I started to do it but once I hit the polynomial long division part I put my pencil down. I would rather watch paint dry than attempt to engage with an activity as tedious as polynomial long division.
The crazy thing is I literally knew all the math to solve the problem, but I don't think I would have come away with a correct answer had I attempted it myself.
@@alkinooskontopodias5919it's like a lego puzzle. When you open the box, you have all the pieces needed for the building. But without the istruction manual is pretty complicated. Especially if we're talking about 500+ pieces (or in this case, advanced math)
Not difficult, just typical JEE Advanced; very lengthy. Substituted tan for x and then some reduction formula. Btw I'm sitting for JEE Advanced this July
The thing about problems on the Putnam is they tend to, if solved correctly, be rather easy on paper. Finding the correct solution however tends to be so hard the median score is still around 0
yup , the poly division is kinda tedious , but usually in romania we kinda get those problems as the last one on the highschool (12 grade) graduation exam
My first instinct was to just expand and integrate that way, but I was like no thats too easy it probably needs a substitution of like cot^2 or some bs
This Problem is not created solely for Putnam held in 1968 instead Its origin is from research paper from 1944 by Dalzell .It has also appeared in university exam of sydney in 1960(Before Putnam).
I feel like an epic bait and switch has taken place here. 🧐 Thumbnail: “Test question: pi < 22/7” Me: “Oooh, oooh, I know that one!!” *clicks video* Video: “Solve for the integral of this 8-term polynomial division and use it to prove pi < 22/7” Me: “WAT”
Yep, felt the same thing. I was thinking, "This is easy; I know the repeating decimal pattern for 1/7", but the problem turned out to be something else entirely.
I'd argue that proving pi is less than 22/7 would have been a much much harder question - you can't take the decimal expansion of pi for granted, so you'd have to find a way to show how that's less than 22/7: which is exactly the integral does! After that pi < 22/7 is demonstrable at the level of a semester or two of calculus.
Looks at video writes pi out to 3 decimal places. Divides 22 by 7 and writes answer to 3 decimal places. Places < between answers for pie and 22/7. "I proved it right?" Foundation maths:" You are on the right track but you didn't show how you calculated pie so you only get 3 marks out of the total marks available." Video:" Not even close."
It's Putnam 1968 Problem. It's Difficulty is higher for people of that time and now It looks easy in present times. See JEE advanced(IIT JEE) Questions of around 1970, even they look like NCERT questions.
imagine if you don't know the decimal values of pi :p then you got this statement: 22/7 = about 3.14285 pi = about 3.????????? is 3.????? smaller than 3.14285? -> do the integral.
-Take the first three odd integers: 1,3,5 -Double them thusly: 113355 -Divide the last three by the first three thusly: 355/113 There ya go, Pi accurate to 6 decimal places!
With 355/113, the error is about 0.00000849136714% against Pi, that's more or less 1 ft every 2250 miles. That means you can calculate a circle the size of the Earth, with an difference of approx. 11.151 ft That's enough for human purposes. But if you want to go beyond, the fraction 103993/33102 is even more accurate, with a difference of 0.000000018395% with Pi. That's 1 ft every 1,050,000 miles. Yo can calculate a circle the size of the Solar System, about 69 trillion miles, with a difference of 12,700 miles.
@@plustwoplaylist5648 I've always wondered if there are people who see these terms and have their solution memorized (in this example integral 1/(1+x²) = tan-¹x). Because i think the math in this video isn't very complex but i would have to look up that term in an integration chart. If you know that from memory you must have a lot to do in that area...
People are better at different things! If you're not skilled in calculus and algebra, then I'm sure you're very good at something else. Don't let it devalue you
I regret having a difficulty in math and not speaking up to get more help when I needed it. I feel very lacking not knowing how to do “basic” algebra. One day I hope to be able to do calculus
I agree, every part of the solution was kinda basic and easy to understand. My problem would have been that I wouldn't know how to start if I would be given this problem.
Ya, it was obvious to me it was going to be arctan, I thought it could be Polynomial division, I checked if it was cause I couldn’t be bothered to do it and get it wrong, but ya it’s much easier than other Putnam questions
It's going to be different for everyone. I understand all the math and could easily do it myself. Having said that, it would probably take me a moment or two to realize the first few steps; I wonder how much time is given to answer the question; would probably take me about 3-5 minutes on a good day.
I'm rusty on my integrals resolutions, I'd definitively not find the tan^-1 part. I haven't done polynomial division in a while but that part isn't very difficult if you think about it and try it a few times.
355/113 is mentioned a couple of times, but it’s worth noting that if you used that fraction instead of Pi to calculate the circumference of the Earth, you would be off by only 11 feet. So you really could confidently use that instead of Pi, for nearly any practical application
Brilliant! I confess, I just did the substitution x=tan(u) and bashed it out the hard way using integrals of tan(u)^n, n=4 to 8 which conveniently MATLAB knows. This is far more elegant, one for the locker
if i wasn’t over a year removed from the calculus courses that taught me that integral work, everything about this problem i’ve already learned-very nice to see it worked out
Proving something Transcendental is way more tricky than proving it irrational because of this we have so many constants in mathematics that are still to be proven or disproven transcendental. But for pi and e we have so many proofs(some are short).
