x^i e^(-x) ) = int (e^(i*log(x) -x) = int (cos(log(x)) e^-x) + i* int (sin(log(x)) e^-x) ≈ 0.50 - 0.15i Pretty horrible. Although (i)! !!!! Is pretty bad too... But repeated factorials slowly brings the answer towards 1+0i. Similarly how repeated factorials of a number between 0 and 1 approaches the fixed point 1.
The Gamma function, and the Pi function, can be used as analytic continuations to the factorial for ALL complex numbers (excluding negative integers). Pi(t) = int(x from 0 to inf) (x^t e^-x) dx which exists for complex numbers. So we can say that i! Exists as does i! ! ! ! ! And is approximately 0.9923-0.0003i Assuming the !!!! Can be assumed to be 4 factorials instead of 2 double factorials. (n!!= n(n-2)(n-4)... (3.5±0.5)(1.5±0.5) depending if n is even or odd... It can be analytically continuated to the complex numbers using x!!=(Pi(x/2)*2^((x+1)/2)) / pi^0.5 (((i!)!!)!!) ≈ 0.99+0.01i )
That's exactly what he did for the first one. You just generalized it, but it's not necessary because he's looking to find one value. Also, he clearly stated that the examples aren't proof, which they are not.
Fascinating, once again. I'm not comfortable with the last evaluation, but I reached my level of incompetence with undergraduate Maths, hence do not over-estimate my competence.
By the logic of the first example, would that make (-1)! Undefined since it would be 1 divided by 0? Is this the logic for why we don't do negative factorials?
Ethanol 314 you can take the derivative or integral of the gamma function, and the Pi function. n! Is not a function to take a derivative or integral of
Yes you can if you use a function, like GAMMA(n), the log derivative is called the digamma function and you can find ALL the minimum and maximum of the GAMMA function.Really powerful function, because you can compute partial harmonic series, Here is an example 1+1/2+1/3+...+1/65536 = ?? well easy with digamma(65537)+ gamma (Euler's number = 0.577215664901)
You know, regarding the 2nd method, I actually came up with a mnemonic for when I was studying for the first actuarial exam, and that's a quote from the Friends episode "The One Where Nobody's Ready", when Rachel very passive-aggressively declared she wasn't going to go with Ross to the museum to hear him give his speech at the benefit, specifically when she told him, "I'm not gonna gooo, so, I think that will accomplish the not going". LOL In short, there's only ONE way to not do something, and that is......TO NOT DO IT!
Same with 0^0. Why bother to leave things undefined? It's just saying, "Screw this challenge, I will not face my fears", but that is against the mission of humanity of breaking away.
Scream ZERO loud enough and it'll turn into a one...
No need to bring politics into this : - )
marbanak what do you mean? It’s because you’re saying it with excitement, so you use explanation points; making 0! = 1. Screaming 0 loud enough makes 1.
@@i_am_anxious02 : I suspect, at the time, I was thinking of character-assassination efforts at the federal level. It was quite a drumbeat. Cheers!
marbanak that’s... weird, I guess. Cheers!
@@marbanak are you *actually* implying that zeroes can't be equal to one? that is literally numbers-ist. reported for numerical supremacy and hate speech
Honestly my favorite out of these is reason #2 because of how philosophical it is. How many ways can you arrange nothing? Exactly one way: no arrangement at all. The complete absence of an arrangement is itself a valid arrangement when you have nothing to arrange. :D
Calyo Delphi Yeah, but you could add that nothing to all positive factorials, in which case they should be their current value + 1: 2! should then be 2 + 1 = 3, for example. Yet no one uses that fact to conclude that, but it is not logically sound in that case.
J.J The Great: I don't think so: You cannot NOT arrange any number of items when you have one or more items. As long as the items are present, they are arranged. So the additional +1 is NOT an option for n! as long as n>=1. But when you have NO items, then the option NOT to arrange them is the ONLY arrangement. It's an option that appears ONLY when there are zero items.
You can metaphisically arge that in absence of anything to arrange there is no arrangement possible. And that is a totally valid approach also.
I disagree, having no object to arrange means you have no arrangement of anything whatsoever. So it should be 0 !!
just like null set is a part of a super set.
Obviously 0!=1
(Get it programmers)
no programers
Lmfao oh god.
That's.. that is so underrated.
Are you looking for null?
and 0!=e
0! is infinite recursion
aka
undefined just like all negative values
In reason 4, what I really like is that, if you use one of the other reasons to accept 0!=1, than, you can turn the other way around to prove the «convention» 0^0=1.
The fact that last no becomes 0\infinity
@@lolme2646... This guy keeps assuming open 0 in the limit as x ---> 0 for y = x^x being y @(0, delta) = 1 for very small delta. His #4 solution shows this. But L'Hopital's Rule showed f(x) = 1 as x approaches 0 at [0, delta)! y= x^x has a limit of 1 at x=0 not only for series expansion reasons of defining e^x = a series limit but for other mathematics where 0^0 appearance problems.
zero is quite a mysterious number
did you just assume zero is a number!?
