ความคิดเห็น •

  • @3blue1brown
    @3blue1brown 6 ปีที่แล้ว +4578

    Fantastic! One of the most accessible proofs of this fact I’ve ever seen.

    • @leandoerblader5003
      @leandoerblader5003 6 ปีที่แล้ว +468

      2 of the best math channels on youtube

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +577

      Glad you like it :)

    • @oldcowbb
      @oldcowbb 6 ปีที่แล้ว +113

      Mathologer ft. Pi creature

    • @neutralcriticism4017
      @neutralcriticism4017 6 ปีที่แล้ว +54

      Where's Brady??

    • @frederickm9823
      @frederickm9823 6 ปีที่แล้ว +21

      Loved the crossover video you uploaded today about topology. And then there was a Mathologer video. Great Christmas presents :)

  • @Mathologer
    @Mathologer 6 ปีที่แล้ว +558

    This video is my best shot at animating and explaining my favourite proof that pi is irrational. It is due to the Swiss mathematician Johann Lambert who published it over 250 years ago.
    The original write-up by Lambert is 58 pages long and definitely not for the faint of heart (www.kuttaka.org/~JHL/L1768b.pdf). On the other hand, among all the proofs of the irrationality of pi, Lambert's proof is probably the most "natural" one, the one that's easiest to motivate and explain, and one that's ideally suited for the sort of animations that I do.
    Anyway it's been an absolute killer to put this video together and overall this is probably the most ambitious topic I've tackled so far. I really hope that a lot of you will get something out of it. If you do please let me know :) Also, as usual, please consider contributing subtitles in your native language (English and Russian are under control, but everything else goes).
    Today's main t-shirt I got from from Zazzle:
    www.zazzle.com.au/25_dec_31_oct_t_shirt-235809979886007646
    (there are lots of places that sell "HO cubed" t-shirts)
    Merry Christmas,
    burkard

    • @completeandunabridged.4606
      @completeandunabridged.4606 6 ปีที่แล้ว +2

      Mathologer Thanks for the video, and a merry christmas from me!

    • @akshat4723
      @akshat4723 6 ปีที่แล้ว

      Mathologer hey, we too have a TH-cam channel named ZORTHU-S.And there we made a video on why sum of all positive natural numbers is -1/12 .please check it out hope u like it

    • @massimilianotron7880
      @massimilianotron7880 6 ปีที่แล้ว

      Mathologer You wrote "cox x" in the description in the part where you talk about the the French book

    • @butterflyspinart
      @butterflyspinart 6 ปีที่แล้ว +5

      The link to the original write-up doesn't seem to work.

    • @lucdegraaf5138
      @lucdegraaf5138 6 ปีที่แล้ว

      Loved it

  • @wrpbeater7987
    @wrpbeater7987 6 ปีที่แล้ว +234

    "Welcome to the last Mathologer video"
    - WHAT
    "... of the year"
    - phew

    • @realbignoob1886
      @realbignoob1886 3 ปีที่แล้ว +5

      Lmfao

    • @giustobuffo
      @giustobuffo 4 หลายเดือนก่อน

      ⁠@@realbignoob1886I came to the video 6 years after that original comment, and had the same thought!

  • @aakash_kul
    @aakash_kul 6 ปีที่แล้ว +1469

    That shirt! 25 base 10 = 31 base 8. In other words, 25 Dec = 31 Oct. Merry Halloween!

    • @ganaraminukshuk0
      @ganaraminukshuk0 6 ปีที่แล้ว +49

      Also, according to a certain Blink 182 song and a movie based on a similar premise, you can have Halloween on Christmas.

    • @alerum3473
      @alerum3473 6 ปีที่แล้ว +19

      Nicely noted😉
      Though, shouldn't it therefore be more like "Merry Halloween"? LAWllll🙂

    • @nilaykulkarni3088
      @nilaykulkarni3088 6 ปีที่แล้ว +5

      😯😯😯

    • @prakhardwivedi3649
      @prakhardwivedi3649 6 ปีที่แล้ว +23

      December = 12 & October = 10

    • @crittinger
      @crittinger 6 ปีที่แล้ว +45

      Soundwave Yes, but deca=10 and oct=8

  • @unvergebeneid
    @unvergebeneid 6 ปีที่แล้ว +1035

    To answer the puzzle: none of those logs is rational. Not even the one claiming it is. I mean, come on, a piece of wood shouting out statements on its own rationality? That's completely bonkers!

    • @tjreynolds685
      @tjreynolds685 6 ปีที่แล้ว +61

      That's quite the irrational statement

    • @paulejking
      @paulejking 6 ปีที่แล้ว +26

      In fact, the log's statement itself sounds pretty irrational 😂

    • @edelcorrallira
      @edelcorrallira 6 ปีที่แล้ว +10

      Logs usually display comments... I believe this is merely an old fashioned :P

    • @kyoung21b
      @kyoung21b 6 ปีที่แล้ว +14

      Re. talking logs it sounds like the overlap between mathologer fans and twin peaks fans is the empty set - oh guess I fit that description, never mind...

    • @unvergebeneid
      @unvergebeneid 6 ปีที่แล้ว +4

      Karl Young, that was a Twin Peaks reference? But yeah, you're right, I wouldn't know...