Short version: by some algebra black magic, we have hermite-lindemann theorem: if α is algebraic, then e^α is transcendental. Since e^πi=-1 which is not transcendental, πi is not algebraic, hence π is transcendental (cf. Appendix 1 of Algebra, S.Lang)
@@vedantthakur8947 The Discussion in comments is not about showing thumbnail Question. It is about transcendental numbers. Your logic is correct but Having an integral to show this result in itself makes it a beautiful proof.
A simpler idea is to use repeatedely the identity csc(2t)+cot(2t)=(1+cos(2t))/sin(2t)= 2cos^2(t)/(2sin(t) cos(t))= cot(t) . Setting t=pi, we can calculate sin(pi/2^n) and tan(pi/2^n) for any natural number n and then bound pi by 2^n*sin(pi/2^n)
I substituted x = tan(theta). Then I derived the recursive formula: tanⁿ(theta) + tanⁿ⁻²(theta) = tanⁿ⁻¹(theta)/(n-1) And used this appropriately to get the answer.
@@futilitariano Ah, I get it. Say if you have a number x, now you have to divide it by 5, so you go x/5. But now if you double it, it will become 2x, then you divide it by 10, which is 2 × 5. So basically what you are doing is mutiplying the numerator and denominator in x/5 by 2. (2x)/(5 × 2) = 2x/10. This is why you always get the same result.
@Sanjida Khan I think multiplying is easier than dividing? Also reduces the room for error as you just have to do one extremely simple multiplication and then shift the decimal one place to the left to get the answer. Doubling a number is relatively easier than dividing by 5.
@Sanjida Khan Depends on what I have to do. Can do 3 digits by 2 digits at most in my head. Just a disclaimer, I have not learnt any system of mental arithmetic yet. We are not allowed to use calculators in our exams and so it has just become a habit to calculate in our head. Or on paper if the numbers are too big.
Generally speaking, you start from the result and manipulate it in ways that are not totally obvious. Here I think they started from the continued fraction for pi or pi/4.
I was wondering why this was "a difficult problem" because it's literally the same as what I am doing in Calc 2 right now. Then I realized that the competitions are time based and I thought "Oh yeah, no I'm good."
Yep. its a combination of integration by parts and trigonometric substitution, followed by a logical deduction about the function to prove the point. For homework? sure. In three minutes? F**k that.
I was so happy when i figured out that fraction myself and then realised its well known 🤦 I dont recall if its less than or greater i think we should try to make 0
The integrand is on a closed interval. You could have used the closed interval. The integrand is zero at x = 0 and it is well defined, since we're not dividing by zero. Thereafter the integrand function is strictly positive and therefore we know the area under the curve is strictly positive and we can state that the area, which is 22/7 subtract pi, is strictly greater than zero and hence obtain the result.
The problem is that it is not well defined in the imaginary plane, since there is a singularity at x = +- i. You could, however, maybe have argued by the residue theorem, but we need more info which we are not given. Your statement does not hold as the integrand is divided by zero at x = +- i.
I'll be honest, this went straight over my head. I've just started to re-school myself in Maths. It's going to be a long journey but I'm hoping by the end of it (3 or 4 years) I'll be able to come back to this and actually know what the hell is going on!
There's no extra credit in a putnam problem, you get 0 through 10 points and it's very hard to get 10 even if you get it completely right because they'll find some tiny detail you didn't mention to justify one of the steps and deduct points even though it didn't affect anything.
"On the mountaintop, there is only one stream, one river." That's the translation you get if you say 3.1415 out loud in Shanghai dialect. 3 means mountain, 4 means water, 5 means river.
In Russian it is pretty simple: "Если очень постараться, то запомнишь всё как есть: три (3), четырнадцать (14), пятнадцать (15), девяносто два (92) и шесть (6)". "Do your best and you'll memorize it as it is: three, fourteen, fifteen, ninety-two and six." The trick is that the numbers rhyme in Russian.
I was out here spending 30 mins doing a u sub with tanx for that integral thinking there would be some neat trick and this man just factored out that numerator... I'm never getting those 30 mins back huh
Just the integral. This whole pi < 22/7 proof is not very common at math tests. Maybe if you are a Math studend yes, but engeneering and physics no I guess.
As far as I remember, the integral would have been solvable with my math knowledge I had in highschool. Every step of the process ringed a pretty loud bell. The conclusion after that may be a little outside the box thinking but is pretty straight forward, isn’t it? So, supposed you know the math of the Integral, is this problem realy that difficult?
Not too difficultdo you have enough guys to solve the whole bracket as x^4?? The question is very lengthy especially if you solve in a time constraint.
@Deathbychocolate They rarely ask anything beyond ncert in math boards. I'm a maths student. Its easy, but its not in ncert so this will be considered hard by a typical school student who isn't into entrance. I'm a jee aspirant and solved the original problem in one of the pyq archives.
@Deathbychocolate i guess you could classify it as hots. But let's be real here, most of the 'hots' are taught in schools and mugged up by students preparing for boards. The concepts are in ncert, but the exact question isn't, and it is not a standard board question that is regularly asked. I studied in a top CBSE school before my drop year and most of my friends who flaunted full marks in maths wouldn't have been able to solve this had this appeared for boards. Yes, it is definitely a joke compared to advanced or for someone who has done a decent preparation of jee. Now imagine if WE were supposed to make up an integral that would help prove the above inequality. That would be a REAL challenge for jee advanced.