@@Blox117 lol
1 - natural (nature)
0 - status (philosophy)
-1 - man-made (science)
Zero is not a number; it is the empty set and acts as a placeholder nothing more. Numbers that are greater than 0 are numbers. All of those numbers with a negative sign in front are the same exact number with the same exact magnitude just that they span out in the opposite direction as they are pointing 180 degrees or PI radians in the other direction, vector computation proves this. Also the negatives are reflected about the perpendicular bisector that is located at zero on the number line or x axis and the point at zero acts as the point of rotation and the point of symmetry. Zero has no quantitative value therefor it is not a number! Yes operations can be performed on it just as any other number, but the outcomes can vary from one operation to another that doesn't hold to the operations of all other numbers either if they are the set of all real numbers or the set of all imaginary and complex numbers.
For example we were all taught that you can not divide by 0 as it is undefined. Let's not look at this as a numerical expression but as a conceptual expression instead. I'll use some regular fractions as examples first to illustrate the point being made. If we take 1 and divide it into 4 such as the fraction (1/4) we are saying that we want to take a slice out of the whole in 4 equal parts and this gives us a ratio proportion of 0.25 or 25% of the whole. These all mean the same thing. If I was to take it's reciprocal and say take 4 and divide it into 1 such as the improper fraction (4/1) we are saying we want 4 equal parts of the original which in turn gives us 4 similar objects of the original.
Let's try to do the same thing this time with 0 and 1. I'll start with 0 in the numerator. Let's divide 0 into 4 such as the fraction (0/4) we are saying that we want 0 parts out of the whole which in turn gives us a value of 0. We can conclude that (0/n) = 0 as long as n does not equal 0. Let's now take 4 and divide it into 0 objects this is saying that we want 4 exact objects but there are no objects to take it from. This could yield two valid results. It could either be 4 as we have 4 equal empty sets or it could be 0 as we have no objects; both interpretations are valid assessments. So we could conclude that (n/0) = n or 0 as long as n does not equal 0. We have one case left and that is when we have (0/0) and with this situation we could have a value of either 0 or 1. When we divide anything by 0 (numerator) we end up with 0 which is true. Also when we divide anything by itself we end up with 1 (identity) which is also true. So when we have the indeterminate form (0/0) the answer can be 0 or 1. A little more complex than this but there are other contexts where 0/0 could also possibly equal +/- infinity but that is beyond this discussion.
The main reason we consider it to be undefined because we always assume that operations on operands must act like a function with a single input and a single output but we also know that this doesn't always hold in nature, there are many times you can input a single value and get multiple results and when this happens if fails the one to one rule as it then becomes a one to many rule. And since we do most of our computations on electronic or digital devices such as calculators, computers, etc. These devices have physical limitations and we don't know how to represent single operations that yield multiple values due to the fact that the transistors or switches that makes up the logic gates are defined as two state boolean devices either on or off, high or low, open or closed 1 or 0, etc.
Yet the evaluation of 0/n, n/0 and 0/0 is well defined and not undefined, but since it has this complexity unlike numbers the empty set or null set or zero vector, zero point or the unit digit place holder it is easier for us to say it is undefined and not have to deal with its complexity. The same situation arises when dealing with the slopes of lines. We know from algebra that a slope with two points p1 and p2 has a slope of m = ( (y2-y2)/(x2-x1)) which is also dy/dx which is also sin(t)/cos(t) with respect to the the angle from the origin above the horizontal or x axis which is also tan(t). We we have 0 slope we have horizontal lines and this holds true because the numerator part the sin(t) or dy = 0. We we have vertical lines we say the slope is undefined because of "division by 0" and in this case the cos(t) or dx = 0 which is exactly where the vertical asymptotes show up in the tangent function.
Now intuitively we say that when the ground is level or flat and there is no change in one's height as you move across the xz plane (y axis vertical) that you have 0 or no slope and this makes sense. However when you have no change in the xz plane but your height is constantly moving up or down in discrete intervals I'd don't like the idea the slope is "undefined" I tend to think of the slope as full slope not partial as (m/n) where m and n are not 0, but as in infinite slope. If you are going up you are approaching + infinity and if you are going down you are approaching - infinity. The reason I state this is because if we closely evaluate the slope of a line when m is in the form of the trig functions sin(t)/cos(t) both functions on their own have a domain that accepts all possible values and their range is between [-1,1] as long as they are standard form (meaning there were no transformations applied to them such as scaling, skewing, or rotations). So when we look at the limit as cos(t) approaches 0 we need to look to see what is happening to sin(t) at the same time, it is approaching either +1 or - 1 which implies +/- infinity just as you can see from the vertical asymptote in the tangent function.
This is just pure reasoning and logic and basic computations to prove these assessments. Also you can apply the dot product using these points to find the actual value needed to calculate the angle in which they from each other (0,0),(1,0), (0,1),(-1,0),(0,-1) and these are the 5 crucial points that make up the unit circle. You can take any 2 of the points and use the dot product in terms of the cosine function and you would get values of 0, +/- 90, +/-180, +/-270, +360 depending on which direction you chose your points to be in but of course you would have to apply the arccos to get the actual angle after using the dot product.
