Making Sense of Ramanujan's Infinite Sum for General Audience.
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- Опубліковано 11 тра 2024
- In this video we will try to Intuitively understand why the weird sum 1+2+3 and so on till infinity or the famous Ramanujan sum. Which seems very counter intuitive at first sight!
The Numberphile video- • ASTOUNDING: 1 + 2 + 3 ...
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Have fun playing around with infinities!
For a proof of this you can watch this video (If I did it, I will just repeat this proof)-
ua-cam.com/video/YuIIjLr6vUA/v-deo.htmlsi=oqNokKAJSxnhUE-Z
In the Video I say beta function regularization
but wanted to say
zeta function regularization
sorry for that!
No ProbleZ
I eat all infinite apples and eat 1/12 of friend's apple too. Hence proved
The logical proof of " 0.999..=1" is that there exists no number between 1 and 0.999...and hence these two are exactly same
An example would be that 1/3 = 0.333… that x 3 would be .9999… but 1/3 x 3 should be 3/3
but if you do the same think for 0,888...=1 its the same but we fond that 0,999...=1 so how can 0,888...=0,999...=1 ?
this wont work for 0.88888
@@adam-denis
This won't work cuz
I can say 0.9 exists between 0.88888 and 1 na
@@mridulacharya8250There exists infinite real number between two distinct real numbers..then how did you concluded there is no real number between 0.999999.... and 1 ??
I discovered ur channel today... Glad I did it
Finally few more intuitive math guys in my list
Welcome aboard!
me 2 lol
Me too
Thank you so much. I've been trying to understand this for ages.
I love your videos! thank you for making this one
Brilliant explanation! Kudos, man!
This is what teaching is about....explaining why it happens....not how it happens....
👏👏👏
Amazing! Really eye-opening to see! Before, I was wondering how this divergent series could converge in such a peculiar way, but now I understand the general idea! I would love to see your version of a simplified mathematical proof in a future video. Once again, great explanation!
-1/12 is the true result under WRONG assumption. The proof is based on the assumption that THE series converges to a number, but it is NOT. End of story.
What type of assumption should be?
First, one need to prove that the series converges to a number, but the series of natural numbers diverges that is infinite as it should be
Very good job on bridging the gap between the intuitive impossibility of adding apples and never getting -1/12 and how that intution is wrong since we truely have no intuition for infinity.
Yes great description very clearly explained and at an easy to digest pace too. Peace and love from Australia
absolutely fantastic video
The way you explain, the accect , the simple example is magnificent
Thanks - very well explained!
i remember i used to talk to you when there was a whatsup group...you are still the best person whom i could talk to abt maths and astrophysics. Keep going! im proud of you!
you can still join the discord server
Bro explains everything so calmly. I love it.
This video really made sense. Great explanation 👍🏻👍🏻
Thank you for the thoughtful explanation.
Problem in handling infinity is that since it's a concept and can never itself be reached, it must be handled by considering the rate of approach to infinity...
Infinity x 1 = infinity
Infinity x 2 = infinity still but it's rate of approach is now 2x faster.
Just like division by zero is not allowed, addition, subtraction, multiplication and division involving infinity is not allowed in maths
Because infinity+infinity is again infinity and not 2 infinity
Infinity is not a number
@@fauzanree1983that's what he is saying.. infinity isn't distributive or associative
@@fauzanree1983 I think Infinite is higher dimension of Numbers
A guy playing with math, breaking all well-agreed definitions and common sense. No, you can not add positive numbers and get anything negative.
There are definitions for convergence.
What you can do is use math in your own way and get absurd results.
Ramanujan turns in his grave being mentioned in this.
@@Param_Hayaran Infinity can be handled as a group or set and yes it does have distributive and associative properties but ONLY as a group.
These do not translate when infinity is used as a number in an equation
This is one of a way for developing and giving a relative mathematical value for the summation of infinitely many terms.
Your explanation is very appreciated 👍
Sir your explanation is 100% correct lekin,
The series S= 1+2+3+4+5...∞ Is considered "divergent"
We can understand it by this example
Imagine you have a piggy bank, and on the first day, you put in ₹1, on the second day you put in ₹2, on the third day you put in ₹3, and so on. Every day, the amount you put in gets bigger.
If you keep doing this forever, the amount of money in your piggy bank will just keep growing and growing without ever stopping. There is no limit to how much money you'll end up with. Means kabhi ye series khtm hi nhi hogi kyunki hum har din piggy bank me paise daal rhe hain.
Same will happen in the series , and it won't give a finite value because it will just grow bigger and bigger without ever stopping and issi ko hum "Divergent" kehte hain.
So, jab hum kehte hain ki ye series -1/12 ke equal hai ye iss principle ke hisab se fit nhi hoti hai jo ki hai
"Divergent series"
Please make a proof video for this as well sharing insights of Sri Ramanujan.
