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You claim to be in the the field of Applied Maths but I don't think there is any difference between pure and applied. It's beacause I see pure Maths as Applied Set theory.
you are clueless . In math Anything can be anything. not a good video in opinion I rather see why its 3.14. we know length is length. its absolute. circumference is a fixed number.
I also find the Ramanujan one here beautiful especially because when you first look at it it just seems like random numbers but there is a deeper logic to it.
Note, at 9:27, the n values should be minus 1 and the expression on the righthandside should be multiplied by 4. What I meant to write was: n = 1: 4(1 - 1/3) ~= 2.66 n = 2: 4(1 - 1/3 + 1/5) ~= 3.42 n = 3: 4(1 - 1/3 + 1/5 - 1/7) ~= 2.89 Also, as promised in the video, the link to my Ramanujan video: ua-cam.com/video/PGJubCoH5as/v-deo.html
By calculating the difference between pi and partial sums for n up to N, you can produce a 2D surface plot showing how the error varies with N and lambda. You can then investigate the pair of values that give the minimum error. You should investigate putting lambda = pi for fun.
Ellie, thank you for recognising, Madhava. However, you missed, Aryabhatta, an extraordinary astronomer and mathematician from the 5th century, whi is often referred to as the “Father of Indian Mathematics.” His seminal work, Aryabhatiya, introduced the concept of zero, decimal notation, and a precise approximation of ‘π’. I hope this helps.
What blinkered short-sightedness Cambridge University had regarding Ramanujan! Ramanujan: "I have this function that solves [this problem]!" Uni: "But you don't have any qualifications, so it can't be any good!" Ramanujan: "But it works!" Uni: "Can you prove it?" Ramanujan: "I'm not good at that part, But it works!" Uni: "We can''t cope with people who can't think in our box!" ... :(
Fascinating presentation, thanks for the effort, Ellie. Euler's formula for π , of absolutely no value to compute it, but the single most important application of π in higher mathematics: I kind of hoped YT would take the LaTeX output from my AI, but it does not $$ \pi = - \sqrt{-1} \mathrm{Log}_e {-1} $$ For non nerds, it is π = - sqrt( -1) Ln( -1)
Small nitpick: I know it would've taken a bit more time, but I would've loved a few matlab or maple numerical computations where you demonstrated what happens when you put in different values of lambda for different truncations. Otherwise, I'm glad you covered this piece of good news! That's always welcome and a good effort.
You have a flair for communicating maths; really clear the way you make it understandable. You def could write excellent maths textbooks in the future imo.
I'd be interested to see the pattern to these minima in lambda plotted over N and lambda. Are there any super minima that give unusually good approximations?
You have forgotten Aryabhatta maths astronomer who calculated Pi up to 3. up to 32 decimal points in 5th century. Madhava,also inventer of calculus did it in15th century.
This could be a bit of smoke. Seeing that large lambdas appear to give better results (provided you add sufficient terms), on a hunch I decided to take the limit on lambda -> infinity, thus removing the terms with lambda in the denominator; the result simplifies to the old Taylor series for 4 * arctan 1 = pi. P.S.: Ah, never mind, my bad. They recognize this fact in the paper, but comment that certain values of lambda allow the series to converge much faster (the Taylor series for arctan is notoriously slow to converge).
É realmente fascinante como o trabalho de Ramanujan transmite essa sensação de mistério e beleza. Suas fórmulas e identidades muitas vezes parecem surgir do nada, com sequências numéricas e relações que parecem aleatórias à primeira vista. No entanto, por trás dessas aparentes “coincidências” está uma lógica profundamente enraizada em conceitos matemáticos avançados.
Well, I get the spirit you are asking in. Lambda is any value, and that's true, but there are still limitations from where you can choose to take it from, but first let's discuss why lambda cannot be -n: because lambda, in context of this summation, does not changes, i.e the sum sees it as a fixed value. Kind of like you can change all your tires in your car, but you don't really do that when your car is running, otherwise you'd get unexpected results. But there can be interesting prospects as well! Lambda cannot be any negative integer, because it will cause undefined output at some point in the iteration. From the equation, you can choose real positive number (that's still infinitely many), also keeping it mind that it does not blows up to infinity (i.e converges when n->infinity)
Fascinating topic and a nice explanation. Curious,, what language do you plan to use to code this in? Will you use a functional programming language? Does your choice pay particular attention to execution performance?
