An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a fourth of a beer. The bartender stops them, pours two beers and says, "You guys should know your limits."
An infinite number of mathematicians walk into a bar, the first one asks for a beer, the second one tells he doesn't want anything, the third one asks for a beer to and the forth one, tells he's fine, and so on... The bartender serves half a beer. They are not happy and trash the place :p
Johnny: Not really, because that would be modelled as 1+0+1+0+... and between them they have ordered infinite pints of beer. More like this: an infinite number of mathematicians walk into a bar and sit around a table. The first gets up, orders a beer and brings it back to the table. The second says he doesn't approve of alcoholic drinks and takes it back to the bar. The third says don't be so wet and brings it back to the table. The fourth says I'm with number 2 and takes it back to the bar again. Counting only the time the beer is on the table and on the bar (thus ignoring the time in transit), since there is only one pint of beer and two locations, on average there is half a pint of beer on the table and half a pint of beer on the bar.
@@encounteringjack5699 it just doesnt go like this. You could apply you reasoning to a banana and still get its even. Evenness/oddness are integer properties. Otherwise there would exist a k ε Z | inf= 2k
@@encounteringjack5699 Okay. I see the problem here. Its all about definitions(as always, right?). We define a number n (integer) to be even if and only if there exists another integer k so that n=2k. i.e. 18 is even because there exists 9 and 18=2*9 No sets or parts involved, just straight division by 2 involving two integers. Infinity, just like our banana(wipe off that smirk) is not an integer so the definition could not apply to it.
I remember watching this video when I was in high school and thinking “man this is amazing”, now I’m a maths major ending his first course of real analysis and boy, this is the stuff of nightmares.
The power consumption of the light has a limit, if it switches at 60Hz, the watts is the same as if it switches at 50Hz or 1Hz or infinite Hz. The sum is meaningless as it's non convergent element n always has a difference of 1 with element n+1. Effectively on/off is related to n directly, not the fallacious idea of the series having an infinite sum.
@@simorote Even though it is a divergent series, the sum mentioned in the video aka the Cesaro Sum is widely accepted as a technique to obtain the sum of a divergent series... Many such techniques are there and Euler himself attempted one such definition of sums of divergent series... So don't just dismiss something by saying that they are lying... The beauty of Mathematics is such that sometimes there are multiple ways of approaching a problem and each way reveals something interesting...
No mystery here! The answer to this question is that the problem is improperly (or incompletely) defined. Therefore, as approached in this presentation, the answer is dependent on additional constraints on the problem imposed by the solution which is applied. Moving the parentheses about depending on the method used is actually adding constraints to the posed question and in so doing CHANGING the posed question. For example, placing the parentheses as (1-1) for ALL summed terms assures that the answer will be zero at infinity. This is a DIFFERENT problem than starting a 1 and then adding (-1+1) terms, assuring that the starting 1 always has a 0 added at each step to infinity. Thus, the answer is INDETERMINATE due to an incompletely defined problem.
The idea is that you try to think a little more deeply beneath the surface and question *why* we have to bring in such definitions in the first place. We use axioms to work around unexpected problems in maths. To truly understand mathematics, you cannot just learn the rules of the game that we have chosen to define. You need to understand the logical reasons why we made the rules as we have. It’s not enough to just state the rules as proof something doesn’t work - that is a fallacy! We made up all the rules in the first place! In general in mathematics, addition is commutative, thus you should be able to move brackets around this sun as you please and get the same result. But the video shows this is not the case! So what do we do? We can cover up and fix the issue by adding extra definitions to the sum like you’ve stated, but that’s just one possibility. Other mathematicians have alternative opinions on how to define such sums. Nobody is wrong or right as there is no correct fundamental answer.
Well yes, but after 2 minutes you don't touch the switch anymore. And if you stop time at any given moment, the light is even on or off. This is a mathematical perfect question, the answer shouldn't be what a human being can process in a few milliseconds close to the end of the super process.
I know this is an old video, but if anyone is interested, the idea of taking the average partial sums is known as "Cesaro summation". And it's just one way of "summing" divergent series.
Thanks for using "quotation marks". The Cesàro "sum" is obtained via an averaging process that doesn't usually produce the same "value", unless the series is strictly convergent (which it isn't, in this example). The quote marks around "value" is a reminder that the so-called "value" of the original series is not any number. There are also fancier ways to produce an associated value, but when the original series is not strictly convergent then none of them produces an actual direct sum of the original series. For example, one method starts with the series in Dirichlet form, uses analytic continuation which changes the values over a complex half-plane, then evaluates the zeta function at a point in that half-plane, which produces a finite value but no longer represents the original series.
Another answer could be -oo (infinity). If you use the parentheses correctly: 1-(1+1)-(1+1)-(1+1)....etc. This will eventually result in -infinity or -oo This includes the parentheses without having to add more to the equation. In his example where he added parentheses: 1(-1+1)(-1+1)(-1+1)....etc. You would actually be multiplying 1*(-1+1)*(-1+1)*(-1+1)....etc. So, whilst he says this will result in an answer or 1, it will actually result in an answer of 0. No matter how you place the parentheses, you can never get an answer of "1" without adding more "+" to the equation.