  • @thecubeur33
    @thecubeur33 6 ปีที่แล้ว +408

    7:26 Small mistake, the second term of the expansion of cos(x) should be of degree 2.

  • @enough_b
    @enough_b 6 ปีที่แล้ว +291

    Regarding the first puzzle :
    log10(2) cannot be rational. The same method can be used, and it is trivial to show that no power of two can be divisible by a power of 10 (Except 10^0, of course).
    log7(8/9) cannot be rational either. log7(8/9) = log7(8) - log7(9); and log7(8) cannot be rational since 7 is odd and 8 is even.
    This leaves the woodlog. We need to go down two paths for this, assuming the drawing represents a real situation :
    1. Either woodlogs are incapable of reason or speech. In which case, this one could be part of the few rational/speaking ones, but it is unlikely such a behavior would evolve so fast without intermediary steps.
    2. Or woodlogs are capable of reason and speech, usually. Yet, they never speak. They get chopped, sawed, burnt, and they still don’t speak. If they are both rational and willing to go through this shutting up, they must have a damn good reason. Yet this log just broke millenias of omerta over a pun. I can’t imagine a situation in which that’s rational.

    • @IllusionzZBxD
      @IllusionzZBxD 6 ปีที่แล้ว +14

      Francesco Malhabile Just to point out if log7(9) is irrational then your proof for log7(8/9) no longer works. (x=Pi,y=Pi + 1, x-y=-1)

    • @korayacar1444
      @korayacar1444 5 ปีที่แล้ว +17

      log7(9) is indeed irrational, you simply can’t find a rational number a/b that satisfies 7^a=9^b.
      And so, the proof is as follows:
      Assume log7(8/9) = a/b where a,b are whole numbers
      7^(a/b)=8/9 | ^b
      7^a=(8/9)^b
      *7^a=(8^b)/(9^b)*
      The left side is always a whole number when a is positive, whereas this doesn’t hold for any signed b on the right side.
      Turn this around:
      1/(7^a)= *7^(-a)=(8^(-b))/(9^(-b))* =(9^b)/(8^b)
      The left side is always a whole number when a is _negative,_ whereas this doesn’t hold for any signed b on the right side.
      Since a can’t be positive or negative, it’s 0 by default, and when a=0, this means that b=0 too. 0/0 is indeterminate, which is a contradiction to the assumption that a/b is rational.
      => log7(8/9) can’t be rational!
      Proofs like these are why general statements about irrationals shouldn’t be used carelessly. With this proof format in mind, all that is needed to be accepted is basic algebraic manipulation, not assumptions that look like they’re begging for a counterexample.
      edit: spelling

    • @pgkrish1
      @pgkrish1 5 ปีที่แล้ว +1

      jesna.j@akbartravels.in1 their first games. The first one. The 5@@korayacar1444

    • @johnchessant3012
      @johnchessant3012 5 ปีที่แล้ว +22

      Koray Acar At 7^a = (8^b)/(9^b), you can re-arrange it to 7^a * 9^b = 8^b, which means that an odd number is equal to an even number; contradiction.

    • @dajaco81
      @dajaco81 4 ปีที่แล้ว +3

      I did log7 8/9 differently.
      7^u/v = 8/9
      7^u = 8^v/9^v
      7^u *9^v =8^v
      Odd / even
      Simpler
      7^u*3^2v = 2^3v

  • @vvmcmurdo
    @vvmcmurdo 4 ปีที่แล้ว +136

    Can I just say that Mathologer is one of the most cranckiest, craziest, wackiest, nerdiest and most likable personalities in TH-cam? Hello?
    Love these amazing videos!

  • @Tiqerboy
    @Tiqerboy 4 ปีที่แล้ว +56

    I can only imagine how awesome Lambert must have felt that night when he finished that proof!

    • @carultch
      @carultch ปีที่แล้ว +2

      Is this the same Lambert of the LambertW function?

    • @bilmag182
      @bilmag182 ปีที่แล้ว +1

      @@carultch Yes

  • @flymypg
    @flymypg 6 ปีที่แล้ว +78

    SPLENDID! More please! You and 3B1B are both doing such great work with math-related animations.

  • @semicharmedkindofguy3088
    @semicharmedkindofguy3088 6 ปีที่แล้ว +7

    Seeing you so cheerful about math on all your videos makes me happy. The joy you exude is infectious! Happy holidays!

  • @vighnesh153
    @vighnesh153 5 ปีที่แล้ว +5

    I have seen this video over 10 times and I still get the chills when it gets proved that PI is irrational. Wonderful work. I know you worked very hard to make this animation and I must tell you, your hard work has been fruitful to many math lovers out there. I hope you never stop making such videos. Love from India.

  • @nujuat
    @nujuat 6 ปีที่แล้ว +155

    Wow that last part of the proof was really nice :)

  • @lukecow2
    @lukecow2 6 ปีที่แล้ว +170

    This proof is far more natural than any proof of this I've ever seen. Please make the video you mentioned at 15:56 :)

  • @IllusionzZBxD
    @IllusionzZBxD 6 ปีที่แล้ว +5

    Not only is this video itself a great example of making a proof accessible, you are a great example of an educator that genuinely enjoys teaching others. You certainly make me feel more confident that I want to teach maths myself to others.

