Math in the Simpsons: Apu's paradox
2015 ж. 1 Қаз.
2 078 706 Рет қаралды
Apu attends a math lecture at MIT which features an amazing calculation. Apu does not get it and neither did 99.99% of the people watching the episode. In this video the Mathologer tells you everything nobody ever wanted to know about this mystery blackboard.
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy :)
One of my favourite gag in Simpsons was an infinite recursion loop. Homer says : "Listen, son, everything I have told is a lie… except for this… and this… and this, and this…"
That's Family Guy
@@stevethea5250 Well that's a tangent and a half.
What happened here lmao
@@stevethea5250 wtf do you mean?
@@Archimedes.5000 I'm from the UK and I have been watching it all unfold since Tuesday. I was holidaying in Florida both in 2016 and 2020 when you were fighting the primaries. You seemed to speak so much sense, I couldn't believe that you weren't chosen as the Democrat candidate. It looks like Joe is going to win which is great but how I would have loved to have seen you as President.
I have no idea what I just learned.
It's pretty basic m80...
The Skeptical Nerd not when you're half asleep.
Julian Swayze Why watch maths videos when you're half asleep?
The Skeptical Nerd It was the last thing I watched before going to sleep.
Not if youre 14
His laugh is so enjoyably real. He just loves maths.
Rob Bozzo mathematics is singular not plural.....so it is math.....
nice job ruining everything
stapuft There’s a nice little thing called British English. Not sure if you’ve heard of it or not.
Yeah, thank god we Americans got it right for you.
Cryosardonic there is also a thing called Latin, which is where mathematics comes from, and in Latin ending a word in I makes it plural, further proof that the Brits keep getting English wrong.
I never really could grasp how the harmonic series was divergent, it just seemed so counter-intuitive when I first heard about it. The proof at 10:15 is so obvious that it instantly clears up all doubts. Thanks for the video.
+Cubik It's definitely a beauty :)
+Mathologer But, if you simplify the second sum to 1/2 + 1/2 ... then at every step it's bigger than the above sum. A = 1/1 + 1/2 + 1/3 + 1/4 ... B = 1/1 + 1/2 + 1/4 + 1/4 ... C = 1/1 + 1/2 + 1/2 + 1/2 ... C = B C < A A > B Therefore both C - A > B and C - A < B are true. Isn't this a contradiction? Or are you not allowed to simplify the sum formally? If that is true then what happens when you do C - B, it's Infinity - Infinity, but one of the Infinity is clearly going to be bigger than the other.
+Magic Gonads Infinity isn't a real value, so using (in-)equality signs with infinite (A, B and C are all 'infinite') isn't really defined.
+Magic Gonads Looking at partial sums, C (derived from B's partial sum) will always have a lot less summands, therefore they are equal in that respect.
giusten In that case, how do you describe A > B? As that was the justification I was questioning.
I'm only watching this because everything else on my feed is "top ten" lists and anti-feminism rants...
hahahah just as mine Leafys here and watchmojo etc... xd
Not top ten, BUT SAMEEEE
i know that feel
Same xD
stop watching bullshit and listening to bullshit right-wing propaganda then
I like how how you use the Limit Comparison Test at the end without actually mentioning it. Math has a lot of fancy names that put off students in the first place, However, making the student see the problem is always better than giving them a tool they do not understand.
👏🏾👏🏾👏🏾👏🏾
It's actually a direct comparison test.
I did well in calculus until I couldn't understand the concepts any more. At that point the students who just memorized the tools passed me. Unfortunately, with a full course load there isn't enough time for most students to properly learn everything.
*Calc II flashbacks*
@@jonhulka Same, I'm in Computer Science now and I would recommend a course in physics ;)
As soon as I heard Flanders say "Infinity plus one" I knew where you were going.
This channel is absolutely great!
The mistake is that the harmonic series is divergent and you can't substract it like that
Yeah you are not allowed to split the sums cause they wäre divergent
yes that was wrong since the beginning
@@dragonflyerstern156 divergente series
You are divergent!
Not to mention the sum is equal to infinity
Great clip! My inner geek is giving you two thumbs up.