@@ryanschardine8458 I agree. I'm sitting for JEE Advanced this year, and I could solve it in 5 minutes, even though I'm sure I won't clear JEE Advanced this year
@@sam08g16 I think you also read the words that I think I won't crack JEE Advanced this year. What leads you to believe, then, that I'm boasting? I'm highlighting the fact that even a smol bren person like me could solve it easily, so easy it was. Why so triggered? Chill, mate
@@kakalimukherjee3297 lol I don't think you are boasting, it was a legit question :) I don't know anyone who takes these kind of super hard maths tests and really admire the intellectual capacity of you guys
Solved it. The integral I = 22/7-pi (which you can determine by expanding and using long division, and the last term has an antiderivative of arctangent, which becomes pi/4 when it has an argument of 1) which means pi = 22/7 - I. I is a positive definite integral, since its only roots are x=0 and 1 in the integrand, while x anywhere in between leads to a positive integrand, therefore the integral over this region is positive. Therefore pi < 22/7.
It’s a common identity that you would be expected to know I suppose, Kahn academy has a video on finding the derivative of arctan(x), and the 4 is just taken out the front www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-4/v/derivative-inverse-tangent
My best fit (approximation)for "pi" (π) is 355/113. It is easy to make. Write the first 3 odd numbers in pairs, in a line: 113355. Draw a line in the middle: 113 | 355. Put the first half in denominator & the next half in the numerator, of the fraction: 355/113. The accuracy is: 2.667618 X 10exp(-7); about quarter of a millionth.
The "1 + x^2" in the denom tricked me into trying a tangent substitution from the start. I hit the expanded string of tangents and decided there must be a better way. Very nice solution!
I love complex-looking integrals that result in a simple answer :) The integral resulting in arctan took some looking-up, the expansion and long division was pretty straight forward. Only time-consuming. (Speaking as an mechanical engineer)
We can sub for x sq wrt to a trignometric function? Parameterise the d∅ interns of dX replace the limits. Another method is using Euler's identity We can also try by using the Taylor series expansion to prove the same.
Wonderful problem. After attempting a few substitutions and one trig attempt, I realized long division was in order. Thus I was able to get the result and knew the integral was obviously positive, thus the upper bound for pi was achieved. Thanks for this nice problem. My time was long about 30 minutes due to failed attempts with substitutions. The correct approach took 10 minutes. And, secant-squared was a dead give away for 1+tan^2 in the last denominator term..
You'll definitely learn this at uni, don't worry. Don't pay attention to peeps telling you that they were able to answer this straight from their mother's womb or something. We all have our own pace. Tbh, the concepts presented in the video were fairly basic. You just need to be very analytic. Long division, basic integration, and knowing the fact that a number squared is always positive. You know these but you're just not used to it. All you need is more exposure to get comfortable
I remember we had a substitute teacher in 5th grade who tried to convince us that 22/7 was the exact value of pi, that it was just a place holder symbol. We were all very confused and tried to explain what “irrational” meant, but to no avail
I'm skilled enough in maths to understand the working and methods used to reach the answer. I also know enough about maths to understand just how screwed I'd be if I ever had to solve this in an exam. So many rules and procedures to use like they're second nature, so many trigonometric identities to memorise. I know enough to know that I know nothing. Truly impressive stuff, glad I watched.
To be fair, its not hard at all. I was too lazy to open the 4th power thing which actually seems to be the most challenging thing here alongside the polinyum devision
@@shadesilverwing0 JEE is held annualy. JEE exams are of two types, JEE mains (an easier exam) and JEE Advanced (a harder exam made for admission into some of the most prestigious colleges in India). JEE asks questions from a wide syllabus from physics, maths and chemistry. There are approximately 108 (i think?) questions in JEEAdv which students have to attempt in 6 hours. You can probably see these questions by searching on google, and btw, questions are updated each year.
"Pause the video if you'd like to give this problem a try..."
No thanks, I am good.
I felt irony in that sentence.
🤣 KAMAEO
I excited that I cant recite first 30 digits of PI , thanks to my daughters, The Pi song! Lol.ASAP science i think
I needed to pause the video to give THE EXPLANATION a try.
I started to do it but once I hit the polynomial long division part I put my pencil down. I would rather watch paint dry than attempt to engage with an activity as tedious as polynomial long division.
The crazy thing is I literally knew all the math to solve the problem, but I don't think I would have come away with a correct answer had I attempted it myself.
Next time you could try to solve it, so that you don't have to speculate.
@@alkinooskontopodias5919it's like a lego puzzle. When you open the box, you have all the pieces needed for the building. But without the istruction manual is pretty complicated. Especially if we're talking about 500+ pieces (or in this case, advanced math)
@@Axem26 nice comment but a bit irrelevant with mine.
Bro JEE advance is one of the toughest exam in world/india.
Questions maden by IIT professors
Not difficult, just typical JEE Advanced; very lengthy. Substituted tan for x and then some reduction formula. Btw I'm sitting for JEE Advanced this July
Me, an engineer: "Nice, so I can use 22/7 instead of Pi"
lol
nah 3 is fine
@@ianthebubbian6182 3 is smaller than 22/7
Hence proved
@@ianthebubbian6182 5 is better
No, e is better
This turned out to be a lot more do-able than I thought it would be.
Kind of bummed I elected not to try it on my own.