  • @withmuchrespect
    @withmuchrespect 6 ปีที่แล้ว +1

    Great video, Mathologer!
    You really went all in on this one..!
    Very clear, light, intuitive and beautiful!!
    For years I'm avoiding looking into these irrationality proofs with fear from all the technical details...
    I value your great work very much!
    Thank you!

  • @lucdegraaf5138
    @lucdegraaf5138 6 ปีที่แล้ว +56

    Again you've made one of the best explanations with incredible animations just to teach people with the same interests on the internet. I'd like to take a moment and just thank you for your work this year all round. I hope you'll have great holidays and a happy new year.
    "Met een vriendlijke groet", (Dutch)
    Luc de Graaff.

  • @vma011
    @vma011 6 ปีที่แล้ว +3

    Absolutely beautiful! Thank you for taking the time to do these videos! These are among the finest quality content I've watched!! Always excited to see them! Happy holidays from Venezuela!

  • @michalbotor
    @michalbotor 5 ปีที่แล้ว +1

    i like how you present proofs sir. showing their sketch first and then filling in the gaps makes it both easy to understand them and remember, and leaves no room to get lost while we fill in the gaps later on. i recall countless times being totally lost after already like a half hour long proof done in a from a to z fashion what are we even proving in the first place..

  • @MrPictor
    @MrPictor 6 ปีที่แล้ว +1

    Thank you for all the work you did in 2017! Looking forward for more exciting videos. Frohe Weihnachten, joyeux Noël ! 🎅🎄

  • @BryceRosenwald
    @BryceRosenwald 6 ปีที่แล้ว +14

    I love how much fun these guys always have filming their videos. It always makes me happy.

  • @bryanshortall787
    @bryanshortall787 5 ปีที่แล้ว +5

    Wow! Great presentation. Bad news is I don't think I could remember how to do this proof on my own in a million years. Good news is that when you were reviewing it, I could very easily follow the logic of each step. The animation definitely made tackling those nasty fractions much more palatable! I can't even begin to imagine what that proof looks like on paper. Ugh!

  • @Ezel17
    @Ezel17 6 ปีที่แล้ว +3

    What a great topic to analyze. Thank you so much for taking the time to create these magnificent animations.

  • @colaurier2594
    @colaurier2594 6 ปีที่แล้ว +16

    These animations are hypnotic. Great stuff !

    • @tomsweeney9580
      @tomsweeney9580 6 ปีที่แล้ว +2

      What software do you use to generate these wonderful animations?

  • @CharlesPanigeo
    @CharlesPanigeo 5 ปีที่แล้ว +14

    21:54 The Well Ordering Principle.
    Every non-empty subset of the natural numbers has a least element, so there can never be an infinitely decreasing sequence of natural numbers.

    • @davidalexander4505
      @davidalexander4505 2 ปีที่แล้ว +1

      Here is an easier way: it is easy to show directly that every subset of the natural numbers which is bounded above is finite. Assuming an infinite, strictly decreasing sequence existed, the first term bounds the sequence above. Thus, there are only finitely many numbers in the sequences. But there are infinitely many numbers in the sequence because it is strictly decreasing. Contradiction.

  • @hyperstone9
    @hyperstone9 6 ปีที่แล้ว +174

    7:25 shouldn't the second term for cos x have x^2 instead of x?

    • @notcaleblim
      @notcaleblim 6 ปีที่แล้ว +4

      hyperstone9 saw that too

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +113

      Every good maths video needs at least one typo :) Luckily this one corrects itself a couple of seconds later.

    • @letspiano3076
      @letspiano3076 6 ปีที่แล้ว

      hyperstone9 yes

    • @dougr.2398
      @dougr.2398 6 ปีที่แล้ว +14

      He’s just checking to see if we are PAYING ATTENTION!! ;-)

    • @jacobhuckins494
      @jacobhuckins494 6 ปีที่แล้ว +3

      Do we get gold stars for spotting errors?

  • @kartoffelmozart
    @kartoffelmozart 6 ปีที่แล้ว +1

    I am truly amazed by what you have achieved in this video. This is the hardest maths I have ever been able to understand, and I only needed to watch the video once. Masterpiece!

  • @Tygearianus
    @Tygearianus 4 ปีที่แล้ว +2

    This animation made following what was happening soooo much easier than just going page by page. Thanks so much.

  • @jonathanwalther
    @jonathanwalther 5 ปีที่แล้ว +11

    I'm not a mathematician, and I love this channel. Thanks for all the sophisticated work.

  • @barutaji
    @barutaji 6 ปีที่แล้ว +121

    Beginning at 7:08 the series for "cos x" has "x/2" instead of "x²/2". When you expand the multiplications of x it is all solved, so it doesn't change the final result.
    Just pointing the typo if you want to correct it with a commentary on youtube.
    Great work ^^

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +44

      Yes, luckily all under control in this respect :)

    • @pgkrish1
      @pgkrish1 5 ปีที่แล้ว +3

      111aa

  • @abhishekchakraborty2284
    @abhishekchakraborty2284 6 ปีที่แล้ว +1

    Best Christmassy Video. Thank you. Love your T Shirt as always. Merry Christmas to you and your team. A big Thank You for all the lessons and hard work which you and your team have put into making Mathematics interesting for Humans. ♥️

  • @rhythmshah4496
    @rhythmshah4496 5 ปีที่แล้ว

    Thanks a lot for uploading this video to share information.
    Your efforts are not wasted but your efforts taught others that there is a lot to explore in mathematics. And this video will also make people who hated mathematics to love it. This all would not have possible without your animations. Please keep making videos and share your knowledge with others.