Yes I agree it's exciting. But it's wrong. Don't believe this. 1 - 1 + 1 - 1... diverges. It's not 'equal' to any number. If you want, check out Sequences and series chapter in calculus.
@@pte1808 You are welcome! If you don't have time to buy a book and read it, you can read articles about 'Convergence and Divergence of Series' on the internet. Few examples will help you understand, according to me.
Okay, for the light experiment, (granted I am not a mathematician and have surely made a mistake) but assuming you could even switch a light on and off, and that the electricity could even reach the light in time to be considered on and off, the maximum number of times you could turn it on and off would be just about 51 times, where the 51st time would equal 4.2*10^-44 seconds away from the goal (which is just under the planck time length). The 51st time would be in the on position, in this case.
_"And I'm hoping it will cause a little bit of debate on the comments, 'cause I know TH-cam is the home of rational and informed debate. So I look forward to that."_ That's exactly my kind of humour. ^^
I'm not a mathematician, but a lot of these questions just seem to be treating infinity as a really big number. If you add 1+½+¼+⅛, then by infinity you reach 2. Well, you don't reach infinity at all, by definition, so you're always a little short of 2. The sum 1-1+1-1... is ½. No, it's either 1 or 0 if you ever reach a point where you can answer the question, depending on whether the last number is 1 or -1. If the sum is infinite, then the answer keeps switching between 1 and 0 forever. If you switch a light on and off progressively faster for 2 minutes, is it on or off at the 2 minute mark? If you're moving the switch at infinite speed then it's probably broken and you may not need a light at that point anyway. A lot of the paradoxes leave me thinking either "I don't think that's how reality works" or "I don't think that's how infinity works".
Years late, but to clarify your concern: infinity when used as a limit, _is_ the limit, there's no beyond it. It's the absolute limit of time one - instinctively - wants to think extends beyond. That's to say : if your sum limit _is_ infinity, then by definition you can and _do_ achieve that limit in your answer, however far away that is. if your final sum was a "little short "of an infinite series, then you'd be a little short of X, but if your limit is infinity, then and only then do you reach X, and because we state the limit _as_ infinity, we're allowed to do that :) Sometimes, such a result is said more in the frame of _"as we approach infinity, the answer approaches X"_ but mathematically, this means the same thing as _"at infinity, the answer actually = X"_ - because, if it _didn't_ mean that, then we'd never be able to calculate anything that contained infinity in the problem, and that would be a problem, and it would really upset physicists :))
@@IngieKerr months later too, but what he was trying to say is that infinity might make sense on math but on real physics, such as turning a light switch on and off, there is no infinity. Everything must happen in a finite amount of time, quantity and space if that makes sense. Anyway, what I’m trying to say is that limits are just mathematical concepts and can’t be taken to reality.
@@gONSOTEThe gaps are getting smaller.. Anyway to anyone thinking the same, we use the concept of infinity and limits to break through the limitations of our dimensional reality(spacetime), which allows us to accept and understand things that otherwise wouldn't make any sense like, that the electron has a radius of zero, and no extent, yet we know it exist, interact, and can measure it's effect. What I'm trying to say is it's an Oxymoron trying to tie infinity, limits and other Applicable Mathematical concepts to our reality(spacetime), But that's not what Mathematicians does when explaining things to us, they give us a "similar" situation in our reality, to grasp the concept not get an answer. Hence the Light switch or that guy from the textbook that bought 109,876,542 Apples from the market.
@@gONSOTE I would say limits very much exist in reality. The number e has never had its value determined exactly because every method of producing it requires infinitely many steps (it would also require infinite space to store all of its digits), with a formula for the value of e being (1 + 1/n)^n with n approaching infinity and another being the infinite sum of 1 + 1 + 1/(2!) + 1/(3!) + 1/(4!) + ... The universe, on the other hand, doesn't have to calculate the value of e. e is a number that simply exists inherently within the physics of the universe (well, maybe. There's some argument that the physical laws might be probabilistic in nature, which could mean that even the universe itself is simply approximating numbers like pi and e). Still, there's plenty of practical applications of the idea of 1 + 1/2 + 1/4 + 1/8 + ... = 2, in engineering, for example, 1.99 will potentially be approximated as 2, so if you have a few dozen terms of that infinite sum in some engineering problem, you will approximate the sum as 2. Even the 1 - 1 + 1 - 1 + ... = 1/2 idea has practical applications. If objects like light bulbs oscillate between two states fast enough, the outcome may be nearly indistinguishable from a static system halfway between the two states. Fluorescent lights in the U.S. work like this; they in fact flicker at high speed due to the alternating voltage of the electrical grid. Some lamps also can very their brightness by flickering at various intervals. This is because our eyes essentially do an exponential moving average of the terms of the sequence, 1, -1, 1, -1, ... which is a bit different than the sequence of averages of partial sums, but the outcome is much the same. A value approximating 1/2. Edit: Finally, the unique property of a limit like 2 for 1 + 1/2 + 1/4 + 1/8 + ... is that 2 is the smallest number that you can say with certainty will not be surpass by any partial sum of those terms. If you told me 1.99, I could find the first few dozen terms of 1 + 1/2 + ... that surpass 1.99. 1.999? I can do that too. 1.9999? Same deal. I can even tell you how many terms it will take without having to actually add everything up. 2 is the very smallest number that I could never find any quantity of terms of 1 + 1/2 + ... that will sum past that number of 2.
