the example with the elastic band can be misunderstood easily, so Brady's argument isn't that wrong actually: if you just say "every second, we add another meter to its circumference", you can always add the additional meter in front of the ant, and of course it will never reach the end that way. instead, you have to emphasize that the band is stretched, i.e. it is uniformly expanded so that every part of it grows the same percentage. that means the distance behind the ant and and the distance in front of the ant grow proportionately to their relative size, and as the distance behind the ant becomes larger and larger in relation to the distance in front of it (it will since the ant travels), more and more of the additional meter is in fact added behind the ant, not in front of it.
ok, let's look at it in detail: 1st second traversed distance = 0 cm, distance ahead = 100 cm, total distance = 100 cm ant moves 1 cm from start: traversed distance = 1 cm, that is 1/100 = 1% of the total distance. 2nd second band is stretched 100 cm: traversed distance = 2 cm (increases with stretching!), distance ahead = 198 cm, total distance = 200 cm ant moves 1 cm: traversed distance = 3 cm, that is 3/200 = 1.5% = 1% + 1/2% of the total distance. 3rd second band is stretched 100 cm: traversed distance = 4.5 cm, distance ahead = 295.5 cm, total distance = 300 cm ant moves 1 cm: traversed distance = 5.5 cm, that is 5.5/300 = 1.83333..% = 1% + 1/2% + 1/3% of the total distance. and so on... every second the ratio of the traversed to the total distance increases, until it finally reaches 100%.
+Amphithryon I feel like the guy in the video failed to emphasise this: the part he has already travelled ALSO stretches, and then the ant moves 1 cm independently from that stretch. The way he explained it sounded like wishy washy -1/12 stuff, while it's actually really logical.
I don't think he failed, he used a rubberband for that sole purpose. Just before the words you quoted he says "we are gonna stretch so that..." and then you go to say he should emphasize the stretching... when 3 word prior to your quote he did. What is probably wrong in this case is the drawing with the ant in the circle, but it should be ok still unless you forget that it's a rubberband and that we are stretching.
@Amphithryon very nice put. Tony clearly gave the correct response to Brady's argument, that what is behind is growing as well. He just didn't mention (or it's cut out) that the growth is exponential to time, and as soon as the ant hits half-way it grows faster than what's in front of the ant. Which is why eventually the 1cm is longer than the growth in front of the ant.
See my question/response above. Thanks Rafael. if the band only expanded in front, the percentage travelled would always be 1%. But because it expands behind as well, the percentage is always growing, albeit very slowly.
I just laughed out loud when he said that. Amazed, yet at the same time really had no idea what that meant for the whole problem. I wish I was smart enough to grasp really amazing ideas like this.
Exactly! It was a mind-blowing moment for sure. I mean, it seems so obvious in hindsight, but I would have been busting my head for days trying to come up with a logical explanation for that.
If you add 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8, you are just adding 1/2 over and over again, so we can clearly see that it diverges. And if you add 1+2+3+4..., you are just adding 1 over and over again, so we can clearly see that it........goes to -1/12
Update: I watched the extra footage, as well as Mathologer's video on it, and things are cleared up now. Both series diverge in the traditional sense, but can be "analytically extended." In other words, if we accept new definitions of sums (Cesaro, Abel, Ramanujan) where traditional sums don't give a finite answer, we can also accept finite answers for divergent sums. Also a little annoyed that "gamma" is different from the Riemann Gamma function, but who cares lol
I can't see any references for Riemann Gamma function.. do you mean Riemann Zeta function? The extended Zeta function includes the Gamma function though
That function is not called Riemann Gamma function, it's just Gamma function. As far I know, Euler originally gave a version of the Gamma function first as an infinite product, then he represented it with an integral. This was in the 18th century. Riemann would come much later 19th century. The Gamma function and Zeta function are just related, but the gamma function is not the work of Riemann. And I couldn't find any references for "Riemann gamma function":
There's an older Numberphile video on this, actually one that got them famous. There was a lot of fuss about it commenting on the same thing (the sum of 1+2+3+4+...=-1/12) and the gripe that echoed around was that they didn't really define the extended sum as a function rather than a summation. So that is what they actually refer to at the beginning of the video saying "We're gonna start in a familiar place".
I worked out the ant band time: The circumference of the band is given by C = t+1 where t is time elapsed. Distance travelled by the ant is s. The speed of the ant seems to be 0.01 m/s, but also has a component given by the band stretching behind it, which gives it further displacement. The rate at which this happens is the proportion of the band that the ant has already travelled across at a given time: s/C = s/(t+1) So we get the differential equation ds/dt = s/(t+1) + 0.01 (ds/dt)/(t+1) - s/(t+1)^2 = 0.01/(t+1) d/dt(s/(t+1)) = 0.01/(t+1) s/(t+1) = 0.01ln(t+1) + c s = 0 when t = 0 so c = 0 s = 0.01(t+1)ln(t+1) The point at which the ant makes it back to the start is when s = C = t+1: t+1 = 0.01(t+1)ln(t+1) 1 = 0.01ln(t+1) t = e^100 - 1
LET'S BRING THIS TO THE TOP This video was more closely related to the armonic series than 0.577. You can't just say "0.577 appears all over physics" and "it knows about primes too" and not expect me to demand a more in depth separate video with .577 as its star.
It's very little what is Know about that constant, in 300 year there was a y significante avance to understand it's Nature, it's only that is appears when You combine theory of numbers and calculusz nobody knows why
Certainly an interesting one. Have seen .577~ pop up from time to time in the maths I work with in computer graphics and physics simulation. Always thought it was just the result of some personal bias, a product of how I do things. Never realized it actually had a more profound meaning.
2:50 "I knew you were going to say that". That's because solving the Reimmer zeta function for a divergent series does not give you an equivalence for n=infinity. It is an "associated" value, not the "answer" of the series.
+D but isn't it amazing that of all the values to be uniquely associated (whether by analytic continuation or ramanujan summation) that it is -1/12 - see our gold nugget video all about this.
Numberphile yes, but please let us stop using the equal sign for it. We are missing the point of the "conversion" that happens when we do this. I think that is a lot more interesting than the illogical use of "=", which is simplistic and lends itself to being disproved by counter example.
+Paul Ahrenholtz The video interprets [Correction: the notation of series like "1+2+3+..."] in two different ways without telling you. If you know which one they're talking about at which time, everything they say is correct. The vast majority of people don't, which leads them to confusion and/or incorrect conclusions.
David Perrier technically that also incorrect... the interesting numbers that *we know of* are finite. But if there's infinite numbers, there's infinite equations that do something interesting
You didn't understand the problem. At that position, 1 meter scaling will be insignificant, because 99.99999999...% of the path is already behind the ant. This means that the Ant will finish the last centimeters without noticing any change in path size.
André Canilho I did understand, I was just making a joke. But you're right the last few meters will be adding less than a cm in front of him since 99.9999% will be added behind him
So it's the % of the meter increase (relative to ant) that gets smaller as ant goes till eventually the increase relative to the ant is smaller than his cm traveled per increase. This was such a cool problem!! I did using excel spreadsheet as ant going 1 cm and circle increasing by 2cm (starts 2cm big) and when circle reaches 22cm the ant has gone fully around the circle.
I don't get it 0cm 2cm 1cm 4cm 2cm 6cm 3cm 8cm 4cm 10cm 5cm 12cm 6cm 14cm 7cm 16cm 8cm 18cm 9cm 20cm 10cm 22cm It seems to get closer to 100% but say in 10000 increaces the ant has gone fully around, but logically thinking the circle 10000 * 2cm = 20000cm and the ant 10000 * 1cm = 10000cm, so only 50% :0
MV AntDist Cir % traveled 0.00 2 0.00% 1 1.00 2 50.00% 2.00 4 50.00% stretch 2 3.00 4 75.00% 4.50 6 75.00% stretch 3 5.50 6 91.67% 7.33 8 91.67% stretch 4 8.33 8 104.17% Mv - ant moves ant Dist- total distance relative to viewer of circle (not ANT) %traveled - the distance viewer sees him travel. I made a mistake, it is only 4 moves. Hope this helps
+erikeeper thats an error in your logical thinking you are at: 0cm 2cm ant walks 1cm 1cm 2cm the circle is expanded and since you are 50% you will be moved(the ant) as well, so 2cm 4cm then ant walks again ending at 3cm 4cm and so on. with this example the circle is completed very quickly, he uses 1/100th to make it only more confusing
Your passion for mathematics is infectious. Gauss, Newton, Leibnez, Pascal, Euclid, Pythagoras and Archimedes are all subscribing to your TH-cam videos.
I love how they know the whole -1/12 affair works, at the very least, as a delicious trolling act. (but of course the video itself was so insightful it went over many people's head)
I am always curious what the original motivations were for investigating things like these. he mentions that he knows of it because of physics and quantum stuff, but Euler and Mascheroni wouldn't have. And yet they calculated it to so many digits. were there older uses for this number? I'd like to hear more background into the origin of euler's work and others
Makes me wonder. So many purely mathematical constants crop up in nature; gamma, root 2, e, pi, golden ratio etc.. It's like the laws of physics are based around mathematical constants that can be derived without needing any physics. Now maths is going to be the same in another universe, so I am sceptical about the laws of physics being different.
1+2+3+4+5... also diverges in the normal sense of the term(meaning the series of partial sums diverges). It's only said to equal -1/12 because of the Riemann zeta function.
Unfortunately, "school mathematics" is generally presented in a boring fashion, oddly disconnected from reality. I can't help but triple facepalm when I hear someone, as they too often do, say "so, what's the use of maths?". That's how poorly it's presented to most people, as they seem to think that accusing the greatest "transferable skill" there is and basis of all science and technology and engineering - and even music - of being "useless" is not only a reasonable, but even a clever, thing to say. It's like handing over a suitcase with a million dollars in it to someone, they look inside and then hand it back to you because it's "just full of paper". Well, yes, but you really have no idea just how much you're totally missing the point there.
Thats true, that russian guy on numberphile has a video about why people hate maths. Maths at school level can be quite dry. Some people think its all about arithmetic, but that's just one of the fundamental tools needed in the majority of math areas. Maths is interesting cause its like a whole other world that exists abstractly that has so much to be discovered. But also, maths is the language we use to describe the universe, and also we can use it to solve problems and build things like computers and particle colliders. So its both interesting and useful and is sort of integrated into reality itself which makes it cool.
Numberphile comments are the only comments on youtube i enjoy reading as there is some level of discourse amongst the subscribers that doesn't devolve into meaningless drivel and name calling. You can actually learn something from the comments, which goes to show you who watches these types of videos...
A fun fact is that it shows up in the 'block stacking problem' (or the Leaning tower of Lire). The idea is that you stack blocks or bricks on top of each other on an edge of a table and make the stack of blocks lean over the edge as much as possible without it falling over. Then you want to know how many blocks you need in order to make the tower lean over the edge, for example 4 times the legth of one block. You can calculate the exact number of blocks you need by rounding (to the closest integer) the value of this formula: e^(2*o-y) where "o" is the number of brick lengths the tower leans over the edge and "y" is the Euler-Mascheroni constant.