I'm no expert but I think you made a small error in your explanation for the Pochhammer notation. According to Wikipedia xₙ is a falling factorial: (x)ₙ = x(x-1)(x-2)...(x-n+1) The definition you wrote is for a rising factorial: x⁽ⁿ⁾ = x(x+1)(x+2)...(x+n-1) In both cases I immediately thought that this was shorthand for the division of factorials. (x)ₙ = x(x-1)(x-2)...(x-n+1) (5)₃ = 5 x 4 x 3 (5)₃ = 3 x 4 x 5 (5)₃ = (1 x 2 x 3 x 4 x 5) / (1 x 2) (5)₃ = 5!/2! In general: (x)ₙ = x! / (x-n)! x⁽ⁿ⁾ = (m+n-1)! / (m-1)! I wasn't reviewing your work btw. I was just checking if my intuition about factorials was correct.
great explainer. Your explanation style is really simple and easy to follow. I have one request, as I understand your previous job involved working at the intersection of physics and comp sci. can you make a video or two about career opportunities that lie in that intersection and some advice for someone who dreams of working in that area? It would be much appreciated 🙏. Either way, thanks for the amazing work you do!!
HI thank you very much for such a nice explanation, but which program do you use to write your notes the you are using in this video ? can you please provide med the name of the app ?
Місяць тому
Once again a brilliant treatise on something which is of no importance, the present values calculated for pi are perfectly adequate. For your next video consider reviving the Middle Ages debate on how many angels can dance on the head of a pin!
100,000 years and 20 billion brains later one brain- Mark McCutcheon:” The Final Theory: Rethinking Our Scientific Legacy “, - discovered/published the CAUSE of gravity, electricity, magnetism, light and well....everything. Genius level event.
@EllieSleightholm Can you please make a video tutorial about how to take effective math notes step by step guide in a Notebook for any math Topics from basic topics such as Arithmetic to advanced Algebra. I think it would help alout of people. If you have time ofc and Btw ❤ your channel❤
In Calculus 4 I learned several convergence or divergence tests to test these series depending on the series to know exactly what value it converges to.
using a constant, which requires a different equation based on the precision you are looking for should immediately dismiss this formula as a use-case.....that's like saying "pi = 4 * lambda" as long as you have the constant lambda precalculated ahead of time, where lambda equals pi/4.......it introduces another transcendental value, making it pointless. But in this case, it has a huge amount of calculation surrounding it to hide the fact that you need a precalculated transcendental value to identify the EXACT value of yet another transcendental. And saying lambda can be used to find different precisions of pi, well you could do that with integers.... 39916649/12705864 gives you 10 decimals of precision. Since there are infinite number of integers, you can find any set that will give you the precision you are looking for, might not be practical but it makes more sense than solving for lambda.
You don't solve for lambda in the newly discovered formula. You chose any lambda you want provided it's larger than -1, say chose lambda = 10 and you replace lambda by 10 in the formula and you get a formula that converges to pi. Replace lambda by 100 and you get another formula that converges to pi.
To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/EllieSleightholm. You’ll also get 20% off an annual premium subscription!
You claim to be in the the field of Applied Maths but I don't think there is any difference between pure and applied. It's beacause I see pure Maths as Applied Set theory.
you are clueless . In math Anything can be anything. not a good video in opinion
I rather see why its 3.14. we know length is length. its absolute. circumference is a fixed number.
Why in the exam they don't give us a formal sheet yet in high school they did 😒
I also find the Ramanujan one here beautiful especially because when you first look at it it just seems like random numbers but there is a deeper logic to it.
Note, at 9:27, the n values should be minus 1 and the expression on the righthandside should be multiplied by 4. What I meant to write was:
n = 1: 4(1 - 1/3) ~= 2.66
n = 2: 4(1 - 1/3 + 1/5) ~= 3.42
n = 3: 4(1 - 1/3 + 1/5 - 1/7) ~= 2.89
Also, as promised in the video, the link to my Ramanujan video: ua-cam.com/video/PGJubCoH5as/v-deo.html
yes it's approaching towards pi
n=29 : ~=3.11010
n=30: ~=3.17161
Will you make a video about how they found the formula and how it ties to their research?
By calculating the difference between pi and partial sums for n up to N, you can produce a 2D surface plot showing how the error varies with N and lambda. You can then investigate the pair of values that give the minimum error. You should investigate putting lambda = pi for fun.
Ellie, thank you for recognising, Madhava. However, you missed, Aryabhatta, an extraordinary astronomer and mathematician from the 5th century, whi is often referred to as the “Father of Indian Mathematics.” His seminal work, Aryabhatiya, introduced the concept of zero, decimal notation, and a precise approximation of ‘π’.
I hope this helps.
Yeah that 3.1416 one right?
Cant believe pi is so conplex for a simple sign lol
I love seeing successive approximations to pi using various formulae. Thanks for doing this for your loyal fans.