This is the best video ive seen about this summation, subscribed
Thank you
I pay big bucks to have your accent! Even though considered heavy by English natives it’s clear yet extremely captivating. When you speak I learn!
Been following your channel for i think 2 years now when you had less than 50k subscribers.
Thanks for the quality content you provide! Learned something new today.
Beacuse of support from people like you this channel is thriving
Simple and easy explained. Thanks sir
Hello. What is the Ramanujan summation?
Question: How is it that the infinite sum of this series diverges only upto -1/12 and not other possibilities if we may have the freedom to handle the sum by the concept of infinity ?
0:04 Gauss equation n(n+1)/2 is sum of numbers from 1 to n. Assume n=infinity. Then n(n+1)/2=-1/12 So infinity= -1/2+square root of 1/12 and -1/2 -square root of 1/12
U should solve quadratic in order to get the value of n
n²+n=-1/6
n²+n+1/6=0
Now by Sridharacharya formula
n= (-1+- rootunder 1/3)/2
Absurd.
@@kartikdd8678 😂 Might be an absurd solution but it makes a lil amount of sense
@@sytherplayz ∞^2+∞-1/6=0 even worse
Well explained.
Very simple and easy to understand explanation! Please make follow-up video to explain how it's intelligently handled by infinity concept/beta function and its implications in the real world.
Can you tell us about more new and informative mathematical concepts and tricks?
Well explained 👍
Keep going.......
Very well explained brother 💚
Thanks for the videos welcome back sir🙏. Can you please make a video on how to excel in maths like a genius would
Amazing 🔥🔥
Sir please do a topic on linear equations graph
Great video
What I derive from this is that since infinity has no relation to reality, any equation containing it is also devoid of reality, as a natural extension. I would also suppose that, in this case the result could be pretty much anything. I assume -1/12 just stands in as an example of this.
Thank u for explain 😊
You just now earned a sub
"Physics is the medium to perceive the Universe;
Mathematics is the language to communicate with the Universe."
- Universe is always speaking, humans only have to listened in the right direction, and with the right tools (mathematics).
Absolutely correct
The best explanation I have seen. Here are couple of things - take the 1 + 2 + 3 infinite series and add 1 to it. Is this allowed? If allowed will the answer be 11/12? What’s the sum of other infinite series - 1 + 4 + 9 + 16 etc.
Amazing this video makes so much sense
beautifully covered
Congratulations for future 500k
So it's about convergent divergent tests this is about?
I saw this in Calc 2
He explained it like he doesn't like the smell of this equcation.
Feels like he proves that infinities bring out mathematical superpositions
Infinity is weird and you should not expect that infinity will behave like a normal finite number. Infinity is not a number it can be a number but using it like a number will give you weird results.
Thanks Sir , How to visualize the Methods like Beta-Regualization , Ramanujan's Summation to make sense
When something is not defined, anyone suggesting anything about that cannot be taken for granted, even if the person is Ramanujan.
...and all delivered while still wearing your pyjamas. Kudos.
It shows up in the zeta function too.
While writing, his nervous hand would have shaken a little, resulting in the minus sign
A infinite so many never runs out
After seeing all comments...I concluded...
Series of natural number is closely linked to -1/12 but the conventional sum is not that connection
Moral of the story is you must not try sum up a divergent series upto infinity. Even if you find the sum of a divergent series in terms of 'n' then don't try to evaluate that sum with the limit 'n' tends to infinity !!!
Thank you!!
OK, infinity is not a number. But then we treat it exactly like a number when we multiply 0.99999 ---> by 10 to get 9.99999 -->. And then we go and perform subtraction on it.
It seems we get to pick and choose when and when not to treat infinity as a number.
I still have a doubt that how did the sum of natural number become negetive and an integer. Like that's not possible in any world!
And if we are handling infinity as any possible number then is it also possible that 2i (an imaginary number) or √2 ( an irrational number) becomes the sum of infinite natural numbers?
Very nice video😃
With infinity, you can prove that anything is equal to anything
When we multiplied the decimal number extending infinitely by 10, is the product we assumed here accurate ?
what do you think?
Bro can you make vidoes on jee advanced maths chapters pls !
Your voice is beautiful bro. 😊
Sir to be very frank...... Your video although informative in its own right raises more questions than it answers. You should have also tackled the how it happens/the proof behind the expression also along with the why it happens in maths.
And you can see those questions in your comments. Please make a video tackling those.
Thank you, my South Asian brother. Your explanation makes so much sense. Far better than the Western rubbish that most YT videos say. Ramanujan would love you.
The Following will give you a little bit of insight into what it is all about.