I’m not bummed at all
The thing about problems on the Putnam is they tend to, if solved correctly, be rather easy on paper. Finding the correct solution however tends to be so hard the median score is still around 0
yup , the poly division is kinda tedious , but usually in romania we kinda get those problems as the last one on the highschool (12 grade) graduation exam
My first instinct was to just expand and integrate that way, but I was like no thats too easy it probably needs a substitution of like cot^2 or some bs
@@jjhhbb9 same. Lol. I was like that's not gonna work. It's Putnam. Just unpauuse.
I love how they managed to hide the derivative of arctan in that mess
Tbh when you see a "x²+1" in the denominator your arctan senses should be mad triggered
@@CiuccioeCorraz I remembered it was arc something but wasn't sure which one ! To my defense it's been a long time.
@@CiuccioeCorraz exactly, it wasn't that hard, when i was in cramming school this was a classic tip.
@@CiuccioeCorraz I'm taking calc 2 next semester and I appreciate this tip
Can you explain why acrtan is used?
Well, think about the person who came up with the idea to draft such a beautiful problem.
Yeah i would want to know how he apporached to build it
@@anandsuralkar2947 I'm pretty sure he is a half life demigod of maths living in the woods of a dense forest
@@anandsuralkar2947 and he probably used a spell to create that question straight from hell itself😂
This Problem is not created solely for Putnam held in 1968 instead Its origin is from research paper from 1944 by Dalzell .It has also appeared in university exam of sydney in 1960(Before Putnam).
@@parasbhardwaj3580 history in slaying people
I feel like an epic bait and switch has taken place here. 🧐
Thumbnail: “Test question: pi < 22/7”
Me: “Oooh, oooh, I know that one!!” *clicks video*
Video: “Solve for the integral of this 8-term polynomial division and use it to prove pi < 22/7”
Me: “WAT”
Yep, felt the same thing. I was thinking, "This is easy; I know the repeating decimal pattern for 1/7", but the problem turned out to be something else entirely.
I'd argue that proving pi is less than 22/7 would have been a much much harder question - you can't take the decimal expansion of pi for granted, so you'd have to find a way to show how that's less than 22/7: which is exactly the integral does! After that pi < 22/7 is demonstrable at the level of a semester or two of calculus.
But then again... this solution takes calculus for granted XD
Looks at video writes pi out to 3 decimal places. Divides 22 by 7 and writes answer to 3 decimal places. Places < between answers for pie and 22/7. "I proved it right?"
Foundation maths:" You are on the right track but you didn't show how you calculated pie so you only get 3 marks out of the total marks available."
Video:" Not even close."
Same with me all the time
In Putnam, you are given 18 minutes to solve this question, while in JEE advanced, you are given 3 minutes.
... and a multiple choice answer, from what I understand.
@@dlevi67 and negative 1 mark for wrong answer, so you have to double check (if u have time)
What are you saying?? This can't be even the simplest question of Putnam 😑
Jee advanced is tough but it can't be compared with Putnam
It's Putnam 1968 Problem. It's Difficulty is higher for people of that time and now It looks easy in present times. See JEE advanced(IIT JEE) Questions of around 1970, even they look like NCERT questions.
Me, an intellectual:
22/7=3.142857143 or 3.143
pi=3.141592654 or 3.142
therefore; pi
:(
Why bothering with that complicated stuff, if there's an easyer way😂
@@mr.gerberus5684 why dont you appreciate the elegant integral :(
@@enejidjsi5939 the only way that leads me to appreciating integral is that they stop using primitives.
Worst thing I had in math up till today
imagine if you don't know the decimal values of pi :p then you got this statement:
22/7 = about 3.14285
pi = about 3.?????????
is 3.????? smaller than 3.14285? -> do the integral.
-Take the first three odd integers: 1,3,5
-Double them thusly: 113355
-Divide the last three by the first three thusly: 355/113
There ya go, Pi accurate to 6 decimal places!
"thusly"
Pi equals to 3.55113????
@@mystey1 no
@@mystey1 ...
With 355/113, the error is about 0.00000849136714% against Pi, that's more or less 1 ft every 2250 miles.
That means you can calculate a circle the size of the Earth, with an difference of approx. 11.151 ft
That's enough for human purposes.
But if you want to go beyond, the fraction 103993/33102 is even more accurate, with a difference of 0.000000018395% with Pi. That's 1 ft every 1,050,000 miles. Yo can calculate a circle the size of the Solar System, about 69 trillion miles, with a difference of 12,700 miles.
Maths in school : 2+2=4
Maths in college : look! there's a number in here!
Yeah I hated when I got to college and had to start using numbers in math...
This is taught to us in school in India 12th grade
No bruh letters lmao full algebra
This is school level maths
Oh damn where lemme see 👀
If you didn't even think about pressing pause from 0:36 to 0:57 you know what silent shame feels like
I literally don't know how to do integrals. Pausing was never an option
integral 1/(1+x²) = tan-¹x
This isn't tricky this is standard equation but in this video he said it is tricky LoL
@@plustwoplaylist5648 it’s harder than the other integrations he was showing
@@plustwoplaylist5648 I've always wondered if there are people who see these terms and have their solution memorized (in this example integral 1/(1+x²) = tan-¹x). Because i think the math in this video isn't very complex but i would have to look up that term in an integration chart. If you know that from memory you must have a lot to do in that area...
I haven't thought to pause because I have seen harder problems in my CS studies.