  • @openid2273
    @openid2273 4 ปีที่แล้ว +9

    While watching this part of the video (4:56), I have come to the following remarkable theorem:
    2L = 1E
    where L is the area measure of Lambert's nose and E is the area measure of Euler's nose.

    • @NoriMori1992
      @NoriMori1992 2 ปีที่แล้ว

      What about their neck circumferences 😂

  • @Tarek172839
    @Tarek172839 6 ปีที่แล้ว +3

    awesome video, thanks for your work! I imagine it must be super hard to create those videos and explain such complex topics in simple terms.

  • @MrVickilatombe
    @MrVickilatombe 6 ปีที่แล้ว +1

    Amazing video! We can see you worked a lot to make it!
    Good job, really good job!
    It blows my mind how deeply you're devoted to explaining all these maths stuff. I study Maths and work for the University of Geneva in popularization (mainly) and I must say not everyone has your talent for explaining stuff so cleverly
    Bravo!
    Can't wait for the next ones!

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +1

      Glad you enjoy my videos so much and thank you very much for saying so :)

  • @longcat
    @longcat 6 ปีที่แล้ว

    Happy Christmas Mathologer x nice to see you again x

  • @Richard_Stroker
    @Richard_Stroker 6 ปีที่แล้ว +5

    log_10(2) = a/b [for some integers a, b, with b not zero]
    2 = 10^(a/b)
    2^b = 10^a
    2^b = (5^a)(2^a)
    5 divides 2
    Contradiction. Therefore we conclude that log_10(2) is irrational.
    log_7(8/9) = a/b
    8/9 = 7^(a/b)
    (8^b)/(9^b) = 7^a
    8^b = (7^a)(9^b) = 2^3b
    7 divides 2
    Contradiction. Therefore we conclude that log_7(8/9) is irrational.

  • @wojteksowinski248
    @wojteksowinski248 6 ปีที่แล้ว +14

    0:29 - I'd love to see Vihart's reaction to that sentence.

  • @realpoems
    @realpoems 6 ปีที่แล้ว

    This is the first one of your videos that I've come across. Absolutely delightful. Thank you for making it.

  • @15october91
    @15october91 6 ปีที่แล้ว +2

    I get so excited whenever you post a video!

  • @acetate909
    @acetate909 5 ปีที่แล้ว +10

    Just finished my proof that shows
    Mathologer=Awesome

  • @Aufenthalt
    @Aufenthalt 6 ปีที่แล้ว +64

    Absolutely amazing!And excellent explained.Uao Lambert was a genius not less than Euler if he succeeded to make all these steps.

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +17

      Yes, did not know much about him before making this video, but the more I find out about him the more interesting it gets :)

    • @kyoung21b
      @kyoung21b 6 ปีที่แล้ว +8

      Yes, very nice proof but “not less than Euler” might be a little strong...

    • @blergblergblerg1343
      @blergblergblerg1343 6 ปีที่แล้ว +4

      You're quick to forget that it was Euler's work that inspired him to utilize infinite fractions, then he used like 2 clever tricks and basic calculation... It definitely is very beautiful, but not genius

  • @dnsaxena8732
    @dnsaxena8732 4 ปีที่แล้ว

    While eveeyone does try to show the toughest concepts to be simpler, you on other hand, mathologer, make us fall in love with it..Millons thanks for all your great efforts.

  • @manla8397
    @manla8397 6 ปีที่แล้ว +1

    Absolutely beautiful proof. Thank you and happy new year

  • @philippeforest8347
    @philippeforest8347 6 ปีที่แล้ว +17

    that smile on mathologer's face at 10:43 is priceless lmao he just achieved the legendary proof by "et caetera"

  • @hafarov8205
    @hafarov8205 6 ปีที่แล้ว +105

    Love his laugh

    • @xamzx9281
      @xamzx9281 6 ปีที่แล้ว

      Эльдар Гафаров даж

    • @pgkrish1
      @pgkrish1 5 ปีที่แล้ว

      Fzabanaci

  • @ice9ify
    @ice9ify 5 ปีที่แล้ว

    This is really great and accessible. I always show this channel to people who are interested in math, and have some grasp of it. Thanks alot mathologer. Your work is appreciated.