Just something to think about how complex the empty set truly is. Depending on the operation being applied and the context in which it is being used it can yield 0,1, +/- infinity and sometimes it could yield more than one in a single evaluation depending on which direction you are taking the limit from.
Cause Indians are mysterious 😅
The fourth reason doesn't really count. The reason 0! works in the power series is BECAUSE it is defined to be 1. So that’s circular reasoning. If it wasnt 1, the sum would be expressed as
1 + Sum(n = 1)(infinity) x^n/n!
Samuel Pierce
yeap, I agree with you and that' s obviously true because the fourth reason doesn't count at all
This video isn't a proof that 0! = 1, it's reasons why we should and do define it that way.
It wouldn't work. Not with every x, at least. When x=0 every term after the 0^0/0! is equal to 0, so there's no way the sum would converge to 1.
Also, I’d asume that for the definiton of exp(x) as a power series you could not define exp(0), but you should do instead lim(x->0) exp(x)... and in this case you are not using the convention of setting 0^0=1
For me this is an interesting topic, yet I am an engineer so maybe I dont know at all what I am talking about :)
I mean output values can go against convergence... you divide by 0... in this case it would mean 0!=0
As an IT guy I see custom thumbnail 0!=1 that reads "zero is not equal to one"
Yaroslaw Kaminsky I didn’t because I always put spaces around most operators
Lol i guess this means we should always use spaces in operations
Yaroslaw Kaminsky ₹ 9
I'm in 8th grade and learning C++. Our teacher has recommended us to put spaces between operators to recognize the differences between operators.
I was thinking that the WHOLE time!!!
I'm a big fan of the empty product. And you can use the empty product to explain 0! = 1.
For nonnegative integers n, you can say that n! is the product of all positive integers less than or equal to n.
For 0!, you are then taking the product of all positive integers less than or equal to 0. But there are no such numbers. Therefore, under this definition, 0! is a product with no factors, i.e., the empty product, which is 1. :)
Well the empty product is the same as 0^0 which is 1.
@@findystonerush9339 0^0 is undefined, but the limit of x^x as x -> 0 is 1. Do not confuse the two.
Although if you use this explanation, (-π)! is also 1, because there are no positive integers less than -π
@@talhochberg5062 "for nonnegative integers n"
@@anon.9303 That's a definition. If you're using that explanation, you can also define the factorial to work on all real numbers (or complex?)
So -1! Must be 1/0 which is undefined?
right. he has a video about it.
it's 0!/0 which is +infinity. (-2!) then is (-1)!/(-1) which is +infinity/(-1) = -infinity. (-3!) is (-2)!/(-2) which is (-infinity)/(-2) = +infinity and so forth.
DerToasti not exactly lim(x to 0+) ((x-1)!) Would be lim (x to 0+) (x)!/x = +inf
But lim x to 0- gives -inf.
If you look at x! For x in (-1,0), (-2,-1), (-3,-2) etc (where (a,b) is the set of numbers between a and b exclusively)
You can see that the sign changes every time you move to the next set, so all negative integars factorials are undefined
Recall that the definition of a "factorial" is only defined for a positive integer(including zero). Otherwise, is undefined. -1 is a negative number(integer) thus -1! is undefined.
No, it's (-1)! that's equal to 1/0, -1! would be -1
Since exponentials and factorials are constructed by multiplication, it makes sense that their foundation would be the multiplicative identity (1).
I really like the reason where you used the pi function, keep up the good work, your videos really help
Another reason, closely related to #2, is a general combination/permutation problem. If the number of ways to choose k elements out of a set of size n is equal to n!/(k!(n-k)!), then when n=k (that is, you choose all the elements of the set, of which there is only one way to do so), it makes sense to say (n-k)!=0!=1.
Yes and that bypasses some peoples' philosophical unease with the question of how many ways there are to arrange "nothing".
B. Xoit "0" way
I didn't understand it so you are probably wromg
@@b43xoit Illogic
@@AlgentAlbrahimi One way, the empty list.
Using the empty product definition concludes both 0⁰ = 1 and 0! = 1.
2nd one is the most obvious way to make understand a total beginner.
I really like number 2 because it helps illuminate a purpose driven use case of factorial. I think ultimately what is useful for factorial depends on what you hope to happen when you hit that "hole" in the function. The use case of what you're actually trying to describe matters.
There is a 5th reason. If we define n! for n >= 1 to be the product of the integers 1 ... n, then 0! is the empty product which by default in mathematics is always 1, just like that the empty sum is always 0.
I like the #4 method the most, because in addition to defining 0! = 1, 0^0 must also be 1 to satisfy that e^0 equals 1
man, your channel is gold - you explain all the stuff I had been wondering about, but never had competence to obtain a valuable answer - thanks!
8:10 mathematicians are connected to the speedforce
Makes sense, anybody that has extensive knowledge and understanding of mathematics and its mysteries can move almost as fast as the speed if light.
Right now Matt Parker is trying to solve a problem really hard. I don't wanna explain it but it's a probability problem about coins with a mild twist. I'd love to see your take on the problem. I know it's really different from what you normally do, but it'd still be cool to see a video on it.