Me who hasn’t done math since Grade 10:
TH-cam: Ah yes this will be perfect
Well, was it any good?
Me who is in grade 9th : Aight let me understand some calculus.
It's always interesting how very complex problems are resolved with them being broken down to simple math.
I'm four minutes in and still trying to figure out why I'm watching something that makes me feel about as intelligent as a potato.
People are better at different things! If you're not skilled in calculus and algebra, then I'm sure you're very good at something else. Don't let it devalue you
I regret having a difficulty in math and not speaking up to get more help when I needed it. I feel very lacking not knowing how to do “basic” algebra. One day I hope to be able to do calculus
It's not a matter of intelligence, just training. This level of math is easy if you keep studying it in college/university/w/e.
the problem itself is as useful as driving a potato
This problem is less useful than a potato in real life. I haven’t really use much of calculus after school for so many years.
Honestly quite straightforward if you can do polynomial division. Everything falls nicely into place without the need of any creative solutions.
I agree, every part of the solution was kinda basic and easy to understand. My problem would have been that I wouldn't know how to start if I would be given this problem.
Ya, it was obvious to me it was going to be arctan, I thought it could be Polynomial division, I checked if it was cause I couldn’t be bothered to do it and get it wrong, but ya it’s much easier than other Putnam questions
It's going to be different for everyone. I understand all the math and could easily do it myself. Having said that, it would probably take me a moment or two to realize the first few steps; I wonder how much time is given to answer the question; would probably take me about 3-5 minutes on a good day.
I'm rusty on my integrals resolutions, I'd definitively not find the tan^-1 part. I haven't done polynomial division in a while but that part isn't very difficult if you think about it and try it a few times.
@NumberisNaN this is probably a real old problem, some of the A1/B1s were really easy back then
This is such a beautiful answer
Yes. 22/7 > 🥧
There was an answer? I could barely find the question...
It's the most straight-forward way to answer the question anyway.
@@nahiyan8 Yeah, it is basically just the computation of an integral, I do not really see "beauty" in the solution at all.
@@chanlaoshi8634 it's the use of different types of calculations that suddenly leads you to the previously shown answer which people find satisfying.
"And that's the Answer", i like his tone than the actual answer..
Me knowing that pi is 3.1416
*"My time has come!"*
That isn't even correct.
@@ovoj5631 Well rounded value is 3.1416, 22/7 is 3.1428
so it is enough information to pass this question.
Rounded off to four decimal places, yes, but that's enough to solve this problem.
@@brownro214 ya it is
Perfectionist: 3.14159265
“Give this problem a tr-“
“I’m gonna stop you right there”
I am preparing for JEE and when he said it is one of the toughest exams in the world, I was like: IMMA SHOW THIS TO MY MOM
Good luck!
Of course it is
But you must be preparing for jee mains right?
Jee advance comes after it
CA student said hi
As student feels overwhelmed, do you want to study maths?
I was busy with substituting something for 2 mins then watch your solution and you just divide this straight way 😁. But Nice Results
Same I was doing
This hurts pretty bad
Only Legends know that today is 'π day'
And that too at 1:59 pm
Pi day - 3-14 🙃 JEE ADV OP
Well it's written on the top of description: #PiDay so...
3.14 at 1:59 p.m.
yep
355/113 is mentioned a couple of times, but it’s worth noting that if you used that fraction instead of Pi to calculate the circumference of the Earth, you would be off by only 11 feet. So you really could confidently use that instead of Pi, for nearly any practical application
Brilliant! I confess, I just did the substitution x=tan(u) and bashed it out the hard way using integrals of tan(u)^n, n=4 to 8 which conveniently MATLAB knows. This is far more elegant, one for the locker
You should familiarize yourself with the method of partial fractions, this is very similar to that.
Pause the video if you’d like to give this problem a try !
Me: forward 10 seconds 2 times
Nice question...I couldn't solve but got to know something new today...Thanks 😃😃
To integrate polynomials use the chain rule: a*(x^k) => a÷(k+1)^(k+1), where a is a real number and k is a natural number.
I get to know new things everyday
@@Jared7873 where is the result x in this chain rule? Are you sure you wrote that correctly?
@@Jared7873 nah. k will be any real number except 0. The power rule works for every such k as I mentioned.
Some steps here that I forgot half a century ago! Thanks Presh.
With respect....
Who are you sir??
I forgot those steps a few weeks ago😅
@@kaustubhdubey2805 zodiac the serial killer
When i saw the question it felt like :
"Your bus driver's age is 69, find the number of children in your school "
type question
69
no?
@@dhruvavikas1632 man of culture 👀
420
😂
if i wasn’t over a year removed from the calculus courses that taught me that integral work, everything about this problem i’ve already learned-very nice to see it worked out
feels so satisfying to understand that after starting university where they teach you all this!
learnt it in Grade 10
After starting uni? man why do we gotta study this in Grade 10 in India :')
This is the reason I won't give jee in my life hopefully
They are lying to look cool. They learnt it in 12th mostly still if you ask them some normal calculas, they will probably won't solve.
@@RohitKumar-vi3zt correct
Suggestion: prove pi is transcendental. I never had the will the research myself, but I think my short attention span can handle a 5 minute video :)
That's more like a 20 minute video
Proving something Transcendental is way more tricky than proving it irrational because of this we have so many constants in mathematics that are still to be proven or disproven transcendental. But for pi and e we have so many proofs(some are short).