    • @Mathologer
      @Mathologer 5 ปีที่แล้ว

      That's great :)

  • @jollyrogererVF84
    @jollyrogererVF84 3 ปีที่แล้ว +1

    Great animations. They definitely make the maths more accessible. Keep them coming please, they're certainly worth the effort 👍

  • @CoasterMagicX2
    @CoasterMagicX2 5 ปีที่แล้ว +9

    I proved that infinite fraction at 6:00!
    Remember the fraction is 1+(1/(1+(1/(2+(1/(1+(1/(2+(1/...). we set the whole fraction equal to x so now x = (the fraction) we now subtract 1 on both sides, and then take the reciprocal of both sides . Leaving us with 1/(x-1)= 1+(1/(2+... Now we set the whole thing equal to z so now z =1/(x-1)= 1+(1/(2+... If you look at the fraction it's just a repeating pattern of 1 +...and 2+..., if we cover the first 1 +... and 2+... It's still the same pattern of 1+... and 2+.... Which is precisely our z! We can actually plug in z so now it's z=1+(1/(2+(1/z))) plug in z for the 1/(x-1) we get 1/(1-x)=1+(1/(2+(1/(1/(x-1))))) unfold bottom to top we get 1/(x-1)= (x+2)/(x+1)
    Cross multiply and rewrite as a quadratic in standard form gets you 0=x^2-3 add 3 on both sides and take the square root gives you x= ±sqrt(3). Throw out the negative sqrt(3) because it isn't a solution for the original equation. Finally by substitution, 1+(1/(1+(1/(2+(1/(1+(1/(2+(1/...).= sqrt(3)
    This proof would be better with actual visuals and math speak but it works.

    • @user-rv9vk8by5i
      @user-rv9vk8by5i 5 ปีที่แล้ว +2

      Lovely proof, but wrong timestamp :^) The fraction is correct, though
      At 6:18, the fraction shown is equal the e. The one equal to root 3 is at 6:00

    • @CoasterMagicX2
      @CoasterMagicX2 5 ปีที่แล้ว +2

      @@user-rv9vk8by5i Thanks! I changed the time stamp.

    • @travellcriner6849
      @travellcriner6849 4 ปีที่แล้ว +2

      One fatal flaw in your proof: You let x be a number equal to that object. The problem is, you haven't yet proved that object really is a number!
      Here is why that matters:
      I will prove -1 = 0
      1) Let x be the number 1 + 1 + 1 + ...
      2) Note that we have -1 + x = -1+(1 + 1 + 1 + ...) = 1 + 1 + 1 + ... = x
      3) That is x-1=x
      4) Subtracting x from both sides yields -1 = 0 as needed.

    • @matn3wman
      @matn3wman 4 ปีที่แล้ว

      @@travellcriner6849 Nice spot! Fortunately it's east to show convergence in the first case whereas 1+1+1+... doesn't converge

  • @lukecox6317
    @lukecox6317 5 ปีที่แล้ว +5

    At 17:27, if we ignore the 1 and set it to x, doesn't that equation have two answers, being both 1 and 2, as the equation simplifies to x = 2/(3-x), and solving the equation results in valid answers for x being equal to both 1 and 2?

    • @maxprofane
      @maxprofane ปีที่แล้ว

      My exact thoughts. I don't how we could show it is not 2 though. Any ideas?

  • @yousteveaaaa
    @yousteveaaaa 4 ปีที่แล้ว

    Lovely !!! I enjoyed every single minute of this video ! And all those animations must have took so much time and effort to make! Incredible ! Thank you !

  • @secretsorcerer
    @secretsorcerer 3 ปีที่แล้ว

    Thanks sir for these kinds of videos, now I'm getting addicted of watching your videos, it's really increasing my curiosity.
    Thanks for the wonderful and unique animation.

  • @SpiffyCheese2
    @SpiffyCheese2 6 ปีที่แล้ว +80

    YES! I found a Mathologer Easter Egg in the intro. 1010011010 = 666. 666 = (36*37)/2 which means it half of a pronic number, which means 1 + 2 + 3 + 4... + 35 + 36 = 666

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +61

      You are only the third person to comment on this since the channel got going :)

    • @SpiffyCheese2
      @SpiffyCheese2 6 ปีที่แล้ว +11

      I love it how you almost pointlessly factored out the "Only" in that statement. Its like equivalent to factoring out 100% in this statement. There is a 100% chance of a 1% chance of "ME" being mathologers favorite fan. By the way I know more digits of pi and √2 then you(2091 digits of pi( Age World Record) and 1024(2^10) digits of 2^1/2) :P.

    • @SpiffyCheese2
      @SpiffyCheese2 6 ปีที่แล้ว +3

      Thank you, sorry I forgot some of my mathematical vocabulary, I will try to improve it when I have some time.

    • @jesselapides4390
      @jesselapides4390 5 ปีที่แล้ว +1

      @@SpiffyCheese2 it's all in the name lol

    • @user-zu1ix3yq2w
      @user-zu1ix3yq2w 4 ปีที่แล้ว +2

      He probably knows when to use "than," though.

  • @maxheadrom3088
    @maxheadrom3088 6 ปีที่แล้ว +5

    That T-shirt is awesome!!!! Btw, I'm still watching the video.

  • @Peshyy
    @Peshyy 6 ปีที่แล้ว +1

    Another awesome video, thank you very much!
    Happy holidays!

  • @SeleniumGlow
    @SeleniumGlow 6 ปีที่แล้ว

    FFS. I couldn't help but notice that 25 Dec = 31 Oct T-shirt and kept thinking about what it meant. I finally figured it out (and banged my head on the dining table few times enough to worry my parents). You have the best collection of T-shirts of all TH-cam community.
    Also, this is a great video on the proof of irrationality of Pi. The 3 step approach is very cool.