4:15 I say the same thing to my students whenever I have to draw a polygon or polyhedra
Loll I said the same thing to my teacher and got cooked over apple wood low and slow
8:41 woah i never understood the sigma sign and how the sum thingy works now i know thx
What a great loss to mathematics it was for Cambridge University to not fund Ramanujan's treatment for Tuberculosis so much more!
What blinkered short-sightedness Cambridge University had regarding Ramanujan!
Ramanujan: "I have this function that solves [this problem]!"
Uni: "But you don't have any qualifications, so it can't be any good!"
Ramanujan: "But it works!"
Uni: "Can you prove it?"
Ramanujan: "I'm not good at that part, But it works!"
Uni: "We can''t cope with people who can't think in our box!"
... :(
Fascinating presentation, thanks for the effort, Ellie.
Euler's formula for π , of absolutely no value to compute it, but the single most important application of π in higher mathematics:
I kind of hoped YT would take the LaTeX output from my AI, but it does not
$$ \pi = - \sqrt{-1} \mathrm{Log}_e {-1} $$
For non nerds, it is
π = - sqrt( -1) Ln( -1)
Small nitpick: I know it would've taken a bit more time, but I would've loved a few matlab or maple numerical computations where you demonstrated what happens when you put in different values of lambda for different truncations.
Otherwise, I'm glad you covered this piece of good news! That's always welcome and a good effort.
You have a flair for communicating maths; really clear the way you make it understandable. You def could write excellent maths textbooks in the future imo.
I would have been genuinely delighted had the Scientists been : Cos(ha) and Sin(ha) 😢
applying to Part III maths soon, you're the person who convinced me it's possible to get in if you don't go to like a top 10
I'd be interested to see the pattern to these minima in lambda plotted over N and lambda. Are there any super minima that give unusually good approximations?
You have forgotten Aryabhatta maths astronomer who calculated Pi up to 3. up to 32 decimal points in 5th century.
Madhava,also inventer of calculus did it in15th century.
This could be a bit of smoke. Seeing that large lambdas appear to give better results (provided you add sufficient terms), on a hunch I decided to take the limit on lambda -> infinity, thus removing the terms with lambda in the denominator; the result simplifies to the old Taylor series for 4 * arctan 1 = pi.
P.S.: Ah, never mind, my bad. They recognize this fact in the paper, but comment that certain values of lambda allow the series to converge much faster (the Taylor series for arctan is notoriously slow to converge).
5:35 or 355/113
It's not just another formula for PI. It is actually giving infinite number of formulas for PI
É realmente fascinante como o trabalho de Ramanujan transmite essa sensação de mistério e beleza. Suas fórmulas e identidades muitas vezes parecem surgir do nada, com sequências numéricas e relações que parecem aleatórias à primeira vista. No entanto, por trás dessas aparentes “coincidências” está uma lógica profundamente enraizada em conceitos matemáticos avançados.
Some people just couldn't grasp the possibility of getting negative result from the latter part of the equation after 4. And these people vote.
That formula was discovered by Madhava of Sangramgama if Kerala School of Mathematics in early sixteenth century. 😊
I'm an idiot but I am curious...
If lambda can be any value, what happens to the approximation when it is equal to -n?
n varies between all the natural numbers from 1 to infinity; Since lambda is a constant, it can't be -n
Well, I get the spirit you are asking in. Lambda is any value, and that's true, but there are still limitations from where you can choose to take it from, but first let's discuss why lambda cannot be -n: because lambda, in context of this summation, does not changes, i.e the sum sees it as a fixed value. Kind of like you can change all your tires in your car, but you don't really do that when your car is running, otherwise you'd get unexpected results.
But there can be interesting prospects as well! Lambda cannot be any negative integer, because it will cause undefined output at some point in the iteration. From the equation, you can choose real positive number (that's still infinitely many), also keeping it mind that it does not blows up to infinity (i.e converges when n->infinity)
pi being pi I'm sure a lot more formulas will turn up.
It is not "pie" but "pi"!
I've seen (and used) 22/7 when doing simple impedance or phase calcs in AC electrical/electronic work. 'A' level stuff for any Brits reading.
Fascinating topic and a nice explanation. Curious,, what language do you plan to use to code this in? Will you use a functional programming language? Does your choice pay particular attention to execution performance?
I'm no expert but I think you made a small error in your explanation for the Pochhammer notation.
According to Wikipedia xₙ is a falling factorial:
(x)ₙ = x(x-1)(x-2)...(x-n+1)
The definition you wrote is for a rising factorial:
x⁽ⁿ⁾ = x(x+1)(x+2)...(x+n-1)
In both cases I immediately thought that this was shorthand for the division of factorials.