The light switch is a bit of a paradox, but the taking the limit of 1/n to be 0 is definitely real. Pretty much all of calculus relies on infinity and taking infinitely large sums of infinitely small numbers
after 2 minutes the light will be changing between on and of "infinitely fast" or having an infinite amount of switching in no time , so what would actually happen is that you would see the half of light intensity
+ahmed Imam No, why would you give the on and off states equal weight? The average intensity right before you reach 2 minutes is 2/3, due to the asymmetrical duty cycle.
+Doug Gwyn Technically not, we're concerned with the state of the lamp at the instant the 2 minutes are up, not with the average amount of light over the course of the 2 minutes. This leads back to a half, as the light just as the timer ticks to 2 minutes it will be alternating infinitely fast.
It is both one and zero because there is no set end, and depending on which point you break it off it could be either... and 1/2 is right between the two. I like that one.
The question is flawed. Averages as a process can bend reality to a result that is impossible. This can be easily seen as the average number of people being a rational result. For example, demographics find the gender of a group of people. You then pull a random person from that group and ask the question of what is the gender of this individual? The data is split into fractions and these fractions represent probabilities. You cannot say for certain the gender of that individual, but you can definitively state the probability. The same logic applies to the question of whether the light is on or off. That said, the 0.5 result is not the representation of a literal result. Since it relied on averages, it innately becomes a probability. That's why it can be accepted as a result to the Grandi's Series. The result of Grandi's Series is a probabilistic question, so it makes sense for it to have a probability as a result.
The equal sign is like humanity's way of forcing the numbers that represent a reality to conform to our hypotheticals. Without it, the numbers are basically acting like a liquid that evens out with whatever other numbers are in the pool until they settle in the middle.
Indeed. In many cases, probabilities are used to used to find values one can expect when the probabilities being compared have the same units. Grandi's Series is one case of this since approaching infinity would lead to 50% of the domain being a value of 0 and the other 50% being a value of 1. To be fair though, in these theoretical scenarios that holds to their premise and never changes their process, equal signs can be used just fine. That is pretty much what mathematical induction is after all. The big issue, when translating it into reality, is that maintaining the soundness of the theory due to how difficult it is to prove a process will never change. A consistent process is important to eliminate negligible values like the trailing -1 or +1 at the end of Grandi's Series or the remainder of 0.3333... when evaluating 1/3 in decimal.
But quantum physics is reality.... particles do happen simultaneously in both ways, and maybe it just means that the lamp will reach a superposition state, when the speed of turning on and off is approaching infinite.
That's right Lin. Superposition can certainly occur. However, if it was in superposition, you would not call it on nor off. You would call it some kind of mix between the two, something new derived from the combination of the two. You know what's interesting actually. In terms of lights turning on and off very quickly, there is a very real-world application of this for lights in electronics. This is called *pulse-width modulation* in which at high enough frequencies of turning a light on and off can change the *brightness* of a light. Perhaps in the real world, the 0.5 refers to the superposition state of on and off being at half-brightness. Half of the time it's on, half of the time it's off. So our eyes perceive it as half of the brightness it's supposed to be when it's on.
It is right if you prolonge the Zeta function, but wrong if you use simple math and convergent rules it is wrong, that's what most people don't understand.
@@DK-xw5hp The value of the zeta function for -1 is -1/12, not 1+2+3+4... The zeta function does exist for negative values, the -1/12 thingy is just an analitic extension.
I swear, that in personal journal writing I wrote and 'started seeing' something like 0:23-0:43 once but everything from 0:43+ is where it's 'starts becoming it's own thing'--the nice feeling of validation when you find words to an experience might sort of apply here now that I know of 'Grandi series'
Actually, it would most likely be that it is like dealing with quantum entanglement. 2 atoms connected, both opposing spins, yet they are both spinning one way and the other at the same time until observed.
If a series doesn’t converge it diverges... that’s just a rule. This series is alternating between 0 and 1 forever and therefore never converges so it is divergent. My opinion (which definitely can be wrong 100%) is that when you try to do math like that with a divergent series, it’s like dividing by 0, you can make anything you want happen. That seems like a logical explanation for why you can get multiple answers, but again, just my opinion.
except it doesnt truly diverge, when a sum diverges it grows indefinitely, there is no growth in this sum only the constant and equal switching of the values (in my opinion of course).
@@fareedabifarraj483 I see the series as a way to show the number 1/2. Imagine 1+1, that's equal to 2. That's one way to show the number 2. And so 1-1+1-1+1-1... series, because it's defined being infinite, instead of seeing it literally as a divergent series, I see it tending to 1/2 because it doesn't want to stick to 0 or 1 values. Think of it as two 'sweethungry' guys that want to eat a single pie and decide to cut it in two, so that both get equal amount. Plus, because there are an infinite amount of -1s and +1s, it can equal 1-1 or 0+1 with how the series works, BUT it's exactly how the series works that the result is not 0 or 1, so it's essentially forced to be a half. Again, that's the way I see it.
the way i see it, he is trying to give the series a value that is meaningful and useful in calculations . you can’t have a square root of -1 but we can say it is = i so that we can have a meaningful value attached to it for calculations. in the same way that we obviously can’t have the square root of a negative number so it is undefined yet we assign it a value of i so that we can use it in calculations, we can assign a meaningful value to this series that we obviously can’t have a value for so that we can use in calculations.