Does his ant on a rubber band example also involve gamma? He doesn't really make that clear in the video. It seems like it just demonstrates a property of the harmonic series. But I've noticed that whenever stretchiness is involved, logs or exponents seem to be involved, so I'm wondering if maybe it does?
It is easy to see see that ant gets to the finish. If ant does not move while rubber is increasing, angle relative to starting position remains same. As long as ant is moving forward, angle of ant relative to starting position is increasing and it will eventually reach the finish line .
The sequence where you continually add on increasingly smaller numbers sums to infinity, but the sequence where you continually add on increasingly larger numbers sums to a negative fraction. Cool
Consider the expansion of the universe during inflation in which at each interval of time (Planck's time = 5.39 x 10^-44).. the universe expands one Planck length.. and that a packet of energy starting at some point circumnavigates around the entire universe.. does it get back to its initial starting point? if so.. how long would it take?
if one planck length per planck time is the speed of light, then how can anything move slower? Nothing cam move less than a planck length. Like, 1 planck lengths per 2 planck times is half the speed of light, so something moved half a planck length in 1 planck time? That can't happen.
You can't measure anything less than one Planck Length. So to measure something that moves at half the speed of light you would have to wait for it to move at least 2 Planck Lengths which would take at least 1 Planck Time. Essentially it is not possible to observe or measure anything less than a Planck Length or a Planck Time. So likewise we cannot say something moved 1 Planck Length in anything less than 1 Planck Time.
Outside of early maths, log very rarely refers to base 10. In pure maths log almost always refers to ln, as base 10 isn't relevant and in computer science log is often used for base 2, because binary.
@@Nebula_ya Engineers still typically use log to mean base 10. And regardless, there's still no reason to have it ambiguous. There's a perfectly unambiguous way to write it.
The sum 1+1/2+1/3….. +1/n is very easy to calculate. It is an integral of 1/X where X changes from 1 to n, which is natural log of n. So if n is infinite, then the sum is also infinite. The question is why the numerical value differ from the integral value by 0.577.
The ant and the elastic band example for the sequence was initially perplexing to me. Then I fired up Excel and it became clear! The following are some of my results with different ant step lengths - the first number is the size of the ant's step and the second is the number of steps needed to complete the circumference: 15=441 14=710 13=1,230 12=2,336 11=4,983 10=12,367 9=37,568 8=150,661 7=898,515. I am interested to learn the formula needed to work out the number of steps required for 6, 5, 4, 3, 2 and 1 step sizes (all the way to 3 tredecillion years!) Is there a formula that can be applied to calculate these from the sequence shown?
Since the ant's steps (in the video, with starting step size 1) are 1/100, 1/200, 1/300, etc. of the circumference of the loop, you just have to see how many terms of 1/100+1/200+1/300+... you need to add up until you get to 1. That's equivalent to adding up terms of 1+1/2+1/3+... until you get to 100. Of course it's too many to compute *exactly*, but you can use the approximation that 1+1/2+...+1/n ~ log(n)+ɣ and set log(n)+ɣ = 100 to find n = exp(100-ɣ) = 1.5×10^43 steps. (I know it says 10^50 in the video but I think that's a mistake, e^100 is more like 10^43.) To do this with starting step size 2, 3, 4, or anything else, just replace 100 with the ratio of the original circumference to the step size, i.e. use the formula n = exp(ratio-ɣ) to find the number of seconds: exp(100/6-ɣ)=9,717,617 exp(100/5−ɣ)=272,400,600 exp(100/4−ɣ)=40,427,833,596 exp(100/3−ɣ)=168,190,380,070,122 exp(100/2−ɣ)=2,911,002,088,526,872,100,231 These will be a little bit off from the true number of steps because the partial sums of the harmonic series are not exactly log(n)+ɣ, so if you wanted to you could redo the math with a better approximation log(n)+ɣ+1/(2n), but it wouldn't change the results by very much and would make it a considerably more difficult calculation. I know this comment is from years ago but I thought people reading it in the future might like to see the method.
Can the ant problem be modelled conversely? Surely, by showing that after every second, the ant is further from the end than it was a second ago, demonstrates that it will never reach the end. Seems like the ant’s infinity is smaller than the rubber band’s infinity.
You say that because your intuition is thinking about inequalities of finite sums. For finite sums, A1 < B1 and A2 < B2 implies that A1 + A2 < B1 + B2. But you shouldn't assume that this property holds true for infinite sums and it's actually not the case. Order is very important for those kind of sums. Actually if you did that : (2 + 1) + (4 + 3) + (6 + 5) + (8 + 7) ..... it's no longer equal to -1/12 (it's actually +5/12)
It isn't. Both are divergent series. And if you have shown, that 1 + 1/2 + 1/3 + ... is divergent, then direct comparison (keyword: direct comparison test) directly implies, that the sum of natural numbers has to diverge as well. In another numberphile video they (including the physicist in this video) spread the word, that the sum of all natural numbers would be -1/12. And while there are ways to assign a finite value to a divergent sum (see f.e. en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF ), in the 'classical' sense both series are simply just divergent and approach infinity.
Actually there's an other example where this happens : every member of 1+2+3+4+.... is greater than those of 0+0+0+0+0.... yet the first sum is negative and thus smaller than the second sum ;)
The problem of this is that in the video it is explained that the geometric series diverges because 1/2+1/2+1/2+... "clearly" diverges. I can see how this can be confusing for people who watched this and also 1+2+3+...= -1/12
Eagerly waiting for the response on the second channel. That strikes me as a glaring flaw in this logic, given previous insistence that the sum of natural numbers is -1/12. Maybe there's some function out there no one's come up with yet that defines the sum of their reciprocals as -1/11?
I notice the comments here talking about the "-1/12" stuff and claiming either it's right or it's "nonsense". There seems to be a lot of confusion on this and a lot of muddle, and I'd like to post this post to provide a definitive PSA to clear up the muddle, once and for all. I hope this does so. There are two different notions called "sum" of a series in play here. There is the "ordinary sum", usually just called "the sum". The ordinary sum is defined by a limit, of course. In this case, 1 + 2 + 3 + 4 + ... diverges. It has no ordinary sum. The limit does not exist. Some say the "ordinary sum" is "infinity", but this is only correct if you are working with the _extended real number line_. If you are using the ordinary real number line, the sum simply does not exist. "Infinity" is not an element of the ordinary real number line. The extended real number line, as the name suggests, "extends" the real number line by adding +/- infinity as new "numbers" at the ends of the line. Now, there is another notion -- actually, several notions grouped under the same rubric, called a "generalized sum". This is _a different notion than the sum_, defined using something other than the limit of partial sums, although it is not an unrelated one, for it is defined in such a way that when an ordinary sum exists, the generalized sum exists as well and coincides with it, hence the name "generalized". But the generalized sum can exist when the ordinary sum does not. Saying "1 + 2 + 3 + 4 + ... = -1/12" is referring to this _generalized sum_, NOT the ordinary sum. Depending on just which notion of generalized sum you are using, some of the manipulations of the series on the left may or may not be valid. That's it. It's just a matter of keeping these concepts straight and separate -- ordinary sums (or just "sums"), generalized sums, real number lines, and extended real number lines. Keeping in mind these things are _related_ but _not the same thing_. Other than that, all these concepts are 100% legit. They're just different, and need distinguishing. This point is often lost here. But I don't blame the audience. The problem is the various presenters and tutors out there who just moosh all this stuff together and be sloppy. Being sloppy with math creates confusion all the time and of course we have generations of people raised on sloppy teaching and so people are no wonder, thoroughly confused about math and make songs about hating math and for those who do eventually come to a clearer understanding and maybe become mathematicians, a not-insubstantial part of their learning effort is wasted on _un_learning all the confused muddle bs pumped into their heads by the bad public school system with its confused teachers. The math isn't wrong. None of these results are wrong. The presenters are bad. Especially when dealing with a lay audience who doesn't necessarily have a tight grasp honed from much experience. An experienced math person can get by with their presentation, but a noob or interested layman will be horribly lost.
5:10 FYI: A tredecillion years is 10^42 years. The sun would become a red giant in fewer than 10^10 years. When the ant cross the finish line, the rubber band, growing at 1 m/s, would be e^100 or about 10^43 meters. Our current observable universe is about 10^27 meters in diameter. Thankfully it expands at about 10^5 m/s/Mps to accommodate the rubber band.
Lazzy is just what scousers/people from Liverpool call elastic. e-LAZZY-stic. Dont think it is used in other regions in North/North west, but is popular in Liverpool
Hello Numberphile, your videos are amazing and I truly enjoy watching them! There's just one thing I'm wondering about: At 5:00 it says that the ant needs about exp(100) seconds to complete the task. exp(100) is about 2.6881e+43. A tredecillion is 10 to the power of 42. So the ant needs about 27 tredecillion seconds. One year has 32,850,000 seconds. If you divide exp(100) by 32850000 you get 8.183e+35. Shouldn't the result be 818 decillion years instead of 3 tredecillion years? Yours, Max
I'm really having trouble with the ant on the rubber band. After e^100 seconds, the band is e^100 meters long. The explanation given, rather matter-of-fact-ly in the video, is that the distance behind the ant also increases. But, the distance in FRONT of the ant also increases over time, and thus at e^100 seconds, the ant must still require (e^100 meters - e^100 centimeters) to travel to reach the end. It really doesn't matter how much distance is behind the ant, if there is still that much distance in front of the ant to traverse.
Well, how about this. An ant that can travel half a meter per second, lets say. Start on the meter long band, just as before. In one second, you are at, naturally, 1/2 of a meter. The band expands by 1 meter, split evenly along its length. So, the 1/2 meter ahead expands to 1 meter, and so does the 1/2 meter behind. Now, you go again, and you are, again, 1/2 meter away from the finish line. This time, though, the half meter ahead only increases by .25 meters, since it is a quarter of the band. Now you are .75 meters away, which is closer. And now, since the band has grown, the bit ahead of you gets a smaller percentage of that growth, allowing you to catch up, faster and faster. The same goes for 1 cm on a 1 meter band, but much, much slower.
stellarfirefly But because the distance behind the ant has been expanding the e^100 cm the ant has travelled has also expanded . Take the first iteration. The ant moves a cm, then the band expands by a meter. There is now 2cm behind the ant because the band has doubled in size.
Think about it this way: even if the ant were to move a certain distance and then stand completely still, it still wouldn't lose its position relatively speaking. Say the band is 10 m in circumference and the ant has already traveled 10% of it, i.e., 1 m, and then decides to take a break. The next second, the new circumference will increase by 1 m. 0.9 m will grow ahead of the ant and 0.1 m will grow behind it. Therefore, the new proportion is (1+0.1)/(10+1)=1/10 again. You can generalize this like so: choose 0 < a < b. If the ant has made it a/b times the total circumference, the next second the proportion will be (a+a/b)/(b+1)=(ab+a)/(b^2+b)=[a(b+1)]/[b(b+1)]=a/b. The ant's progress as a percentage of the total circumference can only increase, and it happens to be the case that it increases just fast enough that it will eventually make it to the end.