It's always a pleasure to watch your videos.
+Steve French Glad you are still enjoying the videos. Actually, I'll have to check out your burr puzzle videos soon because I just managed to lose one of the six pieces of the burr puzzle I showed in the contortionist cubes video.
+Mathologer I have a proposal concerning my burr puzzles that may be of interest you. Since it's much easier for back and forth correspondence I'd rather email you. Would you mind messaging me your email address? Thanks.
+Steve French On second thought, I'll just message you here to let you know what I have on mind. I'll include my email address in case you want to discuss it further.
That's sarcasm if I've ever seen one.
Thanks for these vids. I absolutely loved learning about infinite sums and recursion sequences in college.
+GameMisconduct Great, more of this coming up very soon :)
Really nice video, when the realization hit what you were going for in the last part with the powers of 2, I actually laughed out loud. It really is a thing of beauty, and it makes me wish I had more like this to make me appreciate the beauty in maths back when I was in school.
It's great that you can see the beauty in a proof like this. Not many people can :)
Great video, I found myself referring back to my days in calculus as you went through the problem without even realizing it. Calculus always fascinated me. I was never really good at it, but at the same time I realized how calculus has influenced almost anything significant that had been invented by humans in the modern era. I wish I was able to feel stoked about math.
So fun! Thanks for making this video!
Cool channel. Makes me smile. Your way of explaining math makes me feel like I really get it. So thanks :)
Did anyone else want him to end with "thank you come again"
I thought you weren’t allowed to group together terms in an infinite sum that diverges.
That’s if you are trying to find the actual value of the infinite series. He is trying to find the lower bound of an infinite series that appears to approach infinity. That is entirely okay. If it had converged or gone towards a certain value it wouldn’t have proven or disproven anything, but it would have put the approach to infinity into question. It did however approach infinity. Therefor the infinite series that is larger does as well.
Really enjoyed this, helped me see the philosophical aspect of mathematics that other teachers had never exposed me to before. Thank you!
Please do lots of more of these! ❤️
I always wondered how you accurately point things out on the left screen, but I'm guessing you just project it normally but wash it out with high brightness, hence all the videos are on a bright white background. true?
Oh man, good noticing. That's a really good question.
they project ghosty image via projector for him and then render in actual pictures into video
betabenja when his arm goes up, it goes paler and he is holding a remote for moving a slide show on
Thats literally what i was thinking about the entire video and paid zero attention to whatever jiberish he was talking about. Im def too dumb for this side of youtube
You should watch the weather channel sometime. You'll really be amazed...
The sum of 1/n^3 series is called in the name of Apu - Apury's constant
@@stevethea5250 reported for spam
@@screamsinrussian5773 🚒🚒🚒 Washed my face, so you have a clean place to sit 🚨🚨🚨
@@stevethea5250 ok fartsniffer
@@screamsinrussian5773 💨🐽
Beautiful video, subscribed!
I watched this episode just to relax but when you proved that sum to go to infinity you helped me to understand my recent university math problem :) TY
in the proof at the end that the series adds up to infinity: if you subtract the second series from the 1/x series, does that converge to a finite number?
3 years later... No, it also diverges, because each lot of n/2 terms of the harmonic series - from which you substract 1/2 when you align the terms with those of the series used for comparison - tends towards ln(2), so the difference of these "lots" is not decreasing but actually ever so slowly increasing and tending towards ln(2) - 1/2. Basically you keep on adding 0.19 forever, but needing twice as many terms each time.
A bit of wit from a party I was at. One of the guest was trying to explain the idea of infinity. "Take all the numbers between 0 and infinity. Each of those numbers can also be sub-divided into an infinite number of fractions. half, third quarter and so on up to 1 over infinity, and then one and a half, one and third and so on again. So infinity itself can be sub dived even more times than there are whole numbers between one and inifinity..." or something to that effect. My response, "Mathematically, that is correct. However you're much better at math than you are grammar. "What is that supposed to mean", he asks. "You were never taught in grammar class that you shouldn't split infinities?" Making academics actually laugh with a pun is a special feeling.