You've mathematically described the concept of PWM (pulse width modulation - for lights that would be flashing the light so fast that it looks like it's on continuously at a lower brightness).
There is no answer. This is exactly why infinity does not work as a quantity. Infinite series are good for convergences, but that does not mean there is a literal value of infinity where the convergence will be met. The minute you declare you have found the answer, I'll just go on 1 number further than you looking for it.
I thought James' discussion of the lamp was going to describe PWM (pulse-width modulation) whereby you imitate an analog value with digital values. If you only have digital values (eg voltages) like 0v and 5v, you can "simulate" analog values like 2.5v by rapidly turning it on and off as fast as you can for an equal amount of time. You can simulate a value like 1.25v by turning it on for one time unit and off for three. Of course, that's not really an explanation for infinite series, but it's related.
3:59 "If we pick a nice infinite sum, cause there are nice infinite sums and there are bad infinite sums" Nice infinite sums being convergent and bad being divergent. Convergent ones have a result of summing them, divergent ones either diverge and we don't do math with them because they have no result. That includes our "One minus one plus one minus one" Edit: Look up "Divergence tests" Today to see how to determine if Your sum is divergent
The light is off. Because after infinite switches on and off the light bulb's fuse would go out. Therefore the answer to 1-1+1-1+1.... is 0. Because of fuses.
the chicken-and-egg thing is a non-question. It stems from the problem that there is no strict definition as to where the chicken's ancestors became chicken and not some other animal. The demarcation of species is a human concept, and arbitrary. A chicken just is, nothing came first. A real philosophical question to ponder would be where the universe came from, and that is a real unanswerable question in my view.
+Gabriel Cavendish The egg came first because an egg some other animal laid had a mutation that caused a new species: the chicken. Obviously some other animal wasn’t just walking along and then magically turned into a chicken, that’s not how evolution works.
The light on/off question is actually used in discrete control systems. Circuits can only technically have 2 states--on and off. Sometimes, though, you need to produce an "analog" signal with a varying amplitude. But how do you do that if you only have 2 states? Basically what a digital controller will do is just flip that particular circuit on and off at an extraordinary rate, and the weighted average of the "on"s and "off"s ends up being your attenuated amplitude. So let's say within one second, the flips are such that the circuit is on 80% of the time and off 20% of the time. This will equate to an amplitude attenuated to 80% of your maximum amplitude. In the instance of the specific situation Dr. Grime was talking about, you would get a 50% attenuated amplitude--or the 1/2 you would get from the infinite sum!
I could argue that if you had an incandescent lamp, the result after two minutes would be on but dimmed by half. The power input would fluctuate so fast that the result would be an attenuation of the power input by half, hence the bulb would be half as bright. I thought about this when he was explaining the final part.
***** That's pretty much exactly what I said. On a bit of a tangent, though, you wouldn't need to attenuate an incandescent bulb like that because you can actually give it a true analog signal. But it would still work if you had it hooked up to a digital circuit :)
As you aproach the 2 minute mark, you are getting to a fraction of time of which you have the light on or off. When you have reached the 2 minute mark, you have turned the light on and off an infinite number of times. As infinity is not a number the answer is it is not defined, as strange as it may sound. We can see this with the series as well. The sum is not defined
Yes exactly. You could not have turned the light on and off an infinite number of times in 2 minutes because there is a smallest amount of time it takes to turn the light on and off. You cannot keep doing it faster and faster - in the real world.
@@thischannelhasnocontent8629 no by his logic only divergent infinite series (a series whose partial sums do not tend to a value) are undefined, such as the one in the video. Convergent infinite sums still approach a some limit and have a value.
@@ryanduggan6738 you can use the alternating series test or the geometric series test to prove that this series is divergent - therefore it does not have a value by definition
I’ve seen other expert mathematicians who criticize some of this channel, including this one. According to them, the “sum” is nonsense because it just oscillates between zero and one. That’s the whole answer.
Well regarding the lamp, if it would turn on and off INFINITELY fast, then it would only make sense that it would create an apparition of a dim light let's say- a not fully bright light as compared to the one when it's fully turned on (the one representing 1) since it would create a singular image due to its speed. If you think about it even without knowledge of Grandi's Series, it would makes sense thus also making it a supporting proof for Grandi's Series itself since a dim light, in fact, is a light level in the middle of an on and off state of the lamp (representing 1/2).
The average intensity of the lamp over the entire interval is 2/3, not 1/2. If you start averaging at some moment other than the beginning of the interval (with the toggling still done exactly as specified) then the average from there to the 2-minute mark will be different, somewhere in the range [1/3,2/3] depending on when you start averaging. The fact is that the state of the idealized lamp at precisely 2 minutes is indeterminate, not 0 nor 1 nor 1/2 nor 2/3, etc.