  • @PC_Simo
    @PC_Simo ปีที่แล้ว +8

    This was surprisingly easy to follow; even for me, who took Intermediate Maths in high school, and stopped there. Thank you for making this beautiful proof accessible even for amateur mathematicians, like me. 🙂👍🏻

  • @azizpierre9020
    @azizpierre9020 6 ปีที่แล้ว +8

    Assuming Log2=a/b and a,b are positive integers also b>a because log2 is not bigger than 1 but bigger than 0, then
    2=10^(a/b)
    2^b=10^a
    2^b=(2^a)(5^a)
    2^(b-a)=5^a
    2 is an even number while 5 is odd. So this equation cant be correct while (b-a) and a are postive integers. Therefore log2 cant be rational, it is irrational

    • @yxlxfxf
      @yxlxfxf 6 ปีที่แล้ว +5

      Sımişka Zırıhta you could have ended the proof at 2^b=10^a since a power 2 never ends in 0

    • @rot527
      @rot527 6 ปีที่แล้ว

      But this proof proofs both statements

    • @snnwstt
      @snnwstt 6 ปีที่แล้ว +1

      ... or use modular arithmetic. 2^b = 10^a implies 2^b = 0 (mod 5)... and use the principle of uniqueness of representation of a number through the product of its prime numbers: If it is possible to have 2^n = 0 (mod 5), some integer would have TWO possible representations, one without a 5 as its primes ( 2^n) and one with a 5 among its prime ( to be 0 mod 5). So 2^b = 10^a cannot hold with integers.

  • @GaryFerrao
    @GaryFerrao 4 ปีที่แล้ว

    It was a great, accessible video. Thanks to that, i was able to focus my attention to finding out flaws. And having seen an other of your videos, i'm now very sceptical about the entire adding and subtracting the "infinite series" denominator"; that too after re-arranging the terms.

  • @richardschreier3866
    @richardschreier3866 6 ปีที่แล้ว +1

    Another glorious gem made both accessible and entertaining for modern armchair math enthusiasts around the world. I continue to be amazed how just a few twists and turns of reasoning can illuminate a claim as surprising and seemingly unfathomable as the statement that tan(rational>0) is guaranteed to be irrational. Truly amazing! Thank you for making these delightful videos. If you were to set up Patreon or some other means for us fans to show our support, I'd gladly participate!.

  • @sergeboisse
    @sergeboisse 5 ปีที่แล้ว +6

    So a myth-loger shows us a math-speaking log

  • @gordonchan4801
    @gordonchan4801 6 ปีที่แล้ว +35

    that T-SHIRT

    • @gabrielfraser2109
      @gabrielfraser2109 6 ปีที่แล้ว +3

      "No it doesn't... wait... wait... oh shit it's true"

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +4

      I put a link to where I got it from in the description :)

    • @pgkrish1
      @pgkrish1 5 ปีที่แล้ว +1

      Thanks

    • @PanduPoluan
      @PanduPoluan 3 ปีที่แล้ว

      I had been racking up my brain then it hit me. Brilliant!

  • @handsome_man69
    @handsome_man69 6 ปีที่แล้ว

    Merry Christmas mr Math-person, and your funny heckler in the background

  • @randellrussell2400
    @randellrussell2400 6 ปีที่แล้ว

    Merry Christmas mate. Thanks for the videos. You're my hero.

  • @vladschiopu2885
    @vladschiopu2885 5 ปีที่แล้ว +4

    √2 and π were fighting outside. 8 tried to calm them down. But they are irrational so they kept fighting. 8 came between them, but everything got worse because, unfortunatly √2 ate π.

  • @opl500
    @opl500 6 ปีที่แล้ว +11

    Big animations for you

    • @MagicGonads
      @MagicGonads 6 ปีที่แล้ว

      You're a big animation
      For you

  • @wgm-en2gx
    @wgm-en2gx 6 ปีที่แล้ว

    Great ivideo! Merry Christmas and happy new year!

  • @jakegearhart
    @jakegearhart 4 ปีที่แล้ว

    This channel is wonderful. You making proving things for the sake of proving things fun and exciting. That's hard to do!

  • @shaferai
    @shaferai 6 ปีที่แล้ว +11

    The continued fraction at 17:45 could equal either 1 or 2

    • @mathislove3722
      @mathislove3722 4 ปีที่แล้ว

      No, it can't. You have to refute 2.

    • @ekz9479
      @ekz9479 4 ปีที่แล้ว

      @@mathislove3722 How do you refute 2?

    • @mathislove3722
      @mathislove3722 4 ปีที่แล้ว +1

      ​@@ekz9479 For example, you can show that the fraction is always smaller than a number smaller than 2. However, I proved the limit to be 1 directly. If you look at the terms of the sequence obtained from the continued fraction, you'll see that it follows the pattern (2^n-2)/(2^n-1). This pattern can be proved to be valid. So, the general term for the sequence simultaneously proves the limit to be equal to 1.

    • @maxprofane
      @maxprofane ปีที่แล้ว

      @@mathislove3722 Thanks a lot. It was bugging me.