I'm bad at math but I find these videos intersting. He said that it moves a centimeter and the band stretches a meter per centimeter moved. And then he says the percent he's gone every time it stretches but wouldn't he stay at 1% each time and go nowhere? 1/100, 2/200, 3/300... ???
It will take the ant e¹⁰⁰-1 seconds, which is exactly 851,830,411,826,888,556,414,645,353,448,588,313 years 144 days 18 hours 45 minutes and 42 seconds (1 year = 365.2425 days)
In school on my first test I made a 50%. Then I made a 33%. Then I made a 25%. So I told the teacher, that’s ok because that means eventually I’ll get a 100% for the class, so just give me my A now and save us the time.
Question after some light thinking for several seconds done by yours truly: What happens to all the constants like e, PI etc if you move them from base10 to base12? It might be a spectacularly boring result but I just had this thought rushing through my mind haha! :D
e to the gamma? aahh now you went and peeked my curiosity. There goes the rest of my week. Thanks. :P No really I am super interested in hearing the rest of the explanation of how e / gamma knows about products of primes. Please elaborate!
The ant problem can be interpreted in more than one way since it is only given that the circumference increase 1 m each time the ant move 1 cm. This is because we are not given the details about how the additional increase in length is added, if we are to assume that it is added in such a manner than the distance the ant travel at any intervals does not change each times the circumference of the circle has to be change or changed then the ant will never go round the circle. However from the videos I'm certain that the increase in length in added in a manner that the distance the ant travels by the ant in any interval increase each time a new length is added such that the ant in actuality is having an acceleration each time a new length is added and thus will eventually go around the circle.
Exactly, that's why it can be interpreted in more than one way. In this video the increase in length is add such that the % of the distance the ant travels also increase with each increase in length. However, if we put this in a straight line AOB where A is a starting point and O is a point the ant travels at any intervals and B is the destination and the increase in length is added not as a whole but from point B only then the ant will never reach the final point. Just like this if we assume that the additional length does not change the distance the ant travels then the ant will never go round the circle. I don't think they specified that particular parameters on how the additional length is added which in actuality is quite important or i might missed it where they do specified.
***** I agree, he didn't say it very well, I understood it wrong too at first. I thought about it a bit after and the numbers put on the paper really helped me realize how he meant it. I think it cleared out other possibilities in the end.
the elastic band solution is a trick , each fraction is from a different size elastic band so i dont see how just adding every fraction from a different size band until you reach 100 would get you a whole circuit of a band that seems to grow to infinity
the band does not grow to infinity. 1 - the ant walks 1 cm - 1% of the distance. The band stretches, to 2 meters, the ant is still 1% of the distance around. 2 - the and walks 1cm. this is 1/2% of the band's length (2 meters). it is now 1.5% of the way around. the band stretches to 3 meters, the ant is still 1.5% of the way around. 3 - the and walks 1cm. since the band is 3 meters long at this point, this is 0.33% of the total distance, so the ant has gone 1.833% of the total distance at this point. the band stretches to 4m, and the ant is still at 1.8333% of the distance etc etc. the ant always walks 1cm, when the band stretches this represents a smaller and smaller percentage of the total distance. the thing to realise is that at each step the percentage of the total distance the ant has travelled always increases, and it does so in a way that it can get arbitrarily large (harmonic series), so eventually the ant gets 100% of the way around. he says it takes about e^100 seconds, so the final length of the band is about e^100 m, but the ant did not actually have to walk e^100 meters, each step he gets a little bit of a 'free' boost by the stretching of the band eg at step one he is 1cm along, 1%, then the band stretches to 2m, he is still 1% so after stretching he is 2cm along: he has walked for 1cm but travelled 2cm all up. for step 2, he is at 1.5% and the band stretches from 2 meters to 3 meters, so he walks 1cm but travels from 3cm to 4.5cm.
Each value in the series is consistent with the next. The actual value of the circumference doesn't need to come into the problem as long you have the series in front of you and know that each value in the series is correct and represents the percentage of the circumference walked during that second. If you just had the series and the knowledge that each value in it was 'the percentage of the total circumference walked in that second' you could even assume the circumference constant and that instead the ant is just walking slower and slower each second, you would get the same answer.
H I missed the stretching part behind the Ant, now its understandable why over time the Ant can get around the band b/c it never loses out on any percentage already covered when the band expands but gains a centimetre after each stretch, while it will take ages to cover the band its now explainable why it can. I think he did a poor job explaining which is why even Bradypus did not grasp it.
No, there is no value you can assign to the Harmonic series. In general the infinite sum of 1/f(x) where f(x) is a linear function are the only infinite sums you can't find such a "gold nugget" for. I am not completely sure about that last statement but it should be correct.
Can one say that 'infinity' is the 'gold nugget' for the harmonic series? On the complex plane, there is only 1 'infinity' after all (unlike the real line which has 2 infinities).
I'm sure there is a more rigorous explanation but in complex numbers you don't compare them like z1 < z2. You need to take the modulus |z1| < |z2|, which is true or false for instance. In the same way, it doesn't make sense to say z approaches + infinity, only |z| approaches + infinity (there is more to it than this and I am waving my hands quite a bit). This rules out |z| approaches - infinity because modulus is always positive. Therefore, there is only 1 complex infinity.
Since a lot of people are confused about this, I'll explain a small part of the ant story. Instead of looking at how many meters he's traveled, look at what portion of the circle he's traveled. (I should probably use radians, but degrees have a nice visual property to them.) At the start, the rubber band is 1 m long and the ant travels 1 cm. That means he traveled 1/100 of the circle, or 3.6 degrees. In the next second, the rubber band is expanded to 2 m, so the 1 cm that the ant travels is 1.8 degrees, or half of 3.6. In the third second, the rubber band is 3 m, so the 1 cm that he travels is 1.2 degrees, or 1/3 of 3.6. After n seconds, the ant has traveled 3.6 x (1 + 1/2 + 1/3 + ... + 1/n) degrees. Since the sequence in parentheses has no limit, it will eventually reach 100, at which time the ant will have traveled 360 degrees, or one full rotation.
I was thinking about this, and wondered if it implies that any series whos terms get smaller will diverge but we know this is not true. Im trying to think of an analogous thing that gives us 1 + 1/2 + 1/4... and see why the ant cant get around
Imagine the ant travels 1 cm but the band doubles each time. So he travels around 1% then 0.5% then 0.25% so 1+1/2+1/4+1/8...→2 so ant would only make it 2% of the way around at infinity😅
as a young kid interested in math i was super interested in the euler mascheroni constant, as it seems that no one knows anything about it, but the more you research, the more you realize you need to know complex anaylsis to even get 1% of the knowledge about it.
So the harmonic series diverges because it's bigger than another series that diverges. But since the elements of 1+2+3+4+... are bigger than the elements of the harmonic series, why wouldn't it diverge as well? I've seen the Numberphile video on why it's supposedly -1/12, but there seems to be some inconsistency in the logic here.
it's consistent in a weird way, but the trick is what kind of 'summation' you use. what is inconsistent is that they don't make that clear in this video
1+2+3+..... diverges as well, it just has the value of -1/12 attached to it. Watch the video they made if it really interests you. If you then still think about it: the rabbit hole is deep :)
@Joji Joestar Dude, I posted that comment five years ago. I can't even remember what I was talking about then. Since you've only been on this platform for a couple of years, let me give you a piece of advice: let ancient comments rest in peace.
The conclusion is epic, this is what I love about science, how things that seemed unrelated actualy have a lot in common and finding the bridges between science fields is a true delight !
4:50 - If you look at the path around the rubber band as a linear length and assume the distance behind the ant is not also stretched (that all new length is added to the end), the ant would never reach the end. You're adding 1m to the length, and the ant subtracting 0.01m from it. -0.99 - 0.99 - 0.99 - ... goes to -∞. However, if you _do_ assume the distance behind the ant is also stretched, then yes, the ant will make it for exactly the reasons given. According to the numbers I crunched with WolframAlpha, it would take the ant roughly 15,092,688,622,113,800,000,000,000,000,000,000,000,000,000 (~15 tredecillion) seconds to finish. This is equivalent to about 478,600,000,000,000,000,000,000,000,000,000,000 (478.6 decillion) years, or 350,000,000,000,000,000,000,000,000 (350 octillion) times the _age of the universe_. That's no ordinary ant. EDIT: 5:00 - I paused the video to work out the numbers, so I didn't see him do it himself until afterward. XD
At first glance, the ant problem looks simple enough. After 1 second, the ant has traveled 1cm/100cm which = 1%. After 2 seconds, the ant has traveled 2cm/200cm which = 1%. After 3 seconds, the ant has traveled 3cm/300cm which = 1%. etc. etc. But upon closer examination, we must consider that the band is being stretched at the end of every second and that the distance the ant has traveled is being stretched proportional to the rest of the band. So after 1 second, the ant has traveled (1cm + 1cm(100/200))/200 = 0.75% After 2 seconds, the ant has traveled (2.5cm + 2.5cm(100/300))/300 ~ 1.11% After 3 seconds, the ant has traveled (4+1/3cm + (4+1/3cm)(100/400))/400 ~ 1.354% This would continue until the ant dies of starvation or old age. So the answer is no. An ant would never reach the finish line.
After 1s, the ant has done 1% of the band; after 2 s it has travelled 2cm and the band is 2m, so 1%; after 3s it has travelled 3cm and the band is 3m, so 1% - it never gets more than 1% around the band. How to solve this paradox?
but it does go forward, the distante it already traveled also increases, so in the second 2, that 1cm (1%) has increased to 2cm (1℅) plus the 1cm it traveled on that second.
Ah. Got it. The band isn't only expanding in front of the ant, it is expanding behind as well. So the ant gets some 'free' distance travelled help from the expansion. So although it only travelled 1cm in the first second, during the second second that 1cm expands to 2 cm, PLUS it does another 1 cm. Thanks for explaining.
It doesn't make sense. If the finish line is moving away faster than the ant is walking, it will never get there because its velocity relative to the finish line is negative. What am I not getting?
The question at 5:17-27 is interesting. I will try to address it a bit in a bit more detail. At first glance one might agree that 'every second the finish line is getting a metre further away'. If that were true, the ant clearly would indeed never get at the finish line. But it isn't true, because gradually more and more of that metre is added behind the ant! If at some point the ant has reached, say 30% of its way, then 30 cm will be added behind it and only 70 cm ahead of it. And after stretching, it is still on 30% of its way. That explains why talking about percentages is a natural and correct approach.
I'm working on a proof that gamma is irrational, I have it down to a double infinite sum, that I need to prove is an integer. That sounds easy, but it has to be shown for all possible integer values of b. Ultimately I will probably fail cause it would have been done by now if it was that easy.