I think the 1/2 solution is close to a probability solution. Since the sum is virtually infinite, we cannot discuss where it will end. Therefore, it has a half chance to end in 0, and a half chance to end in 1, thus the answer is 1/2.
An infinite sum does not "end", i.e. you never "end" adding it up. This one simply does not converge (like the guy said, there is no limit). The fact that the average sum converges is interesting though.
That's fun and all with the exception that there is no such thing as infinite, its just an abstraction The first sum (1-1+1-1+1…) will depend entirely on where you stop the sum, the result will loop between 0 or 1, the concept of infinite is not applicable here
A reminder of the golden rules to be adhered to when dealing with divergent series: 1) Do not use brackets. 2) Do not remove any zero (unless you have proven that the divergent series is stable). 3) Do not shuffle around more than a finite number of terms. Not adhering to these rules yields incorrect sums.
Pretty sure flipping a light switch on and off an infinite number of times in the span of two minutes would completely destroy the switch. So yeah, the light would probably be off by the end of it.
I know this is old. But how about looking it like this: You are trying to quantify something which is infinite using our finite methods. It's almost like a quantum bit. It's both 1 and 0 until you observe it, then it falls back to finite state and it's either 1 or 0. It's the same principle here. If you stop the series at infinity in a finite method it will be 0 or 1 depending on when you stopped the series. I think this series might actually be great for explaining physical properties of quantum bits. :)
+arekusanda1 i was looking for this :D exactly what i thought :D and man this might actually be the best comment section on youtube, he doesn`t need to use it sarcastic in his video :D
After a night of thought I came to this same conclusion. Now at 2 minutes the switch is both on and off, but what is happening at 3 minutes? In theory we'very gone past infinity!?!??
+Some Guy. Pretty sure going further than infinity is a larger infinity making it the same problem. I could be wrong though. I doubt I can properly comprehend a lot of the stuff on this channel.
This thing is used in digital circuits to create analogs outputs, and it's called PWM. If I keep switching on and off a pin with a fixed frequency (and that's the assumption), what I get is a square wave with a duty cycle of 50% and therefore the voltage value which is seen from the external world is half of the supply voltage, that is, 1/2. That's my physic interpretation of the value this series converges at.
As described, the "frequency" (duration of 1 cycle) is not constant. Depending on where in a cycle you start, the average could be anywhere from 1/3 to 2/3.
Yes you are right, I was referring to the first half of the video, when they don't speak about the frequency. In the real world their example in which I keep increasing the switching frequency is not feasible, since there is a limit on how fast a micro processor can go. It was an interesting thought experiment, though. Thanks for your answer =)
Yep, this is where the real world tell the math world that it doesn't care what math thinks should be. If you are turning it on and off rapidly enough then it would literally be both 1/2 on and 1/2 off. There is no paradox there. The problem is that math assumes there is only a 1 or a zero, but that assumption doesn't work in practical applications like light switches. You can definitely have a half-state in light switching and math can't claim otherwise. And at the end of 2 minutes you have equal chance of either state being the end, thus 50/50.
If only they tackled the zeta function with this approach in mind emphasizing that "the limit of a series is not always the same as its partial sum (and therefore not really the same as analytic continuation)". More honest presentation.
Damn you, Numberphile and especially you James! I'm at work, plus I have a lot of studying to do for my exams and I'm just sitting here watching your videos O_O
Maybe if you were to infinitely turn a lamp on and off for 2 minutes, the light bulb would be so worn that it only produces half of its original potential.
I think that since you're always in transition between the light on and the light off and as the speed at which you transition between on and off increases and the time of which it's either on or off shortens, it gets closer and closer to impossible to tell if the lights on or if the lights off. If this goes on for an infinite amount of time and what it actually looks like in the room is that the room will look dimly light, with a light intensity that's half way between on and off. Hence the 1/2 answer.
The light is pulse with modulated which with real lights is dimmed. It with be at half brightness with a 50% duty cycle. So I say the best fit is 1/2 or 0.5
i think you kinda misunderstand the meaning. Of course there is no "answer" to infinity. I think, even if they may phrase it different, or smth. its more like they are searching for a value. Because with Values you can work and with "infinities" not.
AzraeltheRightous Two things. First, if you bothered to read my entire post i said the concept of (inf - inf) is fallacy. This has been proven so I'll not go over it. Seond, if the series within the parenthesis is, indeed, infinitely long then by definition it will turn out to be (inf - inf). Thus, the first point holds true. Simple rearrangement is key here. For a short example we show that (1 - 1 + 1 - 1) is also the same as (2 - 2) I simply re arranged the positive and negative ones. With me so far? So, by that measure we have an infinite number of positive ones so we can re arrane them into positive infinity. Likewise we have an infinite number of negative ones so we can rearrange them to get negatve ininity. So, the problem can be rearranged to look like (inf - inf) = S. So, 1 + S = 1 + (inf - inf) Now, the reason it is so important to realize the falacy of inf - inf is because 1 + inf = inf so the above equation of 1 + (inf - inf) can be rewritten to be inf - inf due to th communitive property.
eric mcdowell What you say doesn't make a difference and I'll show you why. The original problem is (1 - 1 + 1 - 1 + .... ) as an infinitely repeating pattern. Using the assosicative property we can rearrange the problem where are the positive ones are on the left and the negative ones on the right to give us : (1 + 1 + 1 + ... + 1 - 1 - 1 - ... - 1) which will give us the formula (infinity - infinity). At this point it no longer matters what you put outside the parenthesis, except zero, because the formula will always revert to (infinity - infinity). Even if you raised the whole thing to the power of infinity you'd still get (infinity - infinity). Thus, the paradox comes out to (infinity - infinity).