  • @tuhinchakrabarty4236
    @tuhinchakrabarty4236 6 ปีที่แล้ว +6

    Awesome....

  • @attilakiss8585
    @attilakiss8585 6 ปีที่แล้ว +1

    The last part was really nice, very precise and strong, still simple. Thank you!

  • @wedusk
    @wedusk 6 ปีที่แล้ว +2

    What a beautiful Christmas present! Thank you

  • @ablebaker8664
    @ablebaker8664 6 ปีที่แล้ว +7

    I've always found it amusing that a ratio can be irrational.
    I guess I'm easily entertained.

    • @masterchief7137
      @masterchief7137 5 ปีที่แล้ว

      That shows how we can't really understand infinity or at least the pythagoreans didn't

    • @niccolopaganini7723
      @niccolopaganini7723 4 ปีที่แล้ว +1

      RATIOnal: can be expressed as a ratio
      You can then simply apply an "ir" to provide the converse, ir-ratio-nal
      *when a dead violin god applies English to math*

  • @Swoost
    @Swoost 6 ปีที่แล้ว +95

    JAWOHL!

    • @andrewxc1335
      @andrewxc1335 6 ปีที่แล้ว +3

      Mathologer: «cringe»
      I laughed, probably too hard.

  • @ryu8148
    @ryu8148 6 ปีที่แล้ว

    A Mathologer video right before Christmas? What a (Trick Or) Treat!

  • @billrussell3955
    @billrussell3955 6 ปีที่แล้ว

    Oh and thank you! Merry christmas & happy holidays!

  • @sinom
    @sinom 6 ปีที่แล้ว +6

    Dir auch fröhliche Weihnachten!

    • @manueloribe9153
      @manueloribe9153 6 ปีที่แล้ว

      hahaha

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +4

      (ha)^3 :)

    • @manueloribe9153
      @manueloribe9153 6 ปีที่แล้ว

      Mathologer
      H*a*h*a*h*a:[Null])
      (ha)^3/[Null])
      (ha)^3/[Null]
      (ha)^3/0
      (ha)^3=0 (Because if not it is=∞)
      h=0 or a=0
      Q.E.D

  • @Malitz101
    @Malitz101 6 ปีที่แล้ว +5

    Wow.

  • @brooksofmaine
    @brooksofmaine 4 ปีที่แล้ว

    Helpful! Thank you for your way of explaining and for the animations

  • @danielrhodes294
    @danielrhodes294 6 ปีที่แล้ว

    One of the best explanations I've seen for pi's irrationality that doesn't include assumptions that seem to come from nowhere. The animation was great in showing this too. I'd love to see more videos simplifying proofs that seem too complicated to understand. Maybe one about pi's transcendence?

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +1

      All the proofs of the transcendence of pi I know are really, really scary :) Anyway, will definitely give it a try eventually :)

  • @vinodkumar-wm3oq
    @vinodkumar-wm3oq 6 ปีที่แล้ว +4

    At 17:40 it is equal to 1 and 2 both.

    • @ReconFX
      @ReconFX 6 ปีที่แล้ว

      vinod kumar Was just about to comment that :D

    • @vinodkumar-wm3oq
      @vinodkumar-wm3oq 6 ปีที่แล้ว

      ReconFX, quadratic :)

    • @enderyu
      @enderyu 5 ปีที่แล้ว

      so what does that mean? it is impossible to solve? It can't be both at the same time, right?

  • @Wompylulz
    @Wompylulz 6 ปีที่แล้ว +7

    The proof of the infinite fraction that yields one is simple:
    1 = 2 divided by 2
    2/2 = 2/(3-1)
    Now substitute repeatedly 1 with 2/2 and the 2 at the denominator with 3-1
    QED

    • @xamzx9281
      @xamzx9281 6 ปีที่แล้ว +8

      Davide Morgante 2=2/1, 2=2/(3-2), 2=2/(3-2/(3-2)), then again and again and you get the same formula as for 1, so 1=2 :)

    • @Harlequin314159
      @Harlequin314159 6 ปีที่แล้ว +4

      if you solve it out algebraically you get S = 2/(3-S) , this yields a quadratic with two solutions: S = 1 , and S = 2. Makes sense given both of your proofs.

    • @xamzx9281
      @xamzx9281 6 ปีที่แล้ว +2

      Harlequin314159 yep

    • @Wompylulz
      @Wompylulz 6 ปีที่แล้ว

      That's quite nice!

    • @algc19
      @algc19 6 ปีที่แล้ว +1

      Quantum I too arrived at that conclusion, so what does it really means? Are we doing this wrong?

  • @MuffinsAPlenty
    @MuffinsAPlenty 6 ปีที่แล้ว

    Merry Christmas, Mathologer. Thanks for sharing that great proof, and your wonderful pun about an infinite descent into mathematical hell ;)
    Also, I loved your solution to 3Blue1Brown's mug puzzle. It's so very "you." It's great.

  • @Miansaleon00
    @Miansaleon00 6 ปีที่แล้ว

    You are awesome and the internet appreciate the hours you spend of this presentation...

  • @marcozz4657
    @marcozz4657 6 ปีที่แล้ว +90

    (HO)³

    • @gluckmac
      @gluckmac 6 ปีที่แล้ว +1

      Marco Zz Great pun!