I agree.
I'm curious about how he'll do that "puzzle".
semi awesomatic The Parker Coin
Did he try using a square? 😃
No, he's using a cylinder.
In explanation #4, you substituted one convention, 0^0 = 1, for another convention, 0! = 1.
Andrew Criswell well he stated that 0^0=1, and from there algebra dictates 0!=1. Really it's a limit problem that you could work through, but I think you'd have to substitute in the pi or gamma function for 0!, and then you may as well just state 0!=1.
@@semiawesomatic6064 I thought 0^0 is undefined, because you get contradictory outcomes depending on how you approach it.
@@carultch Same with 0! as well.
Thank you for saying "should be." This is an axiom, which I am fine with. It grinds my gears when people say this is true in the sense that it can be proven.
It depends on your definitions, though. If you set up your definitions of multiplication and factorial in a certain way, then 0! = 1 _can_ be proven. It's also the "right" value in the sense that it makes _every_ formula where 0! shows up give the correct value in those situations, and the certain-way-of-setting-up-your-definitions I gave above explains how we know it will _always_ work.
I prefer "5 choose 5" as an example, 5!=(0! 5!)
It's a longer formula than permutations of the empty set (0!), but it's a lot more obvious that there is one way to do it without having to philosophize about the empty set to get there.
Love this channel. I think the 2nd explanation makes the most sense to me, shows 0!=1 by the meaning of factorial.
I still like to present the first one as just your requirements for the Pi and Gamma functions:
x! = x(x-1)!
1! = 1
Simply plug in 1 for x, and you get 0!
1! = 1(1-1)!
1 = 1 * 0!
1 = 0!
You can't put in 0 for x because (-1)! is undefined. You *can* substitute 1 for x, though, which is what ZipplyZane did.
Yes, but your "contradiction" relies on -1! having a defined and finite value.
[0! = 0*(0-1)!
= 0*(-1!) which is not zero, but undefined since -1! is undefined in the reals]
+Denis Fluttershy There's a reason why the domain of the factorial is the natural numbers... and that of the Gamma function is the complex numbers.
No we are saying that -1! is undefined because it is... differently from 0! which is DEFINED to be 1 (and this can be justified since no matter what definition of "!" one uses there is no pole or division by zero involved in the value of 0! or Gamma(1))
Sorry - we will disagree deeply on the statement "We know some fundamental things from life and we do math with these thing." Mathematical objects/concepts have no obligation to have a connection to "life" (or the physical world), and very often they do not. What's the physical meaning of the pi-th derivative of e^i? Yet it has a mathematical meaning and a value.
We can write n! = Gamma (n+1). It's the absolute truth for n in N, and if you try to find the value of Gamma for integer negative values of n you will find it's undefined. So Gamma (0) is undefined (in R), as is Gamma (-1), Gamma (-2) and so on. Which is why above I was saying that -1! is undefined because it is: no matter what (commonly used) definition of factorial you try to use, you simply cannot define a value for it that makes sense given other simple mathematical concepts (other "non log-convex" Gamma functions excepted... which aren't necessarily all that simple)
reson 5: m! : n! = (m -n)! with m =n then: m! = n! => m! : n! = 1 = (m -n)! = 0! -> 0! =1
Before anything I would like to say that I love all your videos and the way you explain. Although it is clear that the video is not a proof of why zero factorial is 1 but why we can define it as 1 there are some observations.
The reason #1 uses the definition of factorial and this definition finishes at 1. So using the logic of (n-1)! = n! / n loses meaning if n = 1 because the the factorial is defined until 1. This is a logical argument that MAKES SENSE but it not something that is supposed to find shelter on the factorial definition.
The reason #2 is almost philosophical because there are those who could say that if you have no objects to be arranged than there is no way to arrange what does not exists while there are those who could state that if there is nothing to be arranged than this nothingness is arranged itself in only one possible way.
The reason #3 is elegant, truly beautiful, but it is an extension of factorial and in order to be accepted as extension of factorial it must matches 2 conditions (and it does).
- Working with Gamma Function we need to make sure Gamma (1) matches the 1! and it does AND n! = n x (n-1)!. There is a video of yours (beautiful) profing it. So it matches the conditions.
- The PI function fits also but with a parameter shifted by 1.
Thus it is an extension of factorial and when we set n equals to zero it returns 1. So it is fine but what it says is that it matches a convention established when the factorial function was defined. It is not a proof of zero factorial equals 1.
The reason #4 is also elegant but it is based on o power to 0 which is in some situations an indetermination while in other situations it is defined as 1. Once we overcome this it is very beautiful approach.
However there is one approach that is a reasoning alternative for approach #2 that it is not explored. The combination of a set of n elements where we pick all the n elements. It means (n k) as a column matrix like notation. It is solved as (n! / [k! (n-k)!]). If n and k are equal we get
(n! / [n! (n-n)!]) = (n! / [n! (0)!])
We know that there is only one possible way to arrange a set of elements if we pick all the elements of that set.