1/3 is a rational number that we can express as 0.3+0.1/3 =0.33+0.01/3 =0.333+0.001/3 =0.3333+0.0001/3 and so on infinitely. Each of above expression is an absolute number (1/3) without use of infinity. But abstract concept of infinity enters scene when you express it as 0.333.. It's only in case of absolute value 1/3 that you can go on adding infinite number of 3's.
We know that
Numerator/Denominator =Quotient+(Remainder/Denominator)
12/10=1+(2/10)
In the same way, 10/10=1+(0/10), but we can also express it like 10/10=0.9+(1/10)=0.99+(0.1/10)=0.999+(0.01/10)=0.9999+(0.001/10)=0.99999+(0.0001/10). So 10/10=1=0.99999..
Remember that adding 9's at the end infinite number of times is only possible because of left remainders 0.1, 0.01, 0.001, 0.0001 and so on.
Thanks for clarification. So that means the equal sign in 1=0.99999.... to infinite is only valid when you add the term +0.0000......1 behind the expression right?
Otherwise we should not say 0.9999.... = 1, rather stick to its conventional mathemathical expression: limit n to infinite for 0.9999n approaches 1.
Nice sir🎉😅😅
I would argue only thing in the world is infinity , we just look at a part of it and assume finiteness, infinity doesn't raise any question while finiteness does raise many questions, imagine the universe was of size x which is not infinite then that raises an immediate question why this arbitrary value x out of the infinite other values which has no satisfactory answer, the only satisfactory answer is that universe or existence is infinite and we are looking at a part of it which appears to be of size x, so infinity in truly the only real thing on which we apply our limitations to get illusion of finiteness. But this is philosophical argument unrelated to this sum or its analytical values.
This video doesnt explain why its -1/12 but explains why i could be -1/12.
Because in math theres almost like a phase change there at the end of infinity.
Thanks
how come it can be a negative number..
Well made
my dear brother if put two mirrors front of each other then
the infinite mirrors will appear ?? eheather it can be said as imaginary
I have a question and hope someone can help me with this question. Suppose you want to travel from A to B, If you start by traveling half the distance and then proceed by traveling half of the half of the distance and continue in this manner repeatedly, would you be able to reach B?
this is called the zeno paradox, In our world we can travel from A to B that means there must be a unit of length which cant be halved.
@@mathOgenius Thanks for pointing me to the right direction. I've googled Zeno's paradox and found the answer which confirms my surmise...
This can't be made by humans.. ramanujan was something else he was a fellow of God and God gave some equations that is the key to go in the other world or universe 😌😌
Only one more fellow of God only can find that key lock
We should find that fellow 😞
Only one question Bro!
How can you do 9S/S ??
It is same as infinity/infinty which is not good I think
its 9.999999/9.99999 its not inf.
@@mathOgenius I am not math expert, this is only my view, please correct me if any mistake
Actually it is not infinity in that sense but think in this way , 0.99 and 0.999 both have different level of accuracy so 9S is like one step ahead than S , so both are not comparable
yes thats what was the whole point of the video, the infinity not having a sense of comparison is causing these proofs.
Bro tell me y 1^ infinity is an indeterminate form
The 1 that you are talking about is not exactly equal to 1...it is approaching 1 remember....so you are essentially calculating the limit when the base is very very little less than 1 or little more than 1, Powered to infinity.....cus if it is actually much more than 1 then it'll lead to infinity and in case it is less than 1 then 0....so 1^infinity is essentially indeterminate form
Just playing with numbers very clever but just hocus-pocus
The only explanation is that Ramanujan has Apeirophobia (fear of Infinity)
The equation only valids if you believe that oo + 1 > oo.. ( oo is not an exact number)
As long as you don't believe it (oo + 1 > oo), it means the equation (1+2+..+oo =1/12 is logically wrong...
i have my history exam tomorrow, why am i watching this?
To have a short rest in a differnent place?🙂
Sum of n is by Gauss n(n+1)/2...???
So tell me what number is closest to 1 without being equal to one?
its the last one of the numbers which are between 0 and 1..
At
3:43 you subtracted
9.9999...
-0.9999...
---------------
9.0000......
So here after decimal .9999... are non terminating recurring number right?
So at 4:39
9.990
-0.999
------------
8.991
Why you subtract with 9.990 instead with 9.999 because even removing infinity, the number after decimal should be recurring number just like done at 3:43 since we are following same procedure after removing infinity.
Plz make me understand this.😮
Because the first equation had an infinite number of nines after the decimal point,while the second one had a finite number of nines(three),which are not recurring.
second example was for non reoccurring one, to show what happens when we remove infinity
Wow😮
Nice channel
Bhiya Can You Do Give Away Ones More Time 🙏🥺
This series come from using a numer external of the domanin of a function in the function itself is for this reason that the resoult doesnt make sense
-1/2+square root of 1/6 and -1/2 -square root of 1/6 there was an error before
Top notch
I give you 2 numbers: One number is 0.999 till infinity and the other number is exactly the same with only one random digit not being 9. I don't tell you which one is which and I ask to identify which one of them is equal to 1 and which one isn't. Can you solve it?