Folks who are having a hard time with this, consider this: you could reason that since no numbers multiplied together equal -1, then "what is the square root of -1" has no answer. However, pretending such an answer exists leads to all sorts of interesting results. Similarly, even though it is "clear" that this series "has no answer," pretending that it does leads to all sorts of interesting results. S = -1/12 is no more "wrong" than sqrt(-1) = i
The average of the partial sum works if you have limit because you eventually drown the earlier values into an infinite number of values that get infinitely close to the limit. But if you don't have a limit, that's just an average. It shouldn't be considered as the actual sum of the series.
+Efreeti No. Actually he is doing it wrong if we follow the laws of mathematichs. I do understand where he is going with this video but when he writes 1-1+1-1+1.... this actually means 1+(-1)+1+(-1)+1.... which doesn't have an answer. The answer alternates between 0 and 1. This video confuses people that don't really know math.
On the off chance that you actually read this comment: I saw a number called SSCG(3). Could you do a video on that one, in your big number series? I heard it's larger than TREE(3), and yet I don't know how and why TREE(3) must be this immeasurably big (is it not a sort of lines and graphs after all?), and I know you worked on biggest number imaginable (Rayo's number I believe), although I didn't quite understand it. I struggle in English and my PhD was in Persian literature, so the last time that I HAD TO study (veeeeeeeeeery basic) maths, and in Persian, was about 16 years ago for university's entrance exam, and I didn't quite ace at it (14%. I know. All the shame is mine). But I'm really fascinated by your big number series and your teaching skills, and I'd love to hear about that particular number or function (whatever that is) on your channel. Lots of love from Iran for you, beautiful people and great scientists and outstanding teachers. ❤️
The way you proved the 1-S=S, 1+S=S is also possible to prove. Like this: 1+1-1+1-1+1-1...... 1+(1-1)+(1-1)+(1-1)....... Now just swap the elements inside brackets, which becomes 1-1+1-1+1-1+1........ So 1+S=S Which means 1=0 isn't it.
I'd say that here instead of saying that the limit is 1, 0 or 1/2, we should say that at infinite (or at any step of the serie), we have 1/2=50% chance to have 1 and 1/2=50% chance to have 0, but 0% chance to have 1/2... Or we could say : IF the serie converge to a unique value, the value must be 1/2 But at any given two consecutive step, we have two value 1 and 0, the serie is not convergent...
Guys there is an easier method. This sequence is a geometric progression with a common ratio of r=-1 and the starting term of a=1 A geometric sequence goes like a, ar, ar^2,ar^3..... If you substitute the values of a and r, you get 1,1(-1),1(-1)^2... Which is 1,-1,1 upto infinity. The sum upto infinity of a geometric progression is given by a/(1-r) Substituting, 1/(1-(-1) = 1/2 and there you go. There's the answer.
@@kwertie133 You can't do this. In the geometric series, the convergence radius is 1. when you choose the value a=1, you are taking the value on the boundarie where the series no longer converges.
@@luiz_46487 yes it does. But the answer isn't in math. It's in science. The answer is a superposition where the light is both on and off. Or the answer being both on and off in seperate scenarios. Look up quantum superpostion and the theory of quantum immortality
The only reason he got 1-S=S which leads to S=1/2 is because these are two differntly defined S, so in fact it's 1-S1=S0 which is easy if S1=1 and S0=0 or the other way too. 1-1=0 and 1-0=1 If you convince the audience that 1 is the same as 0, then you can derive whatever false you'd like out of that. End of story.
That’s a sheep’s answer. The question is *why* do we have to have these definitions? Remember that these definitions we made up ourselves using axioms. Just because we’ve decided to say something is false, that doesn’t mean we can then use our own definition as the proof for the definition. What if we changed the rules so that it does = 1/2? The idea I’m getting at here is don’t just rote learn the textbook as gospel. Question why things make sense logically from first principles. That’s what Numberphile have done so beautifully here.
Actually the heating filament in the lamp has thermal capacity, so if you turn it on and off infinite times, in the end you will practically drive it on a 50% duty cycle pulse width modulated signal, thus the lamp will literally be halfway up. Also the coil has parasitic inductance which will average the high frequency signal out further If you make a paradox, make a working one :P
It's reminscent of how a rectifier would work maybe, with the thermal capacity acting like the capacitor in the rectifier. If it behaves like that, I would expect it to work a bit like RMS voltages, where the brightness factor would be sqrt(2). But I am guessing
+Michael Bauers You don't need rectification if you do it with a solid state relay or triak (or even combined with a variable frequency drive), or let's talk about a DC supply and a fast switching transistor. Let save the hassle and go simple :D
In answer to the question about the light, I tried to do the experiment myself just with drawing different types of lines in a sequence rather than using an actual lightbulb. The answer to the puzzle is this: It is actually mathematically impossible to work it out. I tried to draw different lines at different intervals so: after 60 seconds, I would draw a straight line, after 30 seconds a curly line, after 15 seconds a straight line, after 7.5 seconds a curly line, after 3.75 seconds a straight line, after 1.875 seconds a curly line and so on. But, just as it is explained in the video, the number of seconds between each line drawing never quite reaches exactly two minutes, it is always a fraction of a second before no matter how far you go into the sequence. Therefore, we will never find out what will happen after two minutes because that two minutes target will never actually be reached!