I could share it, but its on paper right now. I basically have gamma = 1.5-D, where: D = SUM[n=1,inf]SUM[k=1,inf]g(n,k) Where g(n,k) = h(n,k)/(n^k) h(n,k) = ((-1)^(k+1))/k - (n-1)/(n+1) So D is 0.92278433509 approximately. We can see h(n,k) converges to -1 as n and k tend to infinity. I have a bit more, but its on some paper somewhere, its based on the proof that shows e is irrational
I actually discovered the number gamma myself after plotting the harmonic series and noticing is was similar to ln x, I took the difference, and it approached 0.577. It was about 4 years later that I heard about this number again, and heard no one knew if it was rational or not.
What did you mean by 'first' series? The 1+2+3+4+... everyone agrees that it diverges, so the proof works there. If by first sequence you meant 1+1/2+1/3+1/4+... (and second series 1+1/2+1/4+1/4+1/8+...) then we have to look at what you mean by proof: I would say the second series is used to prove the first, and hence the proof works. But more specifically, to prove a sequence converges, you have to check whether for any number N there is a finite 'stage' in the addition that is bigger then N, where the stages are in our example (1, 1+1/2, 1+1/2+1/3,....). To prove this for the first sequence, one uses the second series and the fact that since the second series is 'smaller' then the first, any stage is also smaller.
it does work for the first: in "normal" mathematics, 1+2+3+4+5+... =infinity, you have to slightly alter some rules in order to obtain 1+2+3+... =-1/12
the problem I have woth this reasoning is in first rule arithmatic system you cant evaluate one expression to two different values and claim it to be consistent. so either the series is divergent or evaluates to a number and if both are true then somehow our arithmetic is flawed
that's clearly not arithmetic. Do you accept 1 + 1/2 + 1/4 + 1/8 ... = 2 easily ? Well this isn't really a normal sum, this is just the smallest number that is bigger than the sum of any subset of this infinite set, we call it the sum of the infinite series and it feels natural and intuitive... Then how about... 1 + 0 + 1 +0 +1 +0 = 1/2 ? Does this make sense??? In a way it clearly does, but the difference between a normal sum and this is more apparent. Still this number is meaningful to this series, and more useful than saying that the sum of this series is "ERROR" or divergent. We are just extending the definition of what it is to sum to be able to use it in more cases, because it's useful. If you keep going like this you end up with stuff like the famous -1/12 one, is just an extension of a function (the sum) to be able to use it in cases where it was before undefined (we extent it's domain).
Really interesting video, thank you Numberphile :) It's a pity that we can associate a finite value to 1+2+... and we cannot do this to the harmonic serie. That makes we wonder lot's of things, maybe one could enlight me on some points ? Thank you again. (a) Are we sure that we can't assign a finite value to the harmonic sum, even if we use a different function from zeta ? (b) I mean, the zeta function is not the only consistent way to attribute a finite value to an infinite sum, or is it ? (c) It seems like there's kind of a "divergent series algebra" (separated or extended from the "classic algebra", i.e. with infinite divergent sums) : does this "extended" algebra have a name ? what is allowed ? what is not ? (d) Are there series (harmonic or others) that cannot be assigned a finite value, even if we use other "zeta" functions ? or the harmonic serie would be the only one ? (e) For example, i've heard that 1+2+4+8+... = -1. The classic real answer would be 2^(n+1)-1 with n->infinity. I guess we can also say that 1+a^2+a^3+... = -1, given that the classic answer would be a^(n+1)-1, is that correct ? (f) Do these last kind of finite values are equivalent/consistent with to the zeta finite values ? (g) This serie (sum(s^n)) looks like to me kind of a zeta dual function , is it related to zeta ? does it have a name ? (h) Is there a simple/intuitive way to understand the trivial (which for me is not) zeros of zeta : sum((2n)^s)=0 ? Sorry for this long list and if you have been, thank you for reading :)
Imagine a photon instead of ant and expanding space as the rubber band. Can the light reach a distant galaxy despite more distance is being created per unit time than the light can cover? Maybe this is still impossible because the space expansion is accelerating and the rubber band grows linearly.
It is theoretically possible IF the space is expanding in the same rate everywhere, so more space is also created before the photon, pushing it forward (relatively).
1:05 i work with music & i've known about the harmonic series for years & make use of it daily but somehow still didnt know that was the equation for it, i've only ever really seen it visualized till now
I remember this from calculus class back in the previous millennium. I had always assumed that this must be irrational and transcendental; I had not realized that the question is unresolved.
4:41 unless the ant has moved to the percent at which it was already at, the ant would be at a lesser percentage at the start of each time frame, making it impossible to reach the end. Tell me if I'm wrong id love to understand the situation
A fact in this video helped me win a silly math contest! Knowing that the number of terms needed to reach a number with the harmonic series is about e to the power of that number allowed me to know if something was under or over 9000 lol
Mr. H Unpopular opinion: The _way we think_ about the universe is mathematical. We often forget to look at the basis of our understanding, which is our mind. Essentially, we are using a coloured glass to interpret that the universe is coloured. I don’t necessarily agree with that, because it is the eyes that are ultimately looking (at the universe), not the glass.
VERY interested in the cosmological speculations at the end (8:50++). This idea that numbers "know" things about each other is certainly not groundless mysticism.
I have a question which is yet to receive a satisfying answer. There are infinite series that are divergent and there are infinite series that are convergent, there appears to be a ratio of decrease between the elements of each set that determines wether or not it converges to a limit. Is the Euler-maschroni constant this ratio? Or is there a separate constant? Essentially what's the number that determines wether an infinite series will or will not converge?
It would be, as said in video, 10e50 meters. For comparison, the observable universe is 8.8×1026 m (thx again wiki). So yeah .. xD is the word here ! :p
I feel like this constant is transcendental because log(n) (assuming it is natural log) is always transcendental, meaning we are subtracting a rational sum by a transcendental number, no matter how large we get. If you continue plugging bigger and bigger numbers into 1+1/2+1/3...1/n-log(n), you will always have a transcendental number, meaning the Euler-Mascheroni constant tends to a transcendental value... right? also this constant is basically what infinity-log(infinity) is equal to (i know this isn't how it works but shut up)
the example with the elastic band can be misunderstood easily, so Brady's argument isn't that wrong actually: if you just say "every second, we add another meter to its circumference", you can always add the additional meter in front of the ant, and of course it will never reach the end that way. instead, you have to emphasize that the band is stretched, i.e. it is uniformly expanded so that every part of it grows the same percentage. that means the distance behind the ant and and the distance in front of the ant grow proportionately to their relative size, and as the distance behind the ant becomes larger and larger in relation to the distance in front of it (it will since the ant travels), more and more of the additional meter is in fact added behind the ant, not in front of it.
I still dont get it :/
ok, let's look at it in detail:
1st second
traversed distance = 0 cm, distance ahead = 100 cm, total distance = 100 cm
ant moves 1 cm from start: traversed distance = 1 cm, that is 1/100 = 1% of the total distance.
2nd second
band is stretched 100 cm: traversed distance = 2 cm (increases with stretching!), distance ahead = 198 cm, total distance = 200 cm
ant moves 1 cm: traversed distance = 3 cm, that is 3/200 = 1.5% = 1% + 1/2% of the total distance.
3rd second
band is stretched 100 cm: traversed distance = 4.5 cm, distance ahead = 295.5 cm, total distance = 300 cm
ant moves 1 cm: traversed distance = 5.5 cm, that is 5.5/300 = 1.83333..% = 1% + 1/2% + 1/3% of the total distance.
and so on...
every second the ratio of the traversed to the total distance increases, until it finally reaches 100%.
+Amphithryon I feel like the guy in the video failed to emphasise this: the part he has already travelled ALSO stretches, and then the ant moves 1 cm independently from that stretch. The way he explained it sounded like wishy washy -1/12 stuff, while it's actually really logical.
I don't think he failed, he used a rubberband for that sole purpose. Just before the words you quoted he says "we are gonna stretch so that..." and then you go to say he should emphasize the stretching... when 3 word prior to your quote he did.
What is probably wrong in this case is the drawing with the ant in the circle, but it should be ok still unless you forget that it's a rubberband and that we are stretching.
@Amphithryon very nice put. Tony clearly gave the correct response to Brady's argument, that what is behind is growing as well. He just didn't mention (or it's cut out) that the growth is exponential to time, and as soon as the ant hits half-way it grows faster than what's in front of the ant. Which is why eventually the 1cm is longer than the growth in front of the ant.
I love when a Numberphile video has a mindblow moment. It grows BEHIND it too!
See my question/response above. Thanks Rafael. if the band only expanded in front, the percentage travelled would always be 1%. But because it expands behind as well, the percentage is always growing, albeit very slowly.
It grows in between. Have you ever thought about that?
I just laughed out loud when he said that. Amazed, yet at the same time really had no idea what that meant for the whole problem.
I wish I was smart enough to grasp really amazing ideas like this.
Exactly! It was a mind-blowing moment for sure. I mean, it seems so obvious in hindsight, but I would have been busting my head for days trying to come up with a logical explanation for that.
Your mind shouldn't really be blown realizing that. It's pretty straight forward
If you add 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8, you are just adding 1/2 over and over again, so we can clearly see that it diverges.
And if you add 1+2+3+4..., you are just adding 1 over and over again, so we can clearly see that it........goes to -1/12
Update: I watched the extra footage, as well as Mathologer's video on it, and things are cleared up now. Both series diverge in the traditional sense, but can be "analytically extended." In other words, if we accept new definitions of sums (Cesaro, Abel, Ramanujan) where traditional sums don't give a finite answer, we can also accept finite answers for divergent sums.
Also a little annoyed that "gamma" is different from the Riemann Gamma function, but who cares lol
I can't see any references for Riemann Gamma function.. do you mean Riemann Zeta function? The extended Zeta function includes the Gamma function though
That function is not called Riemann Gamma function, it's just Gamma function. As far I know, Euler originally gave a version of the Gamma function first as an infinite product, then he represented it with an integral. This was in the 18th century. Riemann would come much later 19th century. The Gamma function and Zeta function are just related, but the gamma function is not the work of Riemann. And I couldn't find any references for "Riemann gamma function":
Whoops yes you're right about that.
I was saying how it's weird that there's a "Gamma function" and there is a "Gamma" as a constant.
There's an older Numberphile video on this, actually one that got them famous. There was a lot of fuss about it commenting on the same thing (the sum of 1+2+3+4+...=-1/12) and the gripe that echoed around was that they didn't really define the extended sum as a function rather than a summation. So that is what they actually refer to at the beginning of the video saying "We're gonna start in a familiar place".
All I know is 1+2=12 and even that might be wrong.
I have got bad news pal.
DidJewNaziMe just add quotation marks
“1”+”2”=“12”
Now it’s correct (:
Tin Can't in Python.
True
1+2=3.
@@super-awesome-funplanet3704 genius
I worked out the ant band time:
The circumference of the band is given by C = t+1 where t is time elapsed.
Distance travelled by the ant is s.