You can't successfully divide from infinity, though. It's undefined. It's like trying to divide 7 from x and saying x can be subdivided even more times than there are whole numbers between one and x. Can't back that claim.
Super Giant hes poorly stating zenos dichotomy paradox. its 1 not infinity.
Whoops. I should've recognized that because I've already heard of the 'paradox', haha.
I'm a professor of chemistry and I would've rolled my eyes. Fuckin' tools
This isn't even mathematically correct... The set of rationals is countable, and therefore can be placed into one-to-one correspondence with the set of natural numbers. You can count it like so: 1/1->2/1->1/2->1/3->2/2->3/1->4/1->3/2->2/3->1/4->1/5->2/4->... . . . . . . . . . . . . 1 2 3 4 5 6 7 8 9 10 11 12 ... Such a method produces every single rational number, and it is clear there's a bijection between the two sets, so the two infinities have the same cardinality.
Great stuff. Keep presenting please. So good that you are in Melbourne.
Love this channel. The hist is great and the info is clear and concise. brilliant
+Paul O'Sullivan Thank you very much for the compliment :)
Mathologer Your welcome I wasnt actually expecting you to see it personally! Apologies for the spelling error, it would seem my phone believes I am more likely to write hist than host???
I understood everything except why one of the 1/8 was out of alignment.
.
mow184 What do you mean by “out of alignment”?
blondaibonsai one of the “1/8” was slightly misaligned with the rest of the numbers on the screen :)
Bruh lol
@@mow184 stamp
Isn't that the harmonic series?
+Anthony Bachour Yes.
+Anthony Bachour I am actually kicking myself for not including a Futurama reference in this video. Just remembered that in the episode Benderama they also featured the harmonic series.
+Mathologer Again and again: Thank you for your videos Herr Professor :D
futurama has lots of great references!
You have a great Channel and explain things very well
this was beautiful! thanks. :)
Glad you liked this one and thank you very much for saying so :)
There are a couple of very nice papers which list many proofs of the divergence of the harmonic series, the first one by Steven J. Kifowitz and Terra A. Stamps titled "The Harmonic Series Diverges Again and Again", and the second one only by Kifowitz titled "More Proofs of Divergence of the Harmonic Series".
+MyOverflow Thanks for that, I just had a look, very nice collections. I found them here scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf and here stevekifowit.com/pubs/harm2.pdf
i didnt understand anything but great video
If you don't understand something just ask :) There are lots of very smart and friendly people roaming these comment sections who are happy to answer questions.
Why towards the end did you replace the sum of fractions with repeated ones like 1/4th 2 times and 1/8th 4 times? How is that = to the sequence above? I just don't see how this helps the proof on the infinite sum on top.
Have another look. It's an inequality that we are producing at this spot not an equality. The new sum is clearly smaller than the one we are interested in and it's really easy to see that it "adds" to infinity. This implies that the sum we are interested also "adds" to infinity. :)
like if you read this explanation with that guy's voice 😅
+Mathologer how can u conclude that what was written on the blackboard was wrong?
How have I only just found this channel!? Subscribed!
what a great way to show the divergence of the harmonic series and to intuitively show why p-series converge and diverge for different values of p. This should be a calc 2 video for students!
also 1/1 + 1/2 + 1/3 + 1/4... - (1/1 - 1/2 + 1/3 - 1/4...) = 2/2 + 2/4 + 2/6 + 2/8... which is just the original sum
Ля, чувак, мб потому что "1/1 - 1/2 + 1/3 - 1/4" - сходящаяся последовательность? Кури матчасть.
@@angelmendez-rivera351 во во братан, и я о том. Та сходящаяся последовательность стремится к ln2. А х->оо, поэтому ln2 не играет роли
Wasn't Apu a Math Genius in season 7? And he got kicked out of college...
That's basically where the clip in this video comes from :)
I mean, why did he got kicked out from college? He was a Genius in season 7 (that clip is from season 25 I think)
Also, he has a PhD in computer science. I think.