You could argue that you could draw a straight line at 0.9375 seconds and then two minutes would come and you could say: "Right, it must be a straight line and the light must be on then." But actually, you have missed loads more line-drawing at intervals such as 0.46875, 0.234375 and so on. Even if you can draw lines or flick switches at the speed of light, time will beat you and two minutes will come and you will never be able to keep going with the infinite series of switch-flicking as it goes onto infinity and never reaches two minutes. Consequently, we can never know.
If you want to be correct about it: for all n element of R: Lim(a -> +inf) n/a = 0 With words: For every rational number n out there; if you divide n by a with a going to infinity, the result will be 0. I think this is what he meant.
i think of it this way: because there is no last step, the + and - keep on going, i think that the number is being added and subtracted at the same time. So: 1-1=0; 0+1=1 and that keeps alternating and there is an equal amount of + and - in the equasion, so the answer is somwhere in the middle, or between (1-1) and (0+1), a half! I'll do another one: 3+1-2+2-2+2-2+2... 3+1=4; 4-2=2; 2+2=4 between 4 and 2 is 3 so 3+1-2+2-2+2-2+2...=3
Okay. 1+1-1+1... does not have a sum and neither is it an infinite series because the mathematical operations performed are being nullified the very next step. BUT! There exists a kind of series of this format: x^0 + x^1 + x^2 + x^3 . . .and so on infinitely. If your "x" is 1 or anything more than 1, it diverges off to infinity (duh). If your x is 0, it causes the operation 0^0 which is not defined. If your x is -1, it produces this series in question (1+1-1+1-1+1-1...) thats becuz every even power of -1 is 1 and every odd power of -1 is -1. And anything less than -1 likewise diverges. For values greater than -1 and less than 1, this sequence converges to a finite value. The value is equal to 1/(1-x). An example would be 0.5^0 + 0.5^1 + 0.5^2 and so on which produced the famous series: 1 plus a half plus a quarter plus an eighth and so on. We know it equals 2. Applying the formula gives 2 as well: 1/(1-.5) = 2. Substituting x with 1 doesnt work as it causes division by 0. We know that when x = (-1) this series diverges so it has no answer but What ramanujan & co did is substitute x = (-1) anyway and they get 1/(1- (-1)) which of course equals a half. People would say well, how can they do it when it doesn't work for -1? Okay. It may not work for -1 but it should work for values *very* close to -1. A tiny fraction more than -1. Like, say, -0.999999. In this case the infinite series where x = -0.999999 would be equal to 1/1-(-0.999999) which is 1/1.99999 which is crazily close to 0.5 So the closer and closer x gets to -1, the closer and closer the sum of all of its powers gets to 0.5. For instance if your x was, say, -1 + (10^(-100)), then your x would look super similar to -1 and the infinite sum of all its powers would look super similar to 0.5. They would just converge very very very slowly.
"Will the light be on or off?" The light will be off. The lamp turned into a quantum particle with undetermined state and therefore the light is off. Got a problem?
I find myself watching these videos and I understand about 20% of them, but I hope that some day someting stuck in my brain and I will use it for a pubquiz or something.
Since it's a natural number sequence imagine it's a wave function oscillating between 1 and -1. Start the sequence from -1 and answer is -1/2. So light on or off depends on where you start measuring in this infinite sequence. Similar to wave function collapse with Heisenberg.
What do u mean by wave function Wouldn't it be a perfect square wave if it was a wave function? The average would be zero.. I think I need more proof than the averages of partial sums to understand that it's 1/2.
I was thinking it would be better to switch it onto a 60VAC service from the 120VAC service after about 1:59.5 so you don't wear out the switch and there is no need to keep turning it on and off.
Actually the Engineers Answer was to build an electronic dimmer, which can make your light half as bright by quickly switching it on and off, and sell that to you. :-) (As an engineer I actually expected that to be the conclusion to the "light" experiment.)
The error is in the question, as it's not a series converging to a limit but something better expressed as a function, so there is a set of sums depending on number of elements in the series. That is: G(1) = 1, G(2) = 0, G(n) = 1 - G(n+1) = 1 - G(n-1) and abs(G(n) - G(n +/- 1)) = 1 ie the difference is always 1 no matter that n nears infinity, therefore the question is fallacious, the sum depends directly on n, where n is the number of elements written. G(n) = 1 - (n+1) modulo 2 = n modulo 2 IOW when you write odd number of terms in the sequence, it's sum is 1 and 0 for even. The games with brackets then fall out as they simply reduce n: G(n) = 1 - G(n-1) or 1 - G(n-1) = G(n)
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An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a fourth of a beer. The bartender stops them, pours two beers and says, "You guys should know your limits."
this is quite possibly the best joke i have ever heard
might the first time i really laughed out loud from a youtube comment
An infinite number of mathematicians walk into a bar, the first one asks for a beer, the second one tells he doesn't want anything, the third one asks for a beer to and the forth one, tells he's fine, and so on...