The speed of the ant seems to be 0.01 m/s, but also has a component given by the band stretching behind it, which gives it further displacement. The rate at which this happens is the proportion of the band that the ant has already travelled across at a given time: s/C = s/(t+1)
So we get the differential equation ds/dt = s/(t+1) + 0.01
(ds/dt)/(t+1) - s/(t+1)^2 = 0.01/(t+1)
d/dt(s/(t+1)) = 0.01/(t+1)
s/(t+1) = 0.01ln(t+1) + c
s = 0 when t = 0 so c = 0
s = 0.01(t+1)ln(t+1)
The point at which the ant makes it back to the start is when s = C = t+1:
t+1 = 0.01(t+1)ln(t+1)
1 = 0.01ln(t+1)
t = e^100 - 1
This deserves more likes.
damnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
Wow!
Is the band expanding smoothly, or in one second increments? Or does it matter?
@@rglrts as far as I guess, bands must expand smoothly
oil and macaroni constant?
Euler is pronounced "oiler"
BigBoatDeluxe In my defense it was kinda late when I read it and wrote my reply, nonetheless I still feel like a dumbass.
Don't say oil, USA will invade then XD
nah we just want the macaroni.
also we want you to stop saying mathS
Also, it is pronounced maskeroni.
LET'S BRING THIS TO THE TOP
This video was more closely related to the armonic series than 0.577.
You can't just say "0.577 appears all over physics" and "it knows about primes too" and not expect me to demand a more in depth separate video with .577 as its star.
It's very little what is Know about that constant, in 300 year there was a y significante avance to understand it's Nature, it's only that is appears when You combine theory of numbers and calculusz nobody knows why
dont think ants live that long
no sh*t sherlock
also, the rubber band would break.
The rubber band wouldn't last a single second.
It would
Yes it would. It'll depend on the band, but would last a few seconds.
Certainly an interesting one. Have seen .577~ pop up from time to time in the maths I work with in computer graphics and physics simulation. Always thought it was just the result of some personal bias, a product of how I do things. Never realized it actually had a more profound meaning.
It pops up in the estimator of a frequency factor for some statistics, as does a transformed version of π^2/6.
@@andrerenault tan 30 degrees = 0.577 so I'm not surprised.
It's also the inverse of the square root of 3.
2:50 "I knew you were going to say that". That's because solving the Reimmer zeta function for a divergent series does not give you an equivalence for n=infinity. It is an "associated" value, not the "answer" of the series.
+D but isn't it amazing that of all the values to be uniquely associated (whether by analytic continuation or ramanujan summation) that it is -1/12 - see our gold nugget video all about this.
Numberphile yes, but please let us stop using the equal sign for it. We are missing the point of the "conversion" that happens when we do this. I think that is a lot more interesting than the illogical use of "=", which is simplistic and lends itself to being disproved by counter example.
So the video is incorrect?
+Paul Ahrenholtz
The video interprets [Correction: the notation of series like "1+2+3+..."] in two different ways without telling you. If you know which one they're talking about at which time, everything they say is correct. The vast majority of people don't, which leads them to confusion and/or incorrect conclusions.
Nicolas Bourbaki what 2 different ways?
A numberphile video on numbers, they are getting really rare these days
There are only so many numbers, man
numbers are infinite...
David Perrier technically that also incorrect... the interesting numbers that *we know of* are finite. But if there's infinite numbers, there's infinite equations that do something interesting
Yeah, those infinite videos would get a ton of views
Inwë Meneldur Wow, clever
I can see motivational poster with that ant story coming up
"Keep slogging away at your Sisyphean task, until you die or the universe is destroyed."
Somehow I don't feel motivated.
suncu91 well, not sure if that will bode well in popular
Just 2 cm left! Almost there!
*travels 1cm*
*band increases another m"
fuck...
You didn't understand the problem.
At that position, 1 meter scaling will be insignificant, because 99.99999999...% of the path is already behind the ant.
This means that the Ant will finish the last centimeters without noticing any change in path size.
André Canilho I did understand, I was just making a joke. But you're right the last few meters will be adding less than a cm in front of him since 99.9999% will be added behind him
So it's the % of the meter increase (relative to ant) that gets smaller as ant goes till eventually the increase relative to the ant is smaller than his cm traveled per increase. This was such a cool problem!! I did using excel spreadsheet as ant going 1 cm and circle increasing by 2cm (starts 2cm big) and when circle reaches 22cm the ant has gone fully around the circle.
Could you show somewhere how did you do that? I've tried it myself, but it didn't work. Maybe I'm doing it wrong
I don't get it
0cm 2cm
1cm 4cm
2cm 6cm
3cm 8cm
4cm 10cm
5cm 12cm
6cm 14cm
7cm 16cm
8cm 18cm
9cm 20cm
10cm 22cm
It seems to get closer to 100% but say in 10000 increaces the ant has gone fully around, but logically thinking the circle 10000 * 2cm = 20000cm and the ant 10000 * 1cm = 10000cm, so only 50% :0
MV AntDist Cir % traveled
0.00 2 0.00%
1 1.00 2 50.00%
2.00 4 50.00% stretch
2 3.00 4 75.00%
4.50 6 75.00% stretch
3 5.50 6 91.67%
7.33 8 91.67% stretch
4 8.33 8 104.17%
Mv - ant moves
ant Dist- total distance relative to viewer of circle (not ANT)
%traveled - the distance viewer sees him travel.
I made a mistake, it is only 4 moves. Hope this helps
That's how I understood it too
+erikeeper thats an error in your logical thinking
you are at:
0cm 2cm
ant walks 1cm
1cm 2cm
the circle is expanded and since you are 50% you will be moved(the ant) as well, so
2cm 4cm
then ant walks again ending at
3cm 4cm
and so on. with this example the circle is completed very quickly, he uses 1/100th to make it only more confusing
Love the excitement he exhibits when he talks about this stuff. Way over my head but fascinating nonetheless.
Hi there
@@emmanuelmanu927 Hi here
@@puppergump4117 Hi where
@General12th Hi when
Your passion for mathematics is infectious. Gauss, Newton, Leibnez, Pascal, Euclid, Pythagoras and Archimedes are all subscribing to your TH-cam videos.
This has got to be one of my favorite Numberphile videos. Just completely thought provoking.
"Yeap, -1/12,totally not controversial"
Someone has been working on his sarcasm skills
I wonder could that someone be
Sarcasm is the English way of communication, it's what the foundations of our society is built on
unintuitive and unexpected results are the best.
I love how they know the whole -1/12 affair works, at the very least, as a delicious trolling act. (but of course the video itself was so insightful it went over many people's head)
5:32 expansion of the universe
I am always curious what the original motivations were for investigating things like these.
he mentions that he knows of it because of physics and quantum stuff, but Euler and Mascheroni wouldn't have. And yet they calculated it to so many digits.
were there older uses for this number? I'd like to hear more background into the origin of euler's work and others
Just commenting to be notified if anyone responds ;)
same
they probably just dealt with patterns and eventually came up with it all the time and went on a binge trying to figure out what it did.
Curiosity.
Makes me wonder. So many purely mathematical constants crop up in nature; gamma, root 2, e, pi, golden ratio etc.. It's like the laws of physics are based around mathematical constants that can be derived without needing any physics. Now maths is going to be the same in another universe, so I am sceptical about the laws of physics being different.
how long would the rubber band be when the ant passes the finish line?
very
E to the 100 meters :P
it grows by 1m per second, and it would take the ant around e^100 seconds so I would say around e^100 m.
as many metres as seconds it takes the ant to cross it, so 3 tredecillion metres
if it's 1m long at the start and expands by 1m every second... hmm.. How many seconds till ant gets to the finish?
Truly thought provoking! This is numberphile at its best, Brady!
It's fun to see Brady's reactions in the window reflection
1+2+3+4+5... also diverges in the normal sense of the term(meaning the series of partial sums diverges). It's only said to equal -1/12 because of the Riemann zeta function.
It saddens me to see such an amazing channel, with less than 2 million subscribers. Where are all the Numberphiles out there?
Unfortunately, "school mathematics" is generally presented in a boring fashion, oddly disconnected from reality.
I can't help but triple facepalm when I hear someone, as they too often do, say "so, what's the use of maths?".
That's how poorly it's presented to most people, as they seem to think that accusing the greatest "transferable skill" there is and basis of all science and technology and engineering - and even music - of being "useless" is not only a reasonable, but even a clever, thing to say.
It's like handing over a suitcase with a million dollars in it to someone, they look inside and then hand it back to you because it's "just full of paper".
Well, yes, but you really have no idea just how much you're totally missing the point there.
Kim Kardashian just got robbed. That's more interesting!
2m is quite a lot.
Thats true, that russian guy on numberphile has a video about why people hate maths.
Maths at school level can be quite dry. Some people think its all about arithmetic, but that's just one of the fundamental tools needed in the majority of math areas.
Maths is interesting cause its like a whole other world that exists abstractly that has so much to be discovered.
But also, maths is the language we use to describe the universe, and also we can use it to solve problems and build things like computers and particle colliders.
So its both interesting and useful and is sort of integrated into reality itself which makes it cool.
ZeanutJam Yes, that's who I was talking about, he seems like a very smart man.
Numberphile comments are the only comments on youtube i enjoy reading as there is some level of discourse amongst the subscribers that doesn't devolve into meaningless drivel and name calling. You can actually learn something from the comments, which goes to show you who watches these types of videos...
A fun fact is that it shows up in the 'block stacking problem' (or the Leaning tower of Lire). The idea is that you stack blocks or bricks on top of each other on an edge of a table and make the stack of blocks lean over the edge as much as possible without it falling over. Then you want to know how many blocks you need in order to make the tower lean over the edge, for example 4 times the legth of one block. You can calculate the exact number of blocks you need by rounding (to the closest integer) the value of this formula: e^(2*o-y) where "o" is the number of brick lengths the tower leans over the edge and "y" is the Euler-Mascheroni constant.
Does his ant on a rubber band example also involve gamma? He doesn't really make that clear in the video. It seems like it just demonstrates a property of the harmonic series. But I've noticed that whenever stretchiness is involved, logs or exponents seem to be involved, so I'm wondering if maybe it does?
It is easy to see see that ant gets to the finish. If ant does not move while rubber is increasing, angle relative to starting position remains same. As long as ant is moving forward, angle of ant relative to starting position is increasing and it will eventually reach the finish line .
The sequence where you continually add on increasingly smaller numbers sums to infinity, but the sequence where you continually add on increasingly larger numbers sums to a negative fraction. Cool
I wonder what the slowest growing infinite sum is.
It´s : "the slowest growing infinite sum"
Take the an infinitely small number and add it to itself an infinite amount of times...?
for d = 1/inf and x=0, calculate x+=d until x=inf
You can always find an infinite sum that grows even slower by increasing the rate by which the denominators grow, so there's no slowest.
The sun of the reciprocals of g_n, where g_64 is the infamous Graham's number.
Consider the expansion of the universe during inflation in which at each interval of time (Planck's time = 5.39 x 10^-44).. the universe expands one Planck length.. and that a packet of energy starting at some point circumnavigates around the entire universe.. does it get back to its initial starting point? if so.. how long would it take?