MIT is hard IDK it's a cartoon
Maybe he got overconfident and didn't study? Hence why he's so reluctant to take a break
Was just learning about this in Calc 2 today, what an awesome video.
it's the first time i find a video on maths on youtube, but i'm loving it!
It is funny, this video coincides with the assignment sheet we go today in our math Pre-course, in which was wrongly stated that the sum from 1 to infinity of (1/x)= 2 by our instructor.
You should present this proof to him then. Maybe you'll get some extra credit :P
+marksmod that is the sum of 1/2^x from 0 to infinity
i never knew you could pull something like that in math! that was amazing! xD
Glad this one worked for you :)
Great channel.
Great fun video! Love your laugh 😄
Since 1:05 I had no idea what he was talking about...
You'll learn about it when you take Algebra 2 or Precalc I think ..
Derek Tong hell no this nonsense is straight up calculus i should know even though it was a miracle i passed that class
JimboZ90210 I had no idea what he was saying since 0:00
exactly what I thought
Thats calculus 2. If you want to learn it, search up "Series Calculus" or something
This one, And that one, And this guy here, And this thing over here, Equals this... Me: 🤕
Im on pc and its says Me: [white box]
Translates to " Me: :( "
Emojipedia says it's a "Face With Head-Bandage Emoji" :-B
Literally the worst proof I have ever seen...
...love this comment
Need to watch this episode a few times I think :)
+Anthony Cook Definitely worth it :)
so it's a matter of orders of magnitude, since 1
Calculates how much Homer has drunk directly from Apu's Slurpee machine...
I couldn't help but laugh at the Simpson versus Flanders debate. "Afraid not." "Afraid so!" "Afraid not, infinity." "Afraid so, infinity plus one!" Then the signature d'oh!
+Pika250 what episode and season please!
+Taras Pokalchuk Season 2 episode 6, Dead Putting Society.
All Homer had to say was "Afraid not, infinity plus x!", where x > 1. Of course, Flanders could come back with "Afraid so, infinity plus y!", where y > x. This then could and (knowing these two) would go on for infinity, since Homer would always give a "x > y" statement and Flanders would always give a "y > x" statement.
ThornHailsnap almost. You're forgetting that it could go on for infinity...plus...one.
8:00 this doesn't necessarily imply that this sum goes to infinity, it only shows that it diverges. If you keep in mind, however, that every summand is positive, we arrive at the result.
And again, this went far over my head... Not sure why I'm subscribed to this channel. All I can say is that I find entertaining.
3:28 ...but then how can it happen that the infinite sum of all natural numbers is a negative number? *(runs for his life)*
It isn't. Don't confuse the actual sum of all natural numbers with the analytic continuation of the riemann zeta function.
Not Broihon that’s why he ran
Lol
*Numberphile Noises*
You should have your hands washed with soap and water for writing that.
*My thoughts while watching the video-* *Ohh....So That being there meant that this should be here and That's how that was done....uh...okaay..*
Very good one, thanks.
Thank you, come again!
the moment you realised that you learned that stuff an knew it already :D
It was driving me so crazy, I knew the intro chords was from a kanye song. I just couldn't put my finger on witch, even knew what'd follow after. IT'S BOUND 2, THE Intro. God damn I was about to go crazy.
Although they're not even the same chords...
It's quite similar to the first chords of Babooshka by Kate Bush
It's just three chords it's probably in hundreds of songs lol
This video resumes why I love Mathologer so much
This guy is the best lecturer... so glad to have had him.
The error occurs on the first line of the Simpson's chalk board. 1/x does not equal 1/(2x-1) + 1/(2x). It's a misuse of infinity to equate the two sums. This leads to the later contradiction within the manipulation.
No, it's not a term-by-term identity. Both the left and right sides are concise ways of noting infinite series. Maybe write down the beginning parts of both series and then check that everything that you see on the left eventually shows up on the right and vice versa. Hope this helps :)
Thanks for the reply but 1/(2x-1) + 1/(2x) diverges faster than 1/x. Is it not a misuse of infinity to equate two different sums just because they are both divergent?
When the first sum is on x=4, the second sum is equal while only on x=2.