Therefore (n! / [n! (0)!]) = 1. We cancel the n! and we get 1 / 0! = 1, so we finally get the definition but it is logical however I think is less philosophical and closer to a math proof but I am not confident that we can call it that way.
I look at it intuitively. Suppose we're adding some terms. Before we begin, the sum is zero, the additive identity. Then we add in each term to get our sum. Now suppose we're multiplying some factors. Before we begin, the product is the multiplicative identity, one; then we multiply by each factor. If there are no terms at all, the sum is zero. Likewise, if there are no factors, the product is one. So zero factorial and zero to the zeroth power should both be defined as one.
You can also look at the recursive definition of n!. If you define 0! to be anything else than 1, you won't be able to compute n! recursively in a way that makes sense.
Janox81 Totally agree. If 0! Isn't 1 then neither is 1!
Essentially that's reason 1. Showing n!=n*(n-1)!
No. If 0 wasn't defined, then we would just start at 1 instead of zero which is just as arbitrary of a place to start. If you wanted you could be explicit and specify it's domain as positive integers.
Bravo, nice short correct explanation
THIS! Yes thank you for pointing that out. I think everyone is forgetting the basis of factorials is to multiply all natural numbers before a number no 0 included- there is a limit already to what factors there are for positive numbers, though a pattern is made factorials themselves are not the basis of other factorials. 0 is a neutral number and can have it's own domain for its factorial.
I like the reason that if you have a list of consecutive numbers raised to the power n, the nth difference is constant and equal to n!
(so 1,4,9,16,25,36,... have differences 3,5,7,9,11,... and second difference 2,2,2,2,... for an annoying example, as 2=2!)
When you raise the numbers to the power of 0, you get 1,1,1,1,1,... which has a constant 0th difference (you take the difference no times) and it's equal to 1. Also works as a reason to say x^0 must be 1
I disagree with the 4th reason. 0! can be defined whereas 0^0 is undefined. Therefore, it's more of a reason why we sometimes define 0^0=1. Here's a better reason to define 0!=1. 0! is by definition an empty product and the result of multiplying 0 factors is the multiplicative identity 1, like the empty sum - the result of adding 0 numbers - is the additive identity 0.
archimidis I see what you mean. To really be rigorous, he should have said the limit as x approaches zero from the positive direction. Since that limit on x^x is equal to 1, I think the proof still stands.
"whereas 0^0 is undefined." I mean, that's why he said it's "conventionally accepted," for convenience's sake. He was very careful during part 4 if you'd notice to avoid saying "equals," even when he was saying 0! should be 1, not 0! equals 1.
archimidis
heap I also argree with you
A^x/A^x = A^x/A^x
When dividing variables with powers, subtracting the exponents gives the correct answer.
So, subtract the exponents on the left.
x-x=0
Now we have,
A^0=A^x/A^x
Anything over itself is 1
Apply this to A^x/A^x
A^0=1
Therefore, any number to the 0th power is 1. This also applies to 0^0, making it 1.
I was thinking the same (almost) about the fourth suggestion. I thought 0^0 was indeterminate number, not necessarily undefined , nice video though!
Dear blackpenredpen,
Hi! I am a subscriber from Hong Kong, I'd like to ask you three questions.
1. What is i! ? ( a Hindu-Arabic number, cool.)
2. What is i^π^e ?
3. What is log(base i)π^e ?
Hope these questions can help you have more theme on the questions about i.
With the third argument you just say "this doesn't matter because it's just 1" (referring to t^0). But on the lower bound it will be 0^0, which is undefined.
WarpRulez
👫😂😂😂😂 that's true
Everything that has an exponent of zero will always be one.
If you can cancel something in the integral out before performing the integration, it's a completely valid mathematical step. t^0 is equal to 1, so it can be removed from the expression entirely.
Additionally, not removing that t^n from the integral while n=0 would mean requiring that you perform integration by parts.
That's why some people think 0! is undefined as well.
We can also say that there exists one and only one function f, that maps the empty set to an arbitrary set Y, that's why 0!=1
I prefer the gamma function explanation as the most important thing is that N! Is produced by the gamma function. The first one is good as well as it's very natural and simple to understand.
Of those I like 2 the most.
another one I like is by defining the factOREO as a productoria (or however is called), defined from 1 to n, therefore when n=0 we get an empty range which give us an empty product which give us the identity of the operator which just 1, much like with summations with empty sum which give us 0
well understood but pls I think the first reason makes it very vivid that 0! is 1 at the expence of others.
: )
I love this very much especially since I came over lot of situations where 0! Must be 1 in order to fulfill something that I was calculating.
For example Cauchy Integral theorem That states : Cn= f^(n) in function of z0/n! = (1/2pi*i)* integral of f(z)/(z-z0)n+1 dz
Which in terms where derivative is " 0 " has n! down which would be 0! And Cauchy placed that without derivative f(z0)=1/(2pi*i) * integral of f(z) / (z-z0) dz (which they also call theorem of average value)
Great video as always.
You ought to try the Digamma function.
2:55 and you want to try (-1)!, you have to calculate: 1/0 and that's impossible. So (-1)! is impossible :)
Wolfgang Wilhelm yes. And I have another video just on negative factoreo
Could there also be a factorial of complex numbers?