Short version: by some algebra black magic, we have hermite-lindemann theorem: if α is algebraic, then e^α is transcendental. Since e^πi=-1 which is not transcendental, πi is not algebraic, hence π is transcendental (cf. Appendix 1 of Algebra, S.Lang)
Man that thumbnail question was very easy..
See pie original value is 3.14156 and 22/7 is actually 3.1428 something that's why it was greater
@@vedantthakur8947 The Discussion in comments is not about showing thumbnail Question. It is about transcendental numbers. Your logic is correct but Having an integral to show this result in itself makes it a beautiful proof.
"Pause the video if you'd like to give this problem a try"
I don't even know what is an integral
Edit: wow 400 likes
I have no idea what “pi” is
@@Yusuf_1992 It's a Greek letter that is used in maths and its value is ≃3.14 but it has infinite digits
It's also the result of the circumference of a circle divided by its diameter
I feel like this is an r/woooosh moment
@@aaronpatel3179 now that I think about it it could be 😂
A simpler idea is to use repeatedely the identity csc(2t)+cot(2t)=(1+cos(2t))/sin(2t)= 2cos^2(t)/(2sin(t) cos(t))= cot(t) . Setting t=pi, we can calculate sin(pi/2^n) and tan(pi/2^n) for any natural number n and then bound pi by 2^n*sin(pi/2^n)
I substituted x = tan(theta). Then I derived the recursive formula:
tanⁿ(theta) + tanⁿ⁻²(theta) = tanⁿ⁻¹(theta)/(n-1)
And used this appropriately to get the answer.
This is a nice question, how do people even create these questions. It blows my mind
@@futilitariano Ah, I get it. Say if you have a number x, now you have to divide it by 5, so you go x/5. But now if you double it, it will become 2x, then you divide it by 10, which is 2 × 5. So basically what you are doing is mutiplying the numerator and denominator in x/5 by 2. (2x)/(5 × 2) = 2x/10. This is why you always get the same result.
@Sanjida Khan I think multiplying is easier than dividing? Also reduces the room for error as you just have to do one extremely simple multiplication and then shift the decimal one place to the left to get the answer. Doubling a number is relatively easier than dividing by 5.
@Sanjida Khan Depends on what I have to do. Can do 3 digits by 2 digits at most in my head. Just a disclaimer, I have not learnt any system of mental arithmetic yet. We are not allowed to use calculators in our exams and so it has just become a habit to calculate in our head. Or on paper if the numbers are too big.
Generally speaking, you start from the result and manipulate it in ways that are not totally obvious. Here I think they started from the continued fraction for pi or pi/4.
I was wondering why this was "a difficult problem" because it's literally the same as what I am doing in Calc 2 right now. Then I realized that the competitions are time based and I thought "Oh yeah, no I'm good."
@Leroy092 same😂😂
hh same
Yep. its a combination of integration by parts and trigonometric substitution, followed by a logical deduction about the function to prove the point. For homework? sure. In three minutes? F**k that.
Now let's proof that π
Damn 😍
I was so happy when i figured out that fraction myself and then realised its well known 🤦
I dont recall if its less than or greater i think we should try to make 0
Here π =3.14159268
And 355/113 =3.14159292
So π< 355/113. 😀😀
@@yjklmnop You are so clever 🤣
@@yjklmnop BRUH, everybody knows that π =3.1415926535... - you were off more than 2 units in the last place !
Typo on the numpad?
The integrand is on a closed interval. You could have used the closed interval. The integrand is zero at x = 0 and it is well defined, since we're not dividing by zero. Thereafter the integrand function is strictly positive and therefore we know the area under the curve is strictly positive and we can state that the area, which is 22/7 subtract pi, is strictly greater than zero and hence obtain the result.
The problem is that it is not well defined in the imaginary plane, since there is a singularity at x = +- i. You could, however, maybe have argued by the residue theorem, but we need more info which we are not given.
Your statement does not hold as the integrand is divided by zero at x = +- i.
I'll be honest, this went straight over my head. I've just started to re-school myself in Maths. It's going to be a long journey but I'm hoping by the end of it (3 or 4 years) I'll be able to come back to this and actually know what the hell is going on!
Legendary exam and Legendary explanation 🔥🔥🔥
How the heck is JEE ADV legendary ?
@@random-td7tf cz it has some great questions which are very fun to solve
@@thecoolring6431 not really
It really has enjoyable questions if you can solve
@@random-td7tf u solve jee advance paper in 3 hours get it online lets see how much u score
For extra credit: Give a simple estimate of the value of the integral (less than 0.003) to improve the estimate of pi.
There's no extra credit in a putnam problem, you get 0 through 10 points and it's very hard to get 10 even if you get it completely right because they'll find some tiny detail you didn't mention to justify one of the steps and deduct points even though it didn't affect anything.
"On the mountaintop, there is only one stream, one river." That's the translation you get if you say 3.1415 out loud in Shanghai dialect. 3 means mountain, 4 means water, 5 means river.
A similar mnemonic exists in standard mandarin as well. "On the mountain (3) top ('.') are a (1) temple (4) and a (1) pot (5) of liquor (9)."
@@robbiechen3707 amazing, never heard of this
山，一寺一壶酒
In Russian it is pretty simple:
"Если очень постараться,
то запомнишь всё как есть:
три (3), четырнадцать (14), пятнадцать (15),
девяносто два (92) и шесть (6)".