My head hurts
I think plank's time/ plank's length =light speed
why would you not just say light speed
if one planck length per planck time is the speed of light, then how can anything move slower? Nothing cam move less than a planck length. Like, 1 planck lengths per 2 planck times is half the speed of light, so something moved half a planck length in 1 planck time? That can't happen.
You can't measure anything less than one Planck Length. So to measure something that moves at half the speed of light you would have to wait for it to move at least 2 Planck Lengths which would take at least 1 Planck Time. Essentially it is not possible to observe or measure anything less than a Planck Length or a Planck Time. So likewise we cannot say something moved 1 Planck Length in anything less than 1 Planck Time.
2 Planck lengths in 1 planck time? That would be twice the speed of light, not half.
I absolutely hate it when natural log is just written as log. It's ambiguous. Please, just use ln.
Outside of early maths, log very rarely refers to base 10. In pure maths log almost always refers to ln, as base 10 isn't relevant and in computer science log is often used for base 2, because binary.
@@Nebula_ya Engineers still typically use log to mean base 10. And regardless, there's still no reason to have it ambiguous. There's a perfectly unambiguous way to write it.
I love the fear in brady's voice as he is given Vietnam flashbacks of the divergent sum of natural numbers
The sum 1+1/2+1/3….. +1/n is very easy to calculate. It is an integral of 1/X where X changes from 1 to n, which is natural log of n. So if n is infinite, then the sum is also infinite. The question is why the numerical value differ from the integral value by 0.577.
I could see Brady's reflection onto the glass pane behind Dr. Padilla.
The ant and the elastic band example for the sequence was initially perplexing to me. Then I fired up Excel and it became clear!
The following are some of my results with different ant step lengths - the first number is the size of the ant's step and the second is the number of steps needed to complete the circumference: 15=441 14=710 13=1,230 12=2,336 11=4,983 10=12,367 9=37,568 8=150,661 7=898,515.
I am interested to learn the formula needed to work out the number of steps required for 6, 5, 4, 3, 2 and 1 step sizes (all the way to 3 tredecillion years!) Is there a formula that can be applied to calculate these from the sequence shown?
Since the ant's steps (in the video, with starting step size 1) are 1/100, 1/200, 1/300, etc. of the circumference of the loop, you just have to see how many terms of 1/100+1/200+1/300+... you need to add up until you get to 1. That's equivalent to adding up terms of 1+1/2+1/3+... until you get to 100. Of course it's too many to compute *exactly*, but you can use the approximation that 1+1/2+...+1/n ~ log(n)+ɣ and set log(n)+ɣ = 100 to find n = exp(100-ɣ) = 1.5×10^43 steps. (I know it says 10^50 in the video but I think that's a mistake, e^100 is more like 10^43.) To do this with starting step size 2, 3, 4, or anything else, just replace 100 with the ratio of the original circumference to the step size, i.e. use the formula n = exp(ratio-ɣ) to find the number of seconds:
exp(100/6-ɣ)=9,717,617
exp(100/5−ɣ)=272,400,600
exp(100/4−ɣ)=40,427,833,596
exp(100/3−ɣ)=168,190,380,070,122
exp(100/2−ɣ)=2,911,002,088,526,872,100,231
These will be a little bit off from the true number of steps because the partial sums of the harmonic series are not exactly log(n)+ɣ, so if you wanted to you could redo the math with a better approximation log(n)+ɣ+1/(2n), but it wouldn't change the results by very much and would make it a considerably more difficult calculation.
I know this comment is from years ago but I thought people reading it in the future might like to see the method.
I absolutely love this kind of stuff.
And here I thought I was the only one using this image. Cheers 👍
@@realitywins6457 Seems we're ... Mandelbros!
@@HungryTacoBoy Ha, that sounds like an eclectic, intellectual, indie-hipster band from Seattle
@@realitywins6457 When they tour they have include other players to fill in the missing parts of their sound.
@@HungryTacoBoy It would have to then be an endless tour, forever parsing their rythms into more complicated patterns
5' 30'' el dibujo lo deja claro para entenderlo. También crece por detrás. Una explicación fantástica del profesor Padilla.
Can the ant problem be modelled conversely? Surely, by showing that after every second, the ant is further from the end than it was a second ago, demonstrates that it will never reach the end.
Seems like the ant’s infinity is smaller than the rubber band’s infinity.
Mascheroni sounds like what you'd get if you successfully pureed mackaroni.
So how do you explain that 1+1/2+1/3+... is bigger than 1+2+3+4+... - when every member of the first row is equal or smaller?
You say that because your intuition is thinking about inequalities of finite sums. For finite sums, A1 < B1 and A2 < B2 implies that A1 + A2 < B1 + B2.
But you shouldn't assume that this property holds true for infinite sums and it's actually not the case.
Order is very important for those kind of sums.
Actually if you did that : (2 + 1) + (4 + 3) + (6 + 5) + (8 + 7) ..... it's no longer equal to -1/12 (it's actually +5/12)
It isn't.
Both are divergent series. And if you have shown, that 1 + 1/2 + 1/3 + ... is divergent, then direct comparison (keyword: direct comparison test) directly implies, that the sum of natural numbers has to diverge as well.
In another numberphile video they (including the physicist in this video) spread the word, that the sum of all natural numbers would be -1/12. And while there are ways to assign a finite value to a divergent sum (see f.e. en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF ), in the 'classical' sense both series are simply just divergent and approach infinity.
Actually there's an other example where this happens : every member of 1+2+3+4+.... is greater than those of 0+0+0+0+0....
yet the first sum is negative and thus smaller than the second sum ;)
The problem of this is that in the video it is explained that the geometric series diverges because 1/2+1/2+1/2+... "clearly" diverges.
I can see how this can be confusing for people who watched this and also 1+2+3+...= -1/12
Eagerly waiting for the response on the second channel. That strikes me as a glaring flaw in this logic, given previous insistence that the sum of natural numbers is -1/12. Maybe there's some function out there no one's come up with yet that defines the sum of their reciprocals as -1/11?
I notice the comments here talking about the "-1/12" stuff and claiming either it's right or it's "nonsense". There seems to be a lot of confusion on this and a lot of muddle, and I'd like to post this post to provide a definitive PSA to clear up the muddle, once and for all. I hope this does so.
There are two different notions called "sum" of a series in play here. There is the "ordinary sum", usually just called "the sum". The ordinary sum is defined by a limit, of course. In this case, 1 + 2 + 3 + 4 + ... diverges. It has no ordinary sum. The limit does not exist.
Some say the "ordinary sum" is "infinity", but this is only correct if you are working with the _extended real number line_. If you are using the ordinary real number line, the sum simply does not exist. "Infinity" is not an element of the ordinary real number line. The extended real number line, as the name suggests, "extends" the real number line by adding +/- infinity as new "numbers" at the ends of the line.
Now, there is another notion -- actually, several notions grouped under the same rubric, called a "generalized sum". This is _a different notion than the sum_, defined using something other than the limit of partial sums, although it is not an unrelated one, for it is defined in such a way that when an ordinary sum exists, the generalized sum exists as well and coincides with it, hence the name "generalized". But the generalized sum can exist when the ordinary sum does not. Saying "1 + 2 + 3 + 4 + ... = -1/12" is referring to this _generalized sum_, NOT the ordinary sum. Depending on just which notion of generalized sum you are using, some of the manipulations of the series on the left may or may not be valid.
That's it. It's just a matter of keeping these concepts straight and separate -- ordinary sums (or just "sums"), generalized sums, real number lines, and extended real number lines. Keeping in mind these things are _related_ but _not the same thing_. Other than that, all these concepts are 100% legit. They're just different, and need distinguishing. This point is often lost here. But I don't blame the audience. The problem is the various presenters and tutors out there who just moosh all this stuff together and be sloppy. Being sloppy with math creates confusion all the time and of course we have generations of people raised on sloppy teaching and so people are no wonder, thoroughly confused about math and make songs about hating math and for those who do eventually come to a clearer understanding and maybe become mathematicians, a not-insubstantial part of their learning effort is wasted on _un_learning all the confused muddle bs pumped into their heads by the bad public school system with its confused teachers.
The math isn't wrong. None of these results are wrong. The presenters are bad. Especially when dealing with a lay audience who doesn't necessarily have a tight grasp honed from much experience. An experienced math person can get by with their presentation, but a noob or interested layman will be horribly lost.
This is truly mind expanding stuff. Thank you.
5:10 FYI: A tredecillion years is 10^42 years. The sun would become a red giant in fewer than 10^10 years. When the ant cross the finish line, the rubber band, growing at 1 m/s, would be e^100 or about 10^43 meters. Our current observable universe is about 10^27 meters in diameter. Thankfully it expands at about 10^5 m/s/Mps to accommodate the rubber band.
Not heard the term "lazzy band" in ages lar.
I've never heard that before. Is it a Northern or Midlands thing? Or slang from a certain profession?
Lazzy is just what scousers/people from Liverpool call elastic. e-LAZZY-stic.
Dont think it is used in other regions in North/North west, but is popular in Liverpool
Ohh, so that's what he's saying at 3:03 "As we call it in Liverpool."
we call em laggy bands here in yorkshire
and down south
Hello Numberphile,
your videos are amazing and I truly enjoy watching them!
There's just one thing I'm wondering about:
At 5:00 it says that the ant needs about exp(100) seconds to complete the task. exp(100) is about 2.6881e+43.
A tredecillion is 10 to the power of 42. So the ant needs about 27 tredecillion seconds.
One year has 32,850,000 seconds. If you divide exp(100) by 32850000 you get 8.183e+35.
Shouldn't the result be 818 decillion years instead of 3 tredecillion years?
Yours, Max
I'm really having trouble with the ant on the rubber band. After e^100 seconds, the band is e^100 meters long. The explanation given, rather matter-of-fact-ly in the video, is that the distance behind the ant also increases. But, the distance in FRONT of the ant also increases over time, and thus at e^100 seconds, the ant must still require (e^100 meters - e^100 centimeters) to travel to reach the end.
It really doesn't matter how much distance is behind the ant, if there is still that much distance in front of the ant to traverse.
Well, how about this. An ant that can travel half a meter per second, lets say. Start on the meter long band, just as before. In one second, you are at, naturally, 1/2 of a meter. The band expands by 1 meter, split evenly along its length. So, the 1/2 meter ahead expands to 1 meter, and so does the 1/2 meter behind.
Now, you go again, and you are, again, 1/2 meter away from the finish line. This time, though, the half meter ahead only increases by .25 meters, since it is a quarter of the band. Now you are .75 meters away, which is closer. And now, since the band has grown, the bit ahead of you gets a smaller percentage of that growth, allowing you to catch up, faster and faster.
The same goes for 1 cm on a 1 meter band, but much, much slower.
stellarfirefly But because the distance behind the ant has been expanding the e^100 cm the ant has travelled has also expanded . Take the first iteration. The ant moves a cm, then the band expands by a meter. There is now 2cm behind the ant because the band has doubled in size.