PS: thx for all the vids - they are very interesting :)
Wolfgang Wilhelm yes if you extend the factorial again. I may work that out in the future
Yes there is a complex factorial (extended) because GAMMA function is meromorphic in the whole complex plane. Beware the negative real part though, you will need the Euler's mirror to compute it (the functional equation) The integral shown only converges for Re(z) >0 where z = a+b*i
Also, (-1)! = GAMMA(0) = undefined (a simple pole with residue 1)
Btw, if we had xe^x we could use something called the Lambert W function on both sides, also called the product log.
According to your 1st explanation, negative one's factorial is undefined cause division of 0! by 0 is undefined...
Rahul Majumder
You are right. I actually have another video on negative factorials
blackpenredpen yeah I watched it, by Pi function...
Yes.
(-1)! = infinity.
Like an empty sum is set to the neutral element of addition (zero), an empty product should be set to the neutral element of multiplication (one). Just define 0! as the product over n from 1 to 0 (hence an as an empty product) …
reason 1: ok yeah i've heard of this before cooI
reason 2: IoI thats nice way to put it I wonder what other ways there ar-
*REASoN 3:* oh dear god...
Reason 4: *dies of a stroke*
reason1: ok
reason2: ya I know this
reason3: WTF
陈明天 not funny the original comment was likes 10 times more funnier!?!?.
@@陈明年 thanks for saying the exact same joke
@@theflaminglionhotlionfox2140 erm I m not talking joke...? I think that is my condition of understanding for 4 proofs.
I prefer 3, then 1, then 4, and lastly 2. 3 doesn't rely the 0^0 convention, and it doesn't rely on keeping a pattern going, unlike 1. With the second method, I think that you could also conclude that 0! is undefined; it doesn't make sense to talk about arranging non-existent things, at least in classical logic.
reason 1 not as clear as it could be. define facorial recursively :- (n+1)! =(n+1)*n! so n! = (n+1)!/(n+1) and if n=0 we have 0!=1!/1=1
I like pi(x) it is a straightforward calculation that has a definite answer and doesn't assume too much
Programmer here.
My first look at the thumbnail was 0 != 1 (0 not equal to 1) :P
: )
I like the simple explanation of #2: it can be explained without any numbers. It also explains why 0! can still have a useful 'real' meaning, but (-1)! doesn't.
As math is all about patterns I prefer explanation #1. So 0! == 1
Bro your way of teaching is just best, your smiling face made me enjoy learning with you.
Always keep smiling in such way😊
Please make videos of the integration of greatest integer functions(GIF) like floor function and ceiling function
The second reason (in my opinion) is wrong since if you wanted to arrange nothing, you wouldn’t arrange it and the ways of arranging it wouldn’t exist meaning 0!=0
Awesome video!!!!!!! Thank you so much!
Your second reasoning was the most intuitive. However, from a mathematical standpoint, your fourth reasoning was the best in my opinion.
user 1 yay. Thank you!
I'd like to know why by convention zero to the zero power equals 1, in infinite series.
Why other things to the 0 power equals 1 is not as big an issue
So that we can put all the terms into one sigma notation.
ruclips.net/video/rJil85GHEyc/видео.html
I personally like to think of it in the 2nd way, of combination, you add 2 values, alpha & omega, alpha must be the first term, and omega must be the last term, you add 0 terms, you get alpha beta, and there is no other way to combine them, i.e. you get 1 solution, same with 1, you get alpha, 1, omega, no other way to combine then, etc.
Isn't 4 th reason not circular reasoning
The factorial function is defined on the IN U {0} but it has a natural extension by the Gamma function defined by Bernoulli which is defined on IR/Z-.
Z- = {-1, -2, -3,.... }. So it is a natural thinking to find the 0! by this way.
Maybe it's not that both 0^0 and 0! are 1 but only 0^0/0! ? :D
AndDiracisHisProphet I'm not sure why stage 4 needed x=0. From the Taylor expansion we get
e^x = 1 + x/1 + x²/(1*2) + ... + x^k/k! +...
So e^x = 1+ sum(n from 1 to inf) (x^n/n!)
But this isn't very neat, we would want 1 in the sum as well. So would want 1 = x^0/0! For all x.
Hence we want 1 = 1/0! (for x not 0 for now), so 0! Should be 1.
Can then deal with 0^0 afterwards, lim (x->0+) x^0 = 1 and lim (x->0-) x^0 = 1, so in this case our 0^0 = 1.
Thank you, Anti-joke-chicken
AndDiracisHisProphet Sorry :(
I was thinking the same thing
@@wolffang21burgers Thanks. You cleared that up for me. I was wondering why 0^0 = 1 by convention?
I like the last one most, because it can not only explain 0^0 = 1 when you know 0! = 1, but explain 0! = 1
Same here!!
Btw, I do not know how to say power series in Chinese.
Google tells me it's "
電源系列" and obviously it's not correct...
blackpenredpen
Power Series had better to be translated as “冪級數”
級數 means the summation of an array
冪 means power, such as 2^2 = 2的2冪
idk why his face at 5:11 made me laugh. he became so serious.