"Do your best
and you'll memorize it as it is:
three, fourteen, fifteen,
ninety-two and six."
The trick is that the numbers rhyme in Russian.
Er I think 5 means pond rather than river
@@OW0974 guess that means a lake
I divided 22/7 and that was 3.1428 🤙
I was out here spending 30 mins doing a u sub with tanx for that integral thinking there would be some neat trick and this man just factored out that numerator...
I'm never getting those 30 mins back huh
Sometimes i regret not trying to solve the problem thinking it is too hard.
No regrets on this one.
When this is part of a normal math test at your school
Just the integral. This whole pi < 22/7 proof is not very common at math tests. Maybe if you are a Math studend yes, but engeneering and physics no I guess.
As far as I remember, the integral would have been solvable with my math knowledge I had in highschool. Every step of the process ringed a pretty loud bell.
The conclusion after that may be a little outside the box thinking but is pretty straight forward, isn’t it?
So, supposed you know the math of the Integral, is this problem realy that difficult?
@@earlgreen2030 Yes it can be in a test, that you have limited amount of time and is under pressure
they wont give you the integral for this question. You have to guess it by yourself.
@@monill2643 Well, of course. That's thr point
These videos make me feel so simpleminded.
Good thing I watched that at 2x. Otherwise, I'd be asking for 5 minutes of my life back.
This is the first time, I've been able to solve a question you posted a video about :D
This question has stole my ❤️
Questions is actually not that difficult but really teaches a lot!
Not too difficultdo you have enough guys to solve the whole bracket as x^4?? The question is very lengthy especially if you solve in a time constraint.
@@rishu_Kumar07 but these type of questions are quite common if you are preparing for jee adv. where time constraints are considerably less
@Deathbychocolate Its definitely harder than boards. Maybe a mains problem. Its easy, but one would be inclined to avoid it under time constraints.
@Deathbychocolate They rarely ask anything beyond ncert in math boards. I'm a maths student. Its easy, but its not in ncert so this will be considered hard by a typical school student who isn't into entrance. I'm a jee aspirant and solved the original problem in one of the pyq archives.
@Deathbychocolate i guess you could classify it as hots. But let's be real here, most of the 'hots' are taught in schools and mugged up by students preparing for boards. The concepts are in ncert, but the exact question isn't, and it is not a standard board question that is regularly asked. I studied in a top CBSE school before my drop year and most of my friends who flaunted full marks in maths wouldn't have been able to solve this had this appeared for boards. Yes, it is definitely a joke compared to advanced or for someone who has done a decent preparation of jee.
Now imagine if WE were supposed to make up an integral that would help prove the above inequality. That would be a REAL challenge for jee advanced.
If you tried working the question you're the best. Thank you Mind Your Decision for helping us.
A concise video explaining why I suffered in my college calculus classes.
This is the best channel!!!!!!!
It's oiling all my algebra and trig. Love how you did the derivatives!
To be honest, this seems a bit too easy for a Putnam exam.
It seriously does. I managed to solve it so it's definitely a little too easy.
@@ryanschardine8458 I agree. I'm sitting for JEE Advanced this year, and I could solve it in 5 minutes, even though I'm sure I won't clear JEE Advanced this year
@@kakalimukherjee3297 do you consider yourself some kind of genius or just very good at maths?
@@sam08g16 I think you also read the words that I think I won't crack JEE Advanced this year. What leads you to believe, then, that I'm boasting? I'm highlighting the fact that even a smol bren person like me could solve it easily, so easy it was. Why so triggered? Chill, mate
@@kakalimukherjee3297 lol I don't think you are boasting, it was a legit question :) I don't know anyone who takes these kind of super hard maths tests and really admire the intellectual capacity of you guys
Easy when you know how to do it. Never had seen polynomial long division like that before, but it makes sense immediately when you see it.
You should upload this video in exact 1:59 PM (GMT) as it is consistent to be π minute
Also 26 seconds
@@whothefisyash should've made this on 3/14/2015, 1:59:26
Solved it. The integral I = 22/7-pi (which you can determine by expanding and using long division, and the last term has an antiderivative of arctangent, which becomes pi/4 when it has an argument of 1) which means pi = 22/7 - I. I is a positive definite integral, since its only roots are x=0 and 1 in the integrand, while x anywhere in between leads to a positive integrand, therefore the integral over this region is positive. Therefore pi < 22/7.
Wow that’s just crazy! Thanks for showing this!
Can’t believe this was in jee advance , this is average problem in TMH about 22 years back.
Tmh?
@@enejidjsi5939 Tata McGraw hill
the only thing i didn't know in this vid was how to find the antiderivative of 4/(x^2+1)
It’s a common identity that you would be expected to know I suppose, Kahn academy has a video on finding the derivative of arctan(x), and the 4 is just taken out the front
www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-4/v/derivative-inverse-tangent
Master your engineering knowledge here👍👍👍
My best fit (approximation)for "pi" (π) is 355/113. It is easy to make.
Write the first 3 odd numbers in pairs, in a line: 113355.
Draw a line in the middle: 113 | 355.
Put the first half in denominator & the next half in the numerator, of the fraction: 355/113.
The accuracy is: 2.667618 X 10exp(-7); about quarter of a millionth.