(x)ₙ = x(x-1)(x-2)...(x-n+1)
(5)₃ = 5 x 4 x 3
(5)₃ = 3 x 4 x 5
(5)₃ = (1 x 2 x 3 x 4 x 5) / (1 x 2)
(5)₃ = 5!/2!
In general:
(x)ₙ = x! / (x-n)!
x⁽ⁿ⁾ = (m+n-1)! / (m-1)!
I wasn't reviewing your work btw. I was just checking if my intuition about factorials was correct.
great explainer. Your explanation style is really simple and easy to follow. I have one request, as I understand your previous job involved working at the intersection of physics and comp sci. can you make a video or two about career opportunities that lie in that intersection and some advice for someone who dreams of working in that area? It would be much appreciated 🙏. Either way, thanks for the amazing work you do!!
Archimedes nearly discovered the limit.
355/113
Is -1/12 going to show up somewhere?
It's pronounced Mah-dha-vah and Ra-mah-nujan. Three syllables each.
Great video! I’m glad it showed up on my feed.
Can anyone tell the name of the app used for notes in this video, please?
Beautifully explained! subscribed.
please new vid showing some calculations using the eqn. thanks for the vid, cheers
Pls explain green integral
HI thank you very much for such a nice explanation, but which program do you use to write your notes the you are using in this video ? can you please provide med the name of the app ?
Once again a brilliant treatise on something which is of no importance, the present values calculated for pi are perfectly adequate. For your next video consider reviving the Middle Ages debate on how many angels can dance on the head of a pin!
Petition to change the term 'formula' to 'recipe' when it's referring to pi
Interesting take
can u show how gamma function and beta function come from and their relation
Pi =2+arctan(-tan2)
Just so you know, "saha and sinha", is indian for sine and cosine.
Casio Rules 🎉🎉😂. At least in the Engineering Universe. 😂🎉 I love engineering maths🎉🎉
100,000 years and 20 billion brains later one brain- Mark McCutcheon:” The Final Theory: Rethinking Our Scientific Legacy “, - discovered/published the CAUSE of gravity, electricity, magnetism, light and well....everything. Genius level event.
IISC ❤❤❤✌✌✌🇮🇳🇮🇳🇮🇳
Voce tem alguns canais sobre matematica que possa indicar? obrigado
@EllieSleightholm Can you please make a video tutorial about how to take effective math notes step by step guide in a Notebook for any math Topics from basic topics such as Arithmetic to advanced Algebra. I think it would help alout of people. If you have time ofc and Btw ❤ your channel❤
That looks like a Ramanuyan formula 😀
You said "Field Theory 'Experiments'...." :)
You should react to find y and his video on "imaginary numbers explained for gen z and his video on string theory
Your mannerisms remind me of Alice Roberts, the British documentary host and presenter.
I do not understand this formula, the 4 + or 4 -- is to big or to small , I think . saludos
In Calculus 4 I learned several convergence or divergence tests to test these series depending on the series to know exactly what value it converges to.
2:39 "then we know through mathematics...", how this lady ever got a math degree is beyond me
Can u pls elaborate
olá, tenho interesse nas novidades da matematica.
Feeling proud Indian army❤
সেই বাঙালি ভারতের বিজ্ঞানী । দুর্দান্ত।
Wait wondering released an hour ago but ur comments is 4 days ago ;-;
using a constant, which requires a different equation based on the precision you are looking for should immediately dismiss this formula as a use-case.....that's like saying
"pi = 4 * lambda"
as long as you have the constant lambda precalculated ahead of time, where lambda equals pi/4.......it introduces another transcendental value, making it pointless. But in this case, it has a huge amount of calculation surrounding it to hide the fact that you need a precalculated transcendental value to identify the EXACT value of yet another transcendental.
And saying lambda can be used to find different precisions of pi, well you could do that with integers.... 39916649/12705864 gives you 10 decimals of precision. Since there are infinite number of integers, you can find any set that will give you the precision you are looking for, might not be practical but it makes more sense than solving for lambda.
You don't solve for lambda in the newly discovered formula. You chose any lambda you want provided it's larger than -1, say chose lambda = 10 and you replace lambda by 10 in the formula and you get a formula that converges to pi. Replace lambda by 100 and you get another formula that converges to pi.
@@fplancke3336 only to a certain precision
image this.
let's put lambda = pi xD.
legendary
cool. saludos
Pi = 3 😛🤪😜😝😅☺️
Veritasium claims Newton is better.
ua-cam.com/video/gMlf1ELvRzc/v-deo.html
🙆♀️
i hate qc
You are looking gorgeous 😊