Charles15 The divergence doesn’t matter. The total sum matters. You get the same result from plugging in 1 and 2 into 1/x as plugging 1 into the second identity, and 3 and 4 into 1/x is the same as 2 in the second identity. When summing to infinity, it doesn’t matter how fast you get there, just the sum of values, where both will be equal to ln(2).
The real error occurs when associativity is used in any _argument_ re: ...∞, with the outcome(s) regarded as proof, or finished operation(s). Association pertains to an actual completed or completable sum of numbers. Otherwise all you have *is* an argument, a pseudo-paradox, a parlor-trick, sleight-of-calc, what have you.
This will come in handy the next time I mow the lawn.
I've got just the article for you (from my life before Mathologer :) www.qedcat.com/archive_cleaned/76.html
Truly this knowledge hast nay bounds.
:)
No, that just feels like it takes infinite time
I knew that equation from Calc. 1. Love your channel, I love math!!
I am happy I found this channel.
I am happy that you are happy :)
Mathologer! Finally getting out of the mathematical rock!
When you said “it’s infinite, it adds up to something but who knows” my mind was blown.
Is your intro/outro music the start menu music from one of the early simcity games?
beautiful explanation for the infinite sum. I particularly enjoyed the ramanujan video.
Cool :)
i see. the first infinity just grows slower than the bottom infinity, but they both will grow.
actually it grows faster... 1/6>1/8
at 4:00 it looks to me more like ln 2 is crossed out
+mrBorkD I think that is the point.
isn't this basically a telescoping series?
No, the terms of the series in this video do not cancel out as they do when you deal with telescoping series :)
+Mathologer thanks for replying. I learnt series back in may and it feels like I already forgot them :-( I better get reviewing . What about running convergence tests, would you use alternating series or ratio test?
Ratio test is inconclusive because you get a limit of 1, the alternating series test will tell you that 1-1/2+1/3+... converges. Having said that, in a calculus course the series in this video are usually dealt with before you hit the various convergence tests :)
+Mathologer Thank you. I enjoyed this video and got me thinking about calculus again, well done.
Absolutely fascinating. Calculus professor didn't go over the proof of why the harmonic series diverged but this makes it so clear (especially after having had experience with the comparison test). IF a small(er) infinite series diverges then the bigger series must also diverge.
No viewer got it because anyone with a brain doesn't watch current Simpsons episodes...
That's what I call very screwed up logic :) 1) The clips in this video are ages old. 2) Why would the "fact" that "anyone with a brain doesn't watch current Simpsons episodes" have any bearing on whether people "get this"?
Mathologer Man its a joke. Im saying the new Simpsons episodes are bad. A lot of people agree. (Although if you're tryin' to get technical with me, I dont get why you're putting the word fact and get this in quotation marks.)
Phew, you meant it as a joke. What a relief! Maybe just read some of the other comments on this and some of the other videos and you'll find a lot of comments very similar to yours that were not meant as jokes. I just don't know anymore what is what in this respect. I put "fact" in quotation marks because I was not sure whether the quoted bit is actually a fact (although it most likely is, I don't watch the Simpsons anymore) and I put "get it" in quotation marks because I am also quoting you there.
Mathologer Ok I see. Dont take every comment so seriously, its a place for half baked thoughts :P
+Mathologer the Joke Yo momma You
Fun =) Infinite series are definitely the bomb.
Nice :) Actually enjoyed maths this time.
Great, mission accomplished :)
I actually quite enjoyed this video :-)
Brilliant and entertaining !
pi is exactly 3
As we all know after Prof. Frink told us, right ? :)
I don't know what YOU are talking about sir, I came about this massively appropriate reference organically ;)
Can we please just re-jig maths a little to make that so? Much more elegant.
Check out this Simpsons clip: kzhead.info/sun/iZ1xo9OHsIONfpE/bejne.html
I was joking. I got it from that clip haha!!
This video is really quite.
Quite what?
DangerZone Sorry. I meant quiet, but that might actually have been my headphones being odd.
the suspense was killing me
The best comment here. you know there is an edit button for your own comment?
mmmm, yes, quite.