😀😀😀😀😀😀😀😀😀😀😀😀 😕
Subfactorial of 0 is one: !0=1
i admit i´m not into thje last reason. that 0^0 business is kind of hard to swallow. yes by convention it is set to one but it just is an undefined headache in most scenarios. So it is not the best way to make a point. the first one definitely is the best though: plain and simple. no need for fancy maths magic. Just intuitive observation of patterns. However if you say it should be defined as one. are there people who think differently? maybe if there are could you also share their perspective?
Good point. Although a slight tweak to the last reason would have also worked. If we calculate e instead (e^1), then we have as the first term 1^0/0!, so we can avoid the 0^0 ambiguity. So e=1/0! + 1/1! + 1/2! + 1/3! + …. Rearranging, we can get 1/0! = 1/1! + 1/2! + 1/3! + … - e. Taking the reciprocal of both sides, we get 0! = 1/(1/1! + 1/2! + 1/3! + … - e). Just doing the first several terms, we will see that the answer converges to 0! = 1.
My favourite reasoning is by the Pi function, as we know the function must have a recursive nature (as seen in one of Matt Parker's videos).
An alternative to reason 1) is to notice that factorial by definition follows (n+1)! = (n+1)*n! which can be rewritten as n! = (n+1)! / (n+1). Therefore substituting in n=0 you get that 0! = (0+1)! / (0 +1) = 1! / 1 = 1/1 = 1 . Similarly this property of factorial shows why -1! is undefined, since -1! = 0! / 0 and dividing by zero is undefined.
Really though the main moral of the video isn’t so much that we should define 0! to be 1 but rather that we should define factorial to be either the pi function at n or the gamma function at n+1. Redefining factorial using those functions includes defining factorial at many numbers that aren’t included in the classic definition including zero.
I never liked the Gauss' pi function, Euler's GAMMA function is so much better, you can compute the finite harmonic series with it, try 1+1/2+1/3+...+1/65536 = ?? easy the answer is 11.66757818323578.... (less than one minute with a calculator)
Doug Rosengard This is not an alternative, it's exactly the first way
It's similar but it's not exactly the same, my explanation above more explicitly uses the formulaic definition of factorial while the video is kind of just hand-waving it.
One would also give the fifth reason: Consider a sum S=Σ_{k=1}^n(a_k) = a_1+a_2+...+a_n. Then for example for n=1 we have of course S=a_1, since we have only one element of this sum. What is S for n=0? We have 0 elements of the sum, so it's natural to put S=0 -- neutral element of addition. For multiplication it's natural to put Π_{k=1}^0 (a_k)=1 - neutral element of the multiplication. So 0! = Π_{k=1}^0 k=1.
Why is -1! not undefined?
It is undefined
@@aleksapetrovic7088 Infinity. Stop disrespecting our creator by calling it "undefined". Infinities gave us our dimensions; we must respect infinities. If we were living in Minecraft, we would call circles "undefined". Since we are living in a world with polar coordinates, the premium package with the spherical bundle, we are accustomed of seeing circles, so they are not "undefined". Also, infinities are everywhere, we cannot move without them, and the Big Bang couldn't happen without them, without them, we would continue to be lumped together in singularity. There are an infinite number of points in a unit line segment alone, and given the fact that infinities are required to extrude something to the next dimension or travel through time, we should divide by 0, spread our wings, learn how to fly, and do the impossible.
Infinity. Stop disrespecting our creator by calling it "undefined". Infinities gave us our dimensions; we must respect infinities. If we were living in Minecraft, we would call circles "undefined". Since we are living in a world with polar coordinates, the premium package with the spherical bundle, we are accustomed of seeing circles, so they are not "undefined". Also, infinities are everywhere, we cannot move without them, and the Big Bang couldn't happen without them, without them, we would continue to be lumped together in singularity. There are an infinite number of points in a unit line segment alone, and given the fact that infinities are required to extrude something to the next dimension or travel through time, we should divide by 0, spread our wings, learn how to fly, and do the impossible.
All teachers should be as passionate as you!
I'm not convinced in the second reason. 😅 btw, thanks for explanation. :)a
Very well explained.
The second one always bugged me.
I disagree that there's one way to arrange nothing, there's no way to arrange nothing because that doesn't make any sense.
I would say 0! is undefined, it's meaningless, like dividing by 0.
Also, the third one doesn't make much sense to me, where you said 1/infinity is 0, I don't think that's true. If 1/100 is a hundredth, than 1/infinity is an infinitesimal. If you take something and split it into infinite groups, those groups aren't empty, otherwise they wouldn't be groups at all.
You dont know about the concept of limits if you say for example 1/infinity its like saying 0,000....(many zeros)001 then you can say its technically 0 because even in real life if you cut 1 apple for example this many times you wouldnt be able to see anything remaining.
Not sure I'm satisfied with the "by convention" part on the 4th. Yeah, I can deduce the reasoning, but I also have to convince students who've never seen it before. The 3rd one is particularly solid for new calc students, so I'll bare that in mind.