    • @frechjo
      @frechjo 6 ปีที่แล้ว +8

      I find the 25 dec = 31 oct better tho

    • @bobengelhardt856
      @bobengelhardt856 6 ปีที่แล้ว

      Yeah, cute. But to be rigorous (!), it should be 3HO. :-)

    • @frechjo
      @frechjo 6 ปีที่แล้ว +8

      3ho = ho+ho+ho
      (ho)^3 = (ho).(ho).(ho) = hohoho
      [ 1 1 1 ] . ho = [ ho ho ho ]
      I think the vector makes more sense, but it's less "t-shirt friendly" ;)

    • @user-un2hf9ve2j
      @user-un2hf9ve2j 5 ปีที่แล้ว +3

      @@frechjo idk but I can't stop laughing xD

  • @seeyouinpolaris7463
    @seeyouinpolaris7463 6 ปีที่แล้ว +18

    but first, please explain WHERE DO YOU GET THESE T-SHIRTS ??

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +9

      A lot I make myself but I am also constantly on the lookout for new maths t-shirts. I put a link to the place where I got the t-shirts in this video from in the description :)

    • @seeyouinpolaris7463
      @seeyouinpolaris7463 6 ปีที่แล้ว +1

      Mathologer omg i didn’t expect your answer but thank you 🤗✨

    • @erikshure360
      @erikshure360 6 ปีที่แล้ว

      I can't stop laughing. That shirt is hilarious!

  • @walterpilco2907
    @walterpilco2907 3 ปีที่แล้ว

    Awesome, continue doing videos like this, don't know how much you helped me.

  • @margarett.newman7574
    @margarett.newman7574 3 ปีที่แล้ว +1

    That is wonderful to see. This way of rendering mathematics is very very pleasant. Well, you make it look joyous and effortless to yourself. The animations allow the imagination and working - mathologising - brain to gen and enjoy. Thank you.

  • @peacemaker42069
    @peacemaker42069 6 ปีที่แล้ว +4

    zeroth

  • @nofanfelani6924
    @nofanfelani6924 6 ปีที่แล้ว +8

    Math is irritational

  • @hamade7997
    @hamade7997 6 ปีที่แล้ว +2

    This was amazing, thank you for visualising such an interesting proof.

  • @dcterr1
    @dcterr1 4 ปีที่แล้ว +1

    Very nice proof! I was able to follow all the logic, though I'd have to muddle through the details in order to reproduce the proof. I've looked at other proofs of the irrationality of pi, but I was never able to follow them. Good job!

  • @donwald3436
    @donwald3436 3 หลายเดือนก่อน +4

    Your shirt is too distracting 😂😂😂

  • @sushantpoudel4372
    @sushantpoudel4372 6 ปีที่แล้ว +4

    what if
    b=a/2
    c=b/2
    d=c/2
    e=d/2
    and so on?
    every term would be less than the previous term and still be rational: a/2^n.
    PS: I am NOT saying pi is rational. I felt that this explanation was a bit incomplete to my expectation.(not saying that it is, I just felt so)

    • @Mathologer
      @Mathologer 6 ปีที่แล้ว +6

      The numbers A, B, C, D, ... are positive integers :)

    • @davidburwell4218
      @davidburwell4218 6 ปีที่แล้ว +1

      yes, but infinitely many, no?

    • @Drtsaga
      @Drtsaga 6 ปีที่แล้ว

      i actually was having the same misunderstanding. the fact the A, B, C, ... are integers solves it for me. :)
      thx

    • @loreleihillard5078
      @loreleihillard5078 6 ปีที่แล้ว +1

      Sushant Poudel they have to be positive integers

  • @gilvaniooliveira8850
    @gilvaniooliveira8850 6 ปีที่แล้ว

    Wow! Thank you for uploading this video. I'm not very good with mathematics, and I'm currently struggling with it in college, but I am so interested by these "mysteries" (for me they are still mysteries lol), and your videos make me want to learn more about it. I wish they taught like you in my University. Love from Brazil.

  • @without9103
    @without9103 5 ปีที่แล้ว

    love it, merry christmas.

  • @alexwang982
    @alexwang982 5 ปีที่แล้ว +4

    GAAAAH
    GAAAHHHH
    1+1=5
    POTATO is a COMPLEX NUMBER
    SQURAE ROOT OF SANDWICH

  • @brandongunnarson7483
    @brandongunnarson7483 4 หลายเดือนก่อน

    Thank you for working so hard on those slides for the infinite fraction, I finally understand how to create them now! I've been trying to learn for months

    • @brandongunnarson7483
      @brandongunnarson7483 4 หลายเดือนก่อน

      Also I absolutely searched for this video because it's pi day

  • @benvendergood1064
    @benvendergood1064 2 ปีที่แล้ว

    LOVE YOUR STUFF‼ Always entertaining and informative....

  • @Reliquancy
    @Reliquancy 6 ปีที่แล้ว +1

    I appreciate the effort to make these over 100 slides thanks! I wonder if the final proof was around 40 pages long how many pieces of paper he went through coming up with it? the economics of the availability of paper might actually have been an issue so long ago! im going through all your videos now thx