Interestingly enough, if you accept the gamma function as the continuation of the factorial, all negative integers are the only numbers that are undefined.
@user-bi3oc2jt4t the factorial function is undefined for all non natural numbers, but there is a function called the Gamma function that extends the factorial function to all numbers except for the negative integers. Lines that connect has a great video on this subject.
I don't know anything about negative factorials, so I think a video on that would be interesting. Also, and I've said this before but I want to say it again, I really appreciate that you go over all the steps and the relevant rules and theorems and what not. Thank you for providing these videos.
All I know about negative factorials is that every negative integer is undefined, so the only solutions that would exist are decimal solutions which would need the use of the Gamma Function
Well, let's think about it. How is the factorial defined? n!= n * (n-1) * (n-2) * ... 3 * 2 * 1, for some natural, positive number. It's easy to see how you can use this definition to get the next factorial: if (n+1)! = (n+1) * n *... 3 * 2 * 1, then (n+1)!=(n+1)*n! So let's see, just for fun, how we can use it to calculate the previous factorial. If we put in k-1=n everywhere, we get k!=k*(k-1)!, or rearranging, (k-1)!=(k!)/k. So what's the lowest factorial we know? 1. The sum of all numbers from 1 down to 1 is... 1. So what's the factorial before that? 0!=(1-1)!=(1!)/1=1/1=1. So 0!=1 which is... strange. Now, can we do it again? (-1)!=(0-1)!=(0!)/0=1/0. This is a problem: anything times zero is zero. If I have zero lots of a million dollars, I have zero dollars. So zero lots of any amount, no matter how big the amount, will never add up to 1. Dividing by zero is a contradiction to how we understand 0 to operate under multiplication. Now, you could try to be clever and say, "well as numbers get smaller, their inverse gets bigger, so the answer is infinity", but that's not true. For one, infinity is a limit; as you do something over and over again, what do you approach? In this case it's unbounded. But more importantly, 1/(-1)=-1
Quicker is to multiply the first term by p/p in the first step rather than all terms by p!. You bring the first term to the same denominator as the others this way, which is more intuitive.
well using the knownledge about positive integers it seems to be every multiple of a said integer, also it might be similar to when a negative number is raised to a positive number ex: -5^2 = -25 and (-5)^2 = 25 so it would be like -5! = -120 and (-5)! = 120
Negative integer factorials are a little tricky. They are, by all reasonable definitions, undefined. However, you can, under certain circumstances, manipulate them algebraically to get a sensible, defined answer. For example if you take n
Here's a fun geometry problem you will love. Start with a 30-60-90 triangle with short leg 1. Construct a 1/6-circle centered at the 60-degree vertex and inside the triangle. Finally, add a semicircle whose diameter is along the long leg of the triangle and which is tangent to the hypotenuse and the 1/6-circle. Find its radius. ** ** ** ** It's an elegant mess, BTW. (2sqrt(3)+1-2sqrt(sqrt(3)+1))/3.
As I understand, there is an extension of the factorial function that has a broader domain, but it only includes noninteger negatives, so it still wouldn't help here.
A little faster if you make your first step "multiply p/(p-1)! by p/p". This gives pp/p! - 13/p! = (p-1)/p!. Then multiply the entire thing by p!, giving pp - 13p = p - 1. Then continue from there.
p=(-3) is still a solution, because earlier you followed through with being able to eliminate all of the factorials, so the resulting equation is only a f(p). Plugging in p=(-3), I get 9 - 13 = -4, which is true. 🙂
No equations p/(p-1)!-13/p!=(p-1)/p! and p^2-13=p-1 are equivalent if and only if all members are defined. Even if considered p! as gamma(p+1) it is undefined for negative integers
3:50 sixteen minus thirteen, not sixteen minus three. Yes, I am "that guy." :D However, you are correct, "mush" is the correct mathematical term, I think I read it in Euclid's Elements. Just teasing, another great video. Oops, I have to go away for a moment...
A video about negative factorials would be exciting
Isn’t all negative factorials just 0? I am not sure what you mean by this
It would require an understanding of analytic continuation, integrals and overall calculus, something which I’m not sure would fit this channel
Interestingly enough, if you accept the gamma function as the continuation of the factorial, all negative integers are the only numbers that are undefined.
@user-bi3oc2jt4t the factorial function is undefined for all non natural numbers, but there is a function called the Gamma function that extends the factorial function to all numbers except for the negative integers.
Lines that connect has a great video on this subject.
@@cartatowegs5080 Thank you, that’s actually a really cool channel
Andy had to go but he came back super quick
He had to p
Really wish my dad did that
okay, now we need a follow-up video about the gamma function
I don't know anything about negative factorials, so I think a video on that would be interesting. Also, and I've said this before but I want to say it again, I really appreciate that you go over all the steps and the relevant rules and theorems and what not. Thank you for providing these videos.
All I know about negative factorials is that every negative integer is undefined, so the only solutions that would exist are decimal solutions which would need the use of the Gamma Function
Well, let's think about it. How is the factorial defined? n!= n * (n-1) * (n-2) * ... 3 * 2 * 1, for some natural, positive number.