Also, will have to practice the two-pen thing. I use really thin markers, and put the lids of three of them between the knuckles of my left hand to draw free whichever market I need.
Is there any practical use of 0!
How to arrange your future
Taylor series
4th reason should really be:
e^x=1+sum(n from 1 to inf) xⁿ/n! .
To make this look nicer, we want to include the 1 within the sum.
So we want x^0/0! = 1, which leads to 0! = 1.
You can only say e^x=sum(n from 0 to inf) xⁿ/n! After you have defined 0! = 1.
I thought of that too
This was my thought too. The definition comes before the sum
Infinite sum is the best
My favorite definition is the #3 through GAMMA function, 0! = GAMMA(1) = 1, I have the proof by Weierstrass representation (starting with Gauss) that 0! =1
Nice one
Similar to dividing by 0, you cannot factor a system containing nothing into any discrete sets
There is only 1 way to factor a 3! system which has 6 discrete sets
12:00 where is the (0^inf)/ (inf!)? Or why is it 0?
Really?
0.
The arrange n objects reason makes the most sense to me
4th reason is the best 😊
vlatko the limit as x-> 0 of x^x is equal to one
The fourth argument is circular because the formula for Exp(x) can be write this way because we already admit that 0! = 1. Anyway, i like very much your videos and also your juvenile enthusiasm. I am over 60 by the way.
(i)! !!!!
Well (i)! = Π(i) = int (x^i e^(-x) )
= int (e^(i*log(x) -x)
= int (cos(log(x)) e^-x) + i* int (sin(log(x)) e^-x)
≈ 0.50 - 0.15i
Pretty horrible.
Although (i)! !!!! Is pretty bad too... But repeated factorials slowly brings the answer towards 1+0i.
Similarly how repeated factorials of a number between 0 and 1 approaches the fixed point 1.
The Gamma function, and the Pi function, can be used as analytic continuations to the factorial for ALL complex numbers (excluding negative integers).
Pi(t) = int(x from 0 to inf) (x^t e^-x) dx
which exists for complex numbers. So we can say that i! Exists as does i! ! ! ! !
And is approximately
0.9923-0.0003i
Assuming the !!!! Can be assumed to be 4 factorials instead of 2 double factorials.
(n!!= n(n-2)(n-4)... (3.5±0.5)(1.5±0.5) depending if n is even or odd... It can be analytically continuated to the complex numbers using x!!=(Pi(x/2)*2^((x+1)/2)) / pi^0.5
(((i!)!!)!!) ≈ 0.99+0.01i )
Neo Matrix defined*
A Wild Triple Factorial has appeared
All your proofs are elegant and good for math. But there is a very simple proof for it.
n! = (n+1)!/(n+1)
Therefore 0! = (0+1)!/(0+1) = 1
That's exactly what he did for the first one. You just generalized it, but it's not necessary because he's looking to find one value. Also, he clearly stated that the examples aren't proof, which they are not.
Steeve Cantave thanks I admire your concern
I knew first 2 but wow the others.
: )
Fascinating, once again. I'm not comfortable with the last evaluation, but I reached my level of incompetence with undergraduate Maths, hence do not over-estimate my competence.
so many choices..........
these are Great ways to explain this!
thank you for making this video so it could help me and so many other people
The beard looks good on you
Thanks!
By the logic of the first example, would that make (-1)! Undefined since it would be 1 divided by 0? Is this the logic for why we don't do negative factorials?
(-1)! is defined to be infinity. 1! is defined to be 0. 0! = infinity * 0, uh-oh...
Can you take the derivative and/or integral of n!
Ethanol 314 you can take the derivative or integral of the gamma function, and the Pi function. n! Is not a function to take a derivative or integral of
Yes you can if you use a function, like GAMMA(n), the log derivative is called the digamma function and you can find ALL the minimum and maximum of the GAMMA function.Really powerful function, because you can compute partial harmonic series, Here is an example
1+1/2+1/3+...+1/65536 = ?? well easy with digamma(65537)+ gamma (Euler's number = 0.577215664901)
Like all four. Use of the gamma function is the neatest method.
First. Nice video.
You know, regarding the 2nd method, I actually came up with a mnemonic for when I was studying for the first actuarial exam, and that's a quote from the Friends episode "The One Where Nobody's Ready", when Rachel very passive-aggressively declared she wasn't going to go with Ross to the museum to hear him give his speech at the benefit, specifically when she told him, "I'm not gonna gooo, so, I think that will accomplish the not going". LOL
In short, there's only ONE way to not do something, and that is......TO NOT DO IT!
Empty product!
Tysm, I was "whattttt" when my tutor told me the answer is 1. I learned something today. Tysm again
Reason #4 is really bizarre as you have that really weird 0^0 in there. But hey, 0! is always equal to 1 because, eh, why not?
Same with 0^0. Why bother to leave things undefined? It's just saying, "Screw this challenge, I will not face my fears", but that is against the mission of humanity of breaking away.
Pi/gamma function is the best reason
I think its a bad logic.
Manoj Shrestha
Explain.