You can get another inequality for the integral to get a lower bound on pi. x*(1-x) (22/7)*[2^8/(1+2^8)]
amazing how I have zero knowledge of calculus, yet i coulf understand every step. Great explainer
The "1 + x^2" in the denom tricked me into trying a tangent substitution from the start. I hit the expanded string of tangents and decided there must be a better way. Very nice solution!
I love complex-looking integrals that result in a simple answer :)
The integral resulting in arctan took some looking-up, the expansion and long division was pretty straight forward. Only time-consuming.
(Speaking as an mechanical engineer)
We can sub for x sq wrt to a trignometric function? Parameterise the d∅ interns of dX replace the limits.
Another method is using Euler's identity
We can also try by using the Taylor series expansion to prove the same.
Wonderful problem. After attempting a few substitutions and one trig attempt, I realized long division was in order. Thus I was able to get the result and knew the integral was obviously positive, thus the upper bound for pi was achieved. Thanks for this nice problem. My time was long about 30 minutes due to failed attempts with substitutions. The correct approach took 10 minutes. And, secant-squared was a dead give away for 1+tan^2 in the last denominator term..
Good to see it on the pi day
The annoying part is that I totally could’ve done that, I was just too lazy and watched the vid instead.
I am from India and preparing for JEE Adv. and even I felt this solution was just too good
Iam also preparing for Jee advance
Good luck my dear frnd❤️❤️❤️❤️❤️
I m here too...jee Aspirant
I'm glad I slept through calculus class, and I never needed it.
The moment you thought you were somewhat doing ok in maths preparing for university and then getting your illusion destroyed :D
You'll definitely learn this at uni, don't worry. Don't pay attention to peeps telling you that they were able to answer this straight from their mother's womb or something. We all have our own pace.
Tbh, the concepts presented in the video were fairly basic. You just need to be very analytic. Long division, basic integration, and knowing the fact that a number squared is always positive. You know these but you're just not used to it. All you need is more exposure to get comfortable
π day 3.14159....
14 march
1:59pm
26 seconds too
It's today?
Finally it was helpful to know 49 digits of the number Pi lmao
This was asked in JEE advanced 2010 Mathematics paper. Before that it had appeared in Putnam as well.
The second part can be solved very quickly via simple approximations, which one can trivially calculate mentally.
pi
There is a catch, you are NOT allowed to take the value of pi for granted
Also you strictly have to use the integral for this problem.
Saw the title, divided 22 by 7 in 3 seconds, said yep it’s bigger, watched the video, this guy did a lot of steps to come up with this answer
I am the one who solved it in JEE advanced. I remember.😉
Badhiya bhai
Isn't JEE Advanced an objective exam. But, this is a proving question how can this come in an objective exam
This solution was sweet 🐝
Happy Pi day :)
Congratulations to those who are first and got this comment
Oh thank you
Tq
Wow this is... "too" straight forward. I kept thinking what was the symmetry I was missing, since there is a symmetry of x and (1-x) for 0
If we know pi up to and including the third decimal place, then there is a way easier proof:
There is |e|
You lost me right after, "... in the world." However I already had divided 22 by 7 and knew it was greater than Pi, though less than pie.
Everyone it is greater. You needed to prove with the help of integration given
I remember we had a substitute teacher in 5th grade who tried to convince us that 22/7 was the exact value of pi, that it was just a place holder symbol. We were all very confused and tried to explain what “irrational” meant, but to no avail
There was a time when I could do that and have fun with it. I miss that time
I'm skilled enough in maths to understand the working and methods used to reach the answer.
I also know enough about maths to understand just how screwed I'd be if I ever had to solve this in an exam.
So many rules and procedures to use like they're second nature, so many trigonometric identities to memorise.
I know enough to know that I know nothing.
Truly impressive stuff, glad I watched.
To be fair, its not hard at all. I was too lazy to open the 4th power thing which actually seems to be the most challenging thing here alongside the polinyum devision
Yh I remember watching this video about a year or 2 ago and being baffled but now I realise it is easy but tedious
@@two697 Nice
5:24 Elon Musk? Coool
I also just noticed
Unbox therapy too
Bill Nye
This was a simple one in JEE ADVANCED. Solved with a blow
It was being asked in JEE advanced once indirectly
2010 iit jee
Simplest way to solve this - know the first few digits of pi and divide 22 by 7 using long division to bring both into decimals and compare those
The real hard part must have been coming up with that question.
I wanna advance my math skills until I'm fully able to solve the problem. Although I don't even know what tan really means, my determination is high.
tan is tangent from trigonometry.
You only need Calc 2 to solve this problem
@@genericyoutubeaccount579 oh okay
This is IIT-JEE 2010 question from paper-I.
How often do they update the exam questions?
@@shadesilverwing0 wdym
@@shadesilverwing0 JEE is held annualy. JEE exams are of two types, JEE mains (an easier exam) and JEE Advanced (a harder exam made for admission into some of the most prestigious colleges in India). JEE asks questions from a wide syllabus from physics, maths and chemistry. There are approximately 108 (i think?) questions in JEEAdv which students have to attempt in 6 hours. You can probably see these questions by searching on google, and btw, questions are updated each year.
This video helped me prove that 3
instead of using binomial expansion, I simply subtracted and added 1 from x^4 term in numerator. This makes the division much much easier!!
Me who knew all along that 22/7 was 3.1428 and π was 3.14159 😎😎