I would need the location of the changed digit otherwise its uncertain.
*my teacher talking about adding apples to teach us addition*
me: “what about negative apples?”
teacher: “well you can’t have negative apples”
*starts exponentially mass producing apples*
Negative apples must b the rotting apples added in the beginning... 😅
I'm confused. I can disprove that 1+2+3+...+infinity = -1/12 through just looking at the first 2 digits. 1+2>-1/12 and infinity>-1/12. What is my thinking error?
yes you can in countless ways disprove this, here is another very basic proof we know.. if x- integer , y- integer then x+y is also an integer so this equation is not correct..
@@mathOgenius what is it then? is "-1/12" just any random number, some that ramanujan came up because he thought "why not"? Could it actually be any number like 1234.25? I don't understand the whole concept. My brain would tell me the answer would be infinity and that there's no single grain of doubt about that
@Wrongidea807, There is no error in your thinking. I am glad that there is at least one sane mind in the comment section and that's you. First of all, 1+2+3+... is NOT equal to -1/12 as you correctly pointed out. Secondly, for the most part this video is non-sense. "Adding a symbol infinity at the end" does not make any sense and it is not the reason why things work out. I don't know your mathematical background, but here is the rough idea. Take the series 1+1/2^2+1/3^2+1/4^2+.... You can show that this series is convergent. In fact, you can replace the power 2 by anything slightly bigger than 1. That is, 1+1/2^(1.1)+ 1/3^(1.1)+... is also convergent and so on. But 1+1/2+1/3+1/4+... diverges to infinity. Therefore, one can make a function f(s)=1+1/2^s+1/3^s+... What we are saying is that f(s) is defined for all s>1. Now comes the non-trivial part, turns out we can take s to be complex number where the real part is >1 and the series is still convergent. This defines a nice function on certain part of the complex plane. This is an extremely nice function. Such functions are called analytic. Turns out this function can be extended to the whole complex plane (almost) by what is called an analytic continuation. The good news is that there is exactly one way to do this analytic continuation. This continued function is defined even when s=-1. When you take s=-1, this function has a value -1/12. But the key thing to note is that we are not saying that this function and 1+1/2^s+1/3^s+... are the same everywhere. These two quantities are same only when real part of s>1. But, if you ignore this and put s=-1 in the series, you get 1+2+3+4+... and hence one often abuses this relationship to say that 1+2+3+...=-1/12. (Ramanujan, I think (I may be wrong here, need to check), did not use the analytic continuation technique.)
@@Wrongidea807It's not a random number. It is a constant in regularization. So 1+2+3+4+..+∞ diverges the sum will blow up into infinity. But if you remove the infinity there will be a constant (-1/12) that will remain.
Biggest no is either zero or one.
3:00 video started 😊😊
I am waiting for your 500k subscribe
Why
Because of telescope giveaway 😅
Good video.
I swear most advanced mathematicians don’t understand infinity 😂
Infinity is nonending (c) - 🤣
But to accomplish these types of math tricks, they start by treating infinity as ending.
Still, like a child, I love the magic tricks.
0.999... = 1 because you can make the number arbitrarily close to 1.
Agar hm puri real number line ko as a closed circular form mein assume karien and define infinity as a point which is diametrically opposite to zero in that circle, tb Ramanujan sum kuch sense bna skta hai...
can u explain? why you assumed real line to be circular? any reason for it? and how u defining addition on circle such that sum of all weights adds to negative? naively its simply a divergent seq and there is theorem ,grouping and rearranging will change values to any contradicting value. thats why rigrous proof uses beta fn to proof zetta sum as -1/12( which uses analytic continuation) it will make sense when dealing with zetta fn.
@@AmanKumar-sk3nk I can't explain everything on the comment section... Assuming real number line as circular is an idea to connect what's happening in the negative side of the number line to that of positive when we are dealing with infinities we got some patterns which suggests some connection between the two and this assumption is made on that context...
@@divyanshkaundal.369 im an iitbhilai undergrad working on related things(as my project) it will be nice if you could tell me the theory of extending real line to circle and any context of it ,thanks. as far as i know i haven't encountered any such thing and what pattern ur talking about (in zetta function?).i think that is only a intuition. same intution is provided by many youtubers like mathologer and all but they simply clearified that each defination has their own profit in their context but universally saying -1/12 does not make sense ? you are welcome to make me correct if am wrong ,thanks
@@AmanKumar-sk3nk brother I don't know but youtube finds some suspicion in my text that I send you 😅... The word that you should look for the context is the (person who gave the zeta function ) Spheres....
@@divyanshkaundal.369 thanks ,have a nice day