It's easy to see how you can use this definition to get the next factorial: if (n+1)! = (n+1) * n *... 3 * 2 * 1, then (n+1)!=(n+1)*n!
So let's see, just for fun, how we can use it to calculate the previous factorial. If we put in k-1=n everywhere, we get k!=k*(k-1)!, or rearranging, (k-1)!=(k!)/k.
So what's the lowest factorial we know? 1. The sum of all numbers from 1 down to 1 is... 1. So what's the factorial before that? 0!=(1-1)!=(1!)/1=1/1=1. So 0!=1 which is... strange.
Now, can we do it again? (-1)!=(0-1)!=(0!)/0=1/0. This is a problem: anything times zero is zero. If I have zero lots of a million dollars, I have zero dollars. So zero lots of any amount, no matter how big the amount, will never add up to 1. Dividing by zero is a contradiction to how we understand 0 to operate under multiplication.
Now, you could try to be clever and say, "well as numbers get smaller, their inverse gets bigger, so the answer is infinity", but that's not true. For one, infinity is a limit; as you do something over and over again, what do you approach? In this case it's unbounded. But more importantly, 1/(-1)=-1
@@themathhatter5290 Why division anything by 0 isn't just equal to 0?
Quicker is to multiply the first term by p/p in the first step rather than all terms by p!. You bring the first term to the same denominator as the others this way, which is more intuitive.
Please make a video about negative factorials
A video about negative factorials? How exciting!
Oh, man, I got this one in a heartbeat. Finally!
2:30 i'm pretty sure Andy really had to p
That one took me a moment… 🤣
2:28
If that's the case, then p = 1.
A negative factorials video would be interesting!
Gamma Function 😊
Wouldn't work since it's not defined for negative integers
@@ElderEagle42 not defined for any integers less than or equal to 0
3:10 - you cancelled 4! only to get it back as a common denominator few seconds later.
i’d love some negative factorial stuff. anything that fucks around with the gamma function is fire
well using the knownledge about positive integers it seems to be every multiple of a said integer, also it might be similar to when a negative number is raised to a positive number ex: -5^2 = -25 and (-5)^2 = 25 so it would be like -5! = -120 and (-5)! = 120
Andy yes please tell us about negative factorials
(Also can you explain why the 'x!' in desmos has a graph for all real values of x)
Fatorial is undefined for negative integers. For general complex numbers use the gamma function.
That editing is top tier.
It is fun to see the graph of this equation , it has intersections with X axis on every negative integers but -3
Brother you are fabulous! 👏👌👌👌👍🙌🙌
Negative integer factorials are a little tricky. They are, by all reasonable definitions, undefined. However, you can, under certain circumstances, manipulate them algebraically to get a sensible, defined answer. For example if you take n
Here's a fun geometry problem you will love. Start with a 30-60-90 triangle with short leg 1. Construct a 1/6-circle centered at the 60-degree vertex and inside the triangle. Finally, add a semicircle whose diameter is along the long leg of the triangle and which is tangent to the hypotenuse and the 1/6-circle. Find its radius.
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**
It's an elegant mess, BTW. (2sqrt(3)+1-2sqrt(sqrt(3)+1))/3.
As I understand, there is an extension of the factorial function that has a broader domain, but it only includes noninteger negatives, so it still wouldn't help here.
hi andy. welcome back
how did you came back that quickly😮😮, btw a guy named 'Lines that connects' made a video abt factorials
Negative factorials would be interesting. I have never done that before.
Negative factorials
finally some easy mental math questions
Oooh, let's do The Gamma Function!!
A little faster if you make your first step "multiply p/(p-1)! by p/p". This gives pp/p! - 13/p! = (p-1)/p!. Then multiply the entire thing by p!, giving pp - 13p = p - 1. Then continue from there.
Video on negative factorials please
yes, please make a video about negative factorials
Make a video about negative factorials
Now I know why simple mathematics
that was fun
Please make a negative factorial video
What do you use to make your videos? The animation seems too smooth for ppt
It’s really sexy that you can do math
Said no one ever. To me.
where did you go
I know nothing about factorials. But seems cool.
Does analytic continuation count as simple math? 🤔
3:48 he said 16-3 is 3 😂😂😂
I'm curious about negative factorials. Make the video!
Don’t want to be negative but a negative factorial video would be nice!
make a video about newton's method @AndyMath
And while we’re on the subject of non-intuitive factorials, what about a video of non-integer factorials?
eye dd ths n my head : )
I got it once I remembered that factorial is about multiplication not addition
p=(-3) is still a solution, because earlier you followed through with being able to eliminate all of the factorials, so the resulting equation is only a f(p). Plugging in p=(-3), I get 9 - 13 = -4, which is true. 🙂
No equations p/(p-1)!-13/p!=(p-1)/p! and p^2-13=p-1 are equivalent if and only if all members are defined. Even if considered p! as gamma(p+1) it is undefined for negative integers
2:30 He had to go p
3:50 sixteen minus thirteen, not sixteen minus three. Yes, I am "that guy." :D However, you are correct, "mush" is the correct mathematical term, I think I read it in Euclid's Elements. Just teasing, another great video. Oops, I have to go away for a moment...
P=4
3:49 sixteen minus three is equal to three!