Once the ant reaches the halfway mark, the stuff behind the back stretches more than stuff in the front, essentially pushing it forward.
Think about it this way: even if the ant were to move a certain distance and then stand completely still, it still wouldn't lose its position relatively speaking.
Say the band is 10 m in circumference and the ant has already traveled 10% of it, i.e., 1 m, and then decides to take a break. The next second, the new circumference will increase by 1 m. 0.9 m will grow ahead of the ant and 0.1 m will grow behind it. Therefore, the new proportion is (1+0.1)/(10+1)=1/10 again. You can generalize this like so:
choose 0 < a < b. If the ant has made it a/b times the total circumference, the next second the proportion will be (a+a/b)/(b+1)=(ab+a)/(b^2+b)=[a(b+1)]/[b(b+1)]=a/b.
The ant's progress as a percentage of the total circumference can only increase, and it happens to be the case that it increases just fast enough that it will eventually make it to the end.
I'm bad at math but I find these videos intersting. He said that it moves a centimeter and the band stretches a meter per centimeter moved. And then he says the percent he's gone every time it stretches but wouldn't he stay at 1% each time and go nowhere? 1/100, 2/200, 3/300... ???
It will take the ant e¹⁰⁰-1 seconds, which is exactly 851,830,411,826,888,556,414,645,353,448,588,313 years 144 days 18 hours 45 minutes and 42 seconds
(1 year = 365.2425 days)
my favourite numberphile video yet. Thankyou!
In school on my first test I made a 50%. Then I made a 33%. Then I made a 25%. So I told the teacher, that’s ok because that means eventually I’ll get a 100% for the class, so just give me my A now and save us the time.
Question after some light thinking for several seconds done by yours truly:
What happens to all the constants like e, PI etc if you move them from base10 to base12?
It might be a spectacularly boring result but I just had this thought rushing through my mind haha! :D
I think irrational numbers are irrational in any rational base
@@Septimus_ii Irrational numbers are, but there is some interesting math behind how normality acts in different bases.
e to the gamma? aahh now you went and peeked my curiosity. There goes the rest of my week. Thanks. :P No really I am super interested in hearing the rest of the explanation of how e / gamma knows about products of primes. Please elaborate!
Would be a great video but I suspect it would end up being a bit too complicated.
search for Mertens' theorems if still curious
Thanks I will check it out.
Piqued.
The ant problem can be interpreted in more than one way since it is only given that the circumference increase 1 m each time the ant move 1 cm. This is because we are not given the details about how the additional increase in length is added, if we are to assume that it is added in such a manner than the distance the ant travel at any intervals does not change each times the circumference of the circle has to be change or changed then the ant will never go round the circle. However from the videos I'm certain that the increase in length in added in a manner that the distance the ant travels by the ant in any interval increase each time a new length is added such that the ant in actuality is having an acceleration each time a new length is added and thus will eventually go around the circle.
Exactly, that's why it can be interpreted in more than one way. In this video the increase in length is add such that the % of the distance the ant travels also increase with each increase in length.
However, if we put this in a straight line AOB where A is a starting point and O is a point the ant travels at any intervals and B is the destination and the increase in length is added not as a whole but from point B only then the ant will never reach the final point. Just like this if we assume that the additional length does not change the distance the ant travels then the ant will never go round the circle.
I don't think they specified that particular parameters on how the additional length is added which in actuality is quite important or i might missed it where they do specified.
***** I agree, he didn't say it very well, I understood it wrong too at first. I thought about it a bit after and the numbers put on the paper really helped me realize how he meant it. I think it cleared out other possibilities in the end.
It's amazing how fascinating this stuff can be when you're in the right mood.
the elastic band solution is a trick , each fraction is from a different size elastic band so i dont see how just adding every fraction from a different size band until you reach 100 would get you a whole circuit of a band that seems to grow to infinity
the band does not grow to infinity.
1 - the ant walks 1 cm - 1% of the distance. The band stretches, to 2 meters, the ant is still 1% of the distance around.
2 - the and walks 1cm. this is 1/2% of the band's length (2 meters). it is now 1.5% of the way around. the band stretches to 3 meters, the ant is still 1.5% of the way around.
3 - the and walks 1cm. since the band is 3 meters long at this point, this is 0.33% of the total distance, so the ant has gone 1.833% of the total distance at this point. the band stretches to 4m, and the ant is still at 1.8333% of the distance
etc etc.
the ant always walks 1cm, when the band stretches this represents a smaller and smaller percentage of the total distance. the thing to realise is that at each step the percentage of the total distance the ant has travelled always increases, and it does so in a way that it can get arbitrarily large (harmonic series), so eventually the ant gets 100% of the way around. he says it takes about e^100 seconds, so the final length of the band is about e^100 m, but the ant did not actually have to walk e^100 meters, each step he gets a little bit of a 'free' boost by the stretching of the band eg at step one he is 1cm along, 1%, then the band stretches to 2m, he is still 1% so after stretching he is 2cm along: he has walked for 1cm but travelled 2cm all up. for step 2, he is at 1.5% and the band stretches from 2 meters to 3 meters, so he walks 1cm but travels from 3cm to 4.5cm.
Once the band stretches the Ant remains at 1cm until it walks another 1cm to get back at 1%
Each value in the series is consistent with the next. The actual value of the circumference doesn't need to come into the problem as long you have the series in front of you and know that each value in the series is correct and represents the percentage of the circumference walked during that second.
If you just had the series and the knowledge that each value in it was 'the percentage of the total circumference walked in that second' you could even assume the circumference constant and that instead the ant is just walking slower and slower each second, you would get the same answer.
the band stretches allover not only in front of the ant. when the band becomes 2m the 1cm behind the ant becomes 2cm
H I missed the stretching part behind the Ant, now its understandable why over time the Ant can get around the band b/c it never loses out on any percentage already covered when the band expands but gains a centimetre after each stretch, while it will take ages to cover the band its now explainable why it can.
I think he did a poor job explaining which is why even Bradypus did not grasp it.
So there's no way to find a "gold nugget" for the harmonic series, like there is for the sum of all natural numbers?
No, there is no value you can assign to the Harmonic series. In general the infinite sum of 1/f(x) where f(x) is a linear function are the only infinite sums you can't find such a "gold nugget" for. I am not completely sure about that last statement but it should be correct.
not using the analytic continuation of zeta, but the ramanujan sum is gamma
Can one say that 'infinity' is the 'gold nugget' for the harmonic series? On the complex plane, there is only 1 'infinity' after all (unlike the real line which has 2 infinities).
Euquila Wait, why does the complex plane have only one infinity?
I'm sure there is a more rigorous explanation but in complex numbers you don't compare them like z1 < z2. You need to take the modulus |z1| < |z2|, which is true or false for instance. In the same way, it doesn't make sense to say z approaches + infinity, only |z| approaches + infinity (there is more to it than this and I am waving my hands quite a bit). This rules out |z| approaches - infinity because modulus is always positive. Therefore, there is only 1 complex infinity.
That ant is absolutely amazing.
Since a lot of people are confused about this, I'll explain a small part of the ant story. Instead of looking at how many meters he's traveled, look at what portion of the circle he's traveled. (I should probably use radians, but degrees have a nice visual property to them.) At the start, the rubber band is 1 m long and the ant travels 1 cm. That means he traveled 1/100 of the circle, or 3.6 degrees. In the next second, the rubber band is expanded to 2 m, so the 1 cm that the ant travels is 1.8 degrees, or half of 3.6. In the third second, the rubber band is 3 m, so the 1 cm that he travels is 1.2 degrees, or 1/3 of 3.6. After n seconds, the ant has traveled 3.6 x (1 + 1/2 + 1/3 + ... + 1/n) degrees. Since the sequence in parentheses has no limit, it will eventually reach 100, at which time the ant will have traveled 360 degrees, or one full rotation.
I was thinking about this, and wondered if it implies that any series whos terms get smaller will diverge but we know this is not true. Im trying to think of an analogous thing that gives us 1 + 1/2 + 1/4... and see why the ant cant get around
Ahh yes, even if the ant is always moving around by angle, the angle could always be getting smaller.
Imagine the ant travels 1 cm but the band doubles each time. So he travels around 1% then 0.5% then 0.25% so 1+1/2+1/4+1/8...→2 so ant would only make it 2% of the way around at infinity😅
as a young kid interested in math i was super interested in the euler mascheroni constant, as it seems that no one knows anything about it, but the more you research, the more you realize you need to know complex anaylsis to even get 1% of the knowledge about it.
Oh wow! I have always wondered will 1+1/2+1/3+1/4+… get to infinity or does it have limits. thanks for including it in the video.
So the harmonic series diverges because it's bigger than another series that diverges. But since the elements of 1+2+3+4+... are bigger than the elements of the harmonic series, why wouldn't it diverge as well? I've seen the Numberphile video on why it's supposedly -1/12, but there seems to be some inconsistency in the logic here.
it's consistent in a weird way, but the trick is what kind of 'summation' you use. what is inconsistent is that they don't make that clear in this video
The negatives are greater than infinity.
1+2+3+..... diverges as well, it just has the value of -1/12 attached to it. Watch the video they made if it really interests you. If you then still think about it: the rabbit hole is deep :)
@Joji Joestar Dude, I posted that comment five years ago. I can't even remember what I was talking about then. Since you've only been on this platform for a couple of years, let me give you a piece of advice: let ancient comments rest in peace.
Finally an interesting episode.
I'm glad you found something that interests you.
every one is interesting :)
They're all interesting.
@@General12th Yes most of Numberphile Videos are indeed "Interesting", well that's in "My Own Opinion".
-Q: Do you Agree/ Disagree with Me?
The conclusion is epic, this is what I love about science, how things that seemed unrelated actualy have a lot in common and finding the bridges between science fields is a true delight !
4:50 - If you look at the path around the rubber band as a linear length and assume the distance behind the ant is not also stretched (that all new length is added to the end), the ant would never reach the end. You're adding 1m to the length, and the ant subtracting 0.01m from it. -0.99 - 0.99 - 0.99 - ... goes to -∞.
However, if you _do_ assume the distance behind the ant is also stretched, then yes, the ant will make it for exactly the reasons given. According to the numbers I crunched with WolframAlpha, it would take the ant roughly 15,092,688,622,113,800,000,000,000,000,000,000,000,000,000 (~15 tredecillion) seconds to finish. This is equivalent to about 478,600,000,000,000,000,000,000,000,000,000,000 (478.6 decillion) years, or 350,000,000,000,000,000,000,000,000 (350 octillion) times the _age of the universe_.
That's no ordinary ant.
EDIT: 5:00 - I paused the video to work out the numbers, so I didn't see him do it himself until afterward. XD
they aren't being clear enough that 1 +2 +3.... is divergent unless different rules for sums are introduced....this is def causing confusion.
@ゴゴ Joji Joestar ゴゴ i know because i studied math as undergrad but this video is meant for a larger audience than math majors.