Thank you for the video. But it also seems in a way that the series converges. If you think about adding that series, the rate of increment of the sum decreases and after a point (like after 1 millionth may be) the increment is insignificant and thus negligible, telling us it stops at a finite sum. But on other hand, there is no wrong in the way you proved that its infinite. So which is correct? Are both correct? Or is it relative? Thanks in advance...
Being in third quarter calculus, having just gone over Sums and Series... I'm glad I could follow along and understand as he was saying it!
are you from germany ? :) you kinda have that german accent (i am from germany btw)
Yes, I did grow up and studied in Germany but I've been living in Australia for the past 20 years :)
+Mathologer you don't have any guest lectures lined up in Sydney do you? 😬
@MKMusic There's an accent there, but not strong at all. It is native-like English fluency with a mild Germanic color.
Ah that is why i unterstand you so well.
That's how i do maths There's something here, there's something there... Let us skip that part.. That thing is actually the same as here... Then there's another thing here and here...
It sounds like your proofs are a great read!
Famous equation I remember having an amazing time discovering in my 3rd year of secondary school in my maths class! I wish I could forget and be someone that didn't understand this so I can watch this video and experience that wonder of discovering this and thinking about infinity in this way. Calculus makes maths worth it. Shame in university it is only 1 out of 5 exams... In the UK at least, idk about foreigners.
i watched a lecture given by one of the simpsons writters in liverpool about 3 years ago, PLEASE dicuss more of the hidden maths in the simpsons, this was really fascinating! i'll keep an eye out for more like this.
Actually one of the videos that I am working on at the moment will feature the two references to Fermat's last theorem in Simpsons episodes :) Did you already check out all my other Simpsons and Futurama themed videos?
just this one and the futurama theorum. good stuff. I also watched the 4d rubix cube video, that was on another level(Ba dum tiss). i'll keep an eye out for any others.
Nice video but when he laughs I'm scared for my life haha
:)
I suppose, David X. Cohen, one of the producers, wrote the calculation. He has studied physics.
Yes, David X. Cohen is a likely suspect. However, there are a few others among the Simpsons writers who could be responsible for this one: mathsci2.appstate.edu/~sjg/simpsonsmath/degrees.html
Ich kenn dass seit 20 Jahren. Aber es ist immer wieder super. Danke Hr. Polster.
:)
The harmonic series is also a good example of using the integral test since 1/x is a monotonically decreasing function for positive values of x.
I love the Simpsons and I love math. This is heaven
Do you like Futurama too ?
Mathologer yess oh my god im actually in the middle of rewatching the entire series right now
Then you should definitely check out the video on the Futurama theorem :)
sweet will do cheers :)
sounds like a good tip.
There is no problem. The way math works is you can make anything out of anything as long as all parties present agree.
Used to watch your videos and be blown away. Now that I've taken calc 3 I just have flashbacks.
Well, in that case watch some of the more recent videos, like for example the last on on why pi is irrational :)
very nice video!! Got my subscription! Cheers from Brazil ;-)
I learned this in Calc. 2 though?
Shouldnt summation of 1/x till infinity tend to 2?
You are confusing this with the sum of the geometric series 1/2+1/4+1/8+1/16+ ... = 2 :)
Oh Yeah xD
Rohan Sharma no, cause the numbers are going down not up think backwards or the breakdown of one which will eventually equal zero or 0.9999999999999999999999999999999999999999999999999999 thus making it infinite because the nine carries over to the next set I could be wrong on the numbers but I get it...high school math was fun
1/2 +1/4 +1/8 +... adds to 1 actually. You guys are confusing it with a series which begins with a 1 and hence is 1 higher equating to 2.
Your videos are always good! it would be nice if you added english subtitles: it would be easier to understand for those who do not have English as mother tongue
I really like this guy's laugh, it's so genuine and adorable
Are you prof. Farnsworth?
Some people seem to think so :)
I only know algebra and understood this. Feelsgoodman.
beautiful, magnifico!
This is like the most basic analysis stuff and i still watch it because i love the way he explains stuff