At first glance, the ant problem looks simple enough.
After 1 second, the ant has traveled 1cm/100cm which = 1%.
After 2 seconds, the ant has traveled 2cm/200cm which = 1%.
After 3 seconds, the ant has traveled 3cm/300cm which = 1%. etc. etc.
But upon closer examination, we must consider that the band is being stretched at the end of every second and that the distance the ant has traveled is being stretched proportional to the rest of the band.
So after 1 second, the ant has traveled (1cm + 1cm(100/200))/200 = 0.75%
After 2 seconds, the ant has traveled (2.5cm + 2.5cm(100/300))/300 ~ 1.11%
After 3 seconds, the ant has traveled (4+1/3cm + (4+1/3cm)(100/400))/400 ~ 1.354%
This would continue until the ant dies of starvation or old age. So the answer is no. An ant would never reach the finish line.
Immortal ant
After 1s, the ant has done 1% of the band; after 2 s it has travelled 2cm and the band is 2m, so 1%; after 3s it has travelled 3cm and the band is 3m, so 1% - it never gets more than 1% around the band. How to solve this paradox?
but it does go forward, the distante it already traveled also increases, so in the second 2, that 1cm (1%) has increased to 2cm (1℅) plus the 1cm it traveled on that second.
After 2 s the ant has more than 2cm behind it, because the part behind the ant stretches as well as the part in front of it.
Ah. Got it. The band isn't only expanding in front of the ant, it is expanding behind as well. So the ant gets some 'free' distance travelled help from the expansion. So although it only travelled 1cm in the first second, during the second second that 1cm expands to 2 cm, PLUS it does another 1 cm. Thanks for explaining.
The space behind it stretches too. So after the stretching it has actually traveled 2 centimeters on the new rubber band.
It doesn't make sense. If the finish line is moving away faster than the ant is walking, it will never get there because its velocity relative to the finish line is negative. What am I not getting?
This is a great channel! Keep it up guys!
The question at 5:17-27 is interesting. I will try to address it a bit in a bit more detail. At first glance one might agree that 'every second the finish line is getting a metre further away'. If that were true, the ant clearly would indeed never get at the finish line. But it isn't true, because gradually more and more of that metre is added behind the ant! If at some point the ant has reached, say 30% of its way, then 30 cm will be added behind it and only 70 cm ahead of it. And after stretching, it is still on 30% of its way. That explains why talking about percentages is a natural and correct approach.
I'm working on a proof that gamma is irrational, I have it down to a double infinite sum, that I need to prove is an integer.
That sounds easy, but it has to be shown for all possible integer values of b. Ultimately I will probably fail cause it would have been done by now if it was that easy.
Can you share your work? It's probably wrong or won't lead to anywhere, but that's not a reason to not have a little bit of fun with it.
Czeckie LOL
I could share it, but its on paper right now.
I basically have gamma = 1.5-D, where:
D = SUM[n=1,inf]SUM[k=1,inf]g(n,k)
Where g(n,k) = h(n,k)/(n^k)
h(n,k) = ((-1)^(k+1))/k - (n-1)/(n+1)
So D is 0.92278433509 approximately.
We can see h(n,k) converges to -1 as n and k tend to infinity.
I have a bit more, but its on some paper somewhere, its based on the proof that shows e is irrational
Have you known this problem since before seeing this?
I actually discovered the number gamma myself after plotting the harmonic series and noticing is was similar to ln x, I took the difference, and it approached 0.577. It was about 4 years later that I heard about this number again, and heard no one knew if it was rational or not.
how come the proof works for second series and it does not for the first? :)
Different series behave differently
What did you mean by 'first' series?
The 1+2+3+4+... everyone agrees that it diverges, so the proof works there.
If by first sequence you meant 1+1/2+1/3+1/4+... (and second series 1+1/2+1/4+1/4+1/8+...) then we have to look at what you mean by proof: I would say the second series is used to prove the first, and hence the proof works.
But more specifically, to prove a sequence converges, you have to check whether for any number N there is a finite 'stage' in the addition that is bigger then N, where the stages are in our example (1, 1+1/2, 1+1/2+1/3,....). To prove this for the first sequence, one uses the second series and the fact that since the second series is 'smaller' then the first, any stage is also smaller.
it does work for the first: in "normal" mathematics, 1+2+3+4+5+... =infinity, you have to slightly alter some rules in order to obtain 1+2+3+... =-1/12
the problem I have woth this reasoning is in first rule arithmatic system you cant evaluate one expression to two different values and claim it to be consistent. so either the series is divergent or evaluates to a number and if both are true then somehow our arithmetic is flawed
that's clearly not arithmetic.
Do you accept 1 + 1/2 + 1/4 + 1/8 ... = 2 easily ? Well this isn't really a normal sum, this is just the smallest number that is bigger than the sum of any subset of this infinite set, we call it the sum of the infinite series and it feels natural and intuitive...
Then how about... 1 + 0 + 1 +0 +1 +0 = 1/2 ? Does this make sense??? In a way it clearly does, but the difference between a normal sum and this is more apparent. Still this number is meaningful to this series, and more useful than saying that the sum of this series is "ERROR" or divergent. We are just extending the definition of what it is to sum to be able to use it in more cases, because it's useful.
If you keep going like this you end up with stuff like the famous -1/12 one, is just an extension of a function (the sum) to be able to use it in cases where it was before undefined (we extent it's domain).
Have an awesome day everyone! :)
+ahmadsbRBLX You did it!
Make me.
WongFu4Lyfe thanks
ahmadsbRBLX that's not a nice thing to say to a stranger lol
***** At least it wasn't oncologist.
Numberphile always have the best thumbnails. THEY DON'T MISS.
Amazing stuff!
What is he calling it? A lozy band?
"lazzy" - a shortening of "elastic"
"Lazzy" band. Doctor Padilla is a scouse and that's what scousers call them!
Eleven Bottles Thanks. I feel like I should've known that but it just wasn't in the English to American brain dictionary.
looks like a proper scouse lad as well
Its a "laggy" band, in yorkshire. You're whelk.
Really interesting video, thank you Numberphile :)
It's a pity that we can associate a finite value to 1+2+... and we cannot do this to the harmonic serie. That makes we wonder lot's of things, maybe one could enlight me on some points ? Thank you again.
(a) Are we sure that we can't assign a finite value to the harmonic sum, even if we use a different function from zeta ?
(b) I mean, the zeta function is not the only consistent way to attribute a finite value to an infinite sum, or is it ?
(c) It seems like there's kind of a "divergent series algebra" (separated or extended from the "classic algebra", i.e. with infinite divergent sums) : does this "extended" algebra have a name ? what is allowed ? what is not ?
(d) Are there series (harmonic or others) that cannot be assigned a finite value, even if we use other "zeta" functions ? or the harmonic serie would be the only one ?
(e) For example, i've heard that 1+2+4+8+... = -1. The classic real answer would be 2^(n+1)-1 with n->infinity. I guess we can also say that 1+a^2+a^3+... = -1, given that the classic answer would be a^(n+1)-1, is that correct ?
(f) Do these last kind of finite values are equivalent/consistent with to the zeta finite values ?
(g) This serie (sum(s^n)) looks like to me kind of a zeta dual function , is it related to zeta ? does it have a name ?
(h) Is there a simple/intuitive way to understand the trivial (which for me is not) zeros of zeta : sum((2n)^s)=0 ?
Sorry for this long list and if you have been, thank you for reading :)
Imagine a photon instead of ant and expanding space as the rubber band. Can the light reach a distant galaxy despite more distance is being created per unit time than the light can cover?
Maybe this is still impossible because the space expansion is accelerating and the rubber band grows linearly.
It is theoretically possible IF the space is expanding in the same rate everywhere, so more space is also created before the photon, pushing it forward (relatively).
Space is flat though, so it's not the same.
I must have watched this video 10 times over the years by now and it's always captivating.
1:05 i work with music & i've known about the harmonic series for years & make use of it daily but somehow still didnt know that was the equation for it, i've only ever really seen it visualized till now
it is amazing despite I do not understand anything
Pretty much every single math / science video I watch
I'm Italian and I have to point out that Mascheroni is actually pronounced "Maskeroni".
"Look mom, i found the 20th digit, it took me almost a month"
"woah mascheroni, i bet anyone will ever go as far as you"
250.000.000.000
I remember this from calculus class back in the previous millennium. I had always assumed that this must be irrational and transcendental; I had not realized that the question is unresolved.
One of the best video I ever seen. I really enjoy your channel.
"But what's behind also gets further away" - crystallizes the concept quite elegantly.
I loved the ant puzzle.
I love u
I love u
4:22 "In the second second..."
A lot of your vids contain similar traps
4:41 unless the ant has moved to the percent at which it was already at, the ant would be at a lesser percentage at the start of each time frame, making it impossible to reach the end. Tell me if I'm wrong id love to understand the situation
A fact in this video helped me win a silly math contest! Knowing that the number of terms needed to reach a number with the harmonic series is about e to the power of that number allowed me to know if something was under or over 9000 lol
So everything does eventually come full circle. Hahah
Cosmology might be related to number theory, is this a hint that we're in a simulation?
Mr. H Unpopular opinion: The _way we think_ about the universe is mathematical. We often forget to look at the basis of our understanding, which is our mind. Essentially, we are using a coloured glass to interpret that the universe is coloured. I don’t necessarily agree with that, because it is the eyes that are ultimately looking (at the universe), not the glass.
0.577th
VERY interested in the cosmological speculations at the end (8:50++). This idea that numbers "know" things about each other is certainly not groundless mysticism.
I have a question which is yet to receive a satisfying answer. There are infinite series that are divergent and there are infinite series that are convergent, there appears to be a ratio of decrease between the elements of each set that determines wether or not it converges to a limit. Is the Euler-maschroni constant this ratio? Or is there a separate constant? Essentially what's the number that determines wether an infinite series will or will not converge?
How big would that circle be when the ant finishes? xD
the number of seconds it took to finish, but in meters
It would be, as said in video, 10e50 meters. For comparison, the observable universe is 8.8×1026 m (thx again wiki). So yeah .. xD is the word here ! :p
Well, it takes ~e^100 seconds, so ~e^100 meters
8.8x10e26, *superscript failed to appeared here
The circle would be billions of times larger than the universe itself.
It's an ELASTIC band not a lazzie band
In Pittsburgh, it's a "gum band."
Daniel Dyszkant You'll be telling us next there's no plazzie bag.
This video is suspiciously 10 minutes and 2 seconds long...
I feel like this constant is transcendental because log(n) (assuming it is natural log) is always transcendental, meaning we are subtracting a rational sum by a transcendental number, no matter how large we get. If you continue plugging bigger and bigger numbers into 1+1/2+1/3...1/n-log(n), you will always have a transcendental number, meaning the Euler-Mascheroni constant tends to a transcendental value... right?
also this constant is basically what infinity-log(infinity) is equal to (i know this isn't how it works but shut up)
This is so great - so few things can show so vividly how mathematics too descend into mysticism and quickly.