The bartender serves half a beer. They are not happy and trash the place :p
Johnny: Not really, because that would be modelled as 1+0+1+0+... and between them they have ordered infinite pints of beer. More like this: an infinite number of mathematicians walk into a bar and sit around a table. The first gets up, orders a beer and brings it back to the table. The second says he doesn't approve of alcoholic drinks and takes it back to the bar. The third says don't be so wet and brings it back to the table. The fourth says I'm with number 2 and takes it back to the bar again. Counting only the time the beer is on the table and on the bar (thus ignoring the time in transit), since there is only one pint of beer and two locations, on average there is half a pint of beer on the table and half a pint of beer on the bar.
But at no point is there half a pint on the table. In the same way that a quadratic with a root at +2 and -2 does not have a root at 0.
its like asking whether infinity is even or odd
Anastasis Sfyrides how dare you guess it’s gender lol
I like the way you think
@@encounteringjack5699 for gods sake infinity is not an integer
@@encounteringjack5699 it just doesnt go like this. You could apply you reasoning to a banana and still get its even. Evenness/oddness are integer properties. Otherwise there would exist a k ε Z | inf= 2k
@@encounteringjack5699 Okay. I see the problem here. Its all about definitions(as always, right?).
We define a number n (integer) to be even if and only if there exists another integer k so that n=2k.
i.e. 18 is even because there exists 9 and 18=2*9
No sets or parts involved, just straight division by 2 involving two integers.
Infinity, just like our banana(wipe off that smirk) is not an integer so the definition could not apply to it.
3:33 the best dramatic zoom of all time
Until the 19th century :o
Lol
9:02 too
Enoch Romero 33 the sacred number of the nazarene
I'm so glad someone picked up on this 🤣
I remember watching this video when I was in high school and thinking “man this is amazing”, now I’m a maths major ending his first course of real analysis and boy, this is the stuff of nightmares.
Amazing how they think they can lie to people like that.
Why? It's not accurate or well proven? Or because these problems are so hard for mathematicians that they're nightmares?
@@RCmies Because it's false, the sum in the video is a divergent series, so it has no sum.
The power consumption of the light has a limit, if it switches at 60Hz, the watts is the same as if it switches at 50Hz or 1Hz or infinite Hz.
The sum is meaningless as it's non convergent element n always has a difference of 1 with element n+1.
Effectively on/off is related to n directly, not the fallacious idea of the series having an infinite sum.
@@simorote Even though it is a divergent series, the sum mentioned in the video aka the Cesaro Sum is widely accepted as a technique to obtain the sum of a divergent series... Many such techniques are there and Euler himself attempted one such definition of sums of divergent series... So don't just dismiss something by saying that they are lying... The beauty of Mathematics is such that sometimes there are multiple ways of approaching a problem and each way reveals something interesting...
Look at his pupils. He's clearly on crystal math.
I groaned so hard when I read this.
Not gonna lie, you had me at the first half
0:34 definitly
*maths
It worked for Erdøs
"This is weird. " - Guido Grandi, Italian Mathematician
Stu Mackenzie Possibly the funniest comment I’ve seen this year 😂😂😂😂👍🏻
Technically it translates to 'This is strange'
But lol yeah. 😂
I thought the same thing
3:16 look at the book
It's onf.
xD
NO, it's ofn
Acutally, it's onf only so ofn.
like the anti-ice system
No, it's nof!
No mystery here! The answer to this question is that the problem is improperly (or incompletely) defined. Therefore, as approached in this presentation, the answer is dependent on additional constraints on the problem imposed by the solution which is applied. Moving the parentheses about depending on the method used is actually adding constraints to the posed question and in so doing CHANGING the posed question. For example, placing the parentheses as (1-1) for ALL summed terms assures that the answer will be zero at infinity. This is a DIFFERENT problem than starting a 1 and then adding (-1+1) terms, assuring that the starting 1 always has a 0 added at each step to infinity. Thus, the answer is INDETERMINATE due to an incompletely defined problem.
The idea is that you try to think a little more deeply beneath the surface and question *why* we have to bring in such definitions in the first place.
We use axioms to work around unexpected problems in maths. To truly understand mathematics, you cannot just learn the rules of the game that we have chosen to define. You need to understand the logical reasons why we made the rules as we have. It’s not enough to just state the rules as proof something doesn’t work - that is a fallacy! We made up all the rules in the first place!
In general in mathematics, addition is commutative, thus you should be able to move brackets around this sun as you please and get the same result. But the video shows this is not the case! So what do we do? We can cover up and fix the issue by adding extra definitions to the sum like you’ve stated, but that’s just one possibility. Other mathematicians have alternative opinions on how to define such sums. Nobody is wrong or right as there is no correct fundamental answer.
@srquint Exactly 🎯!
@alexhoward7627 but that just means that the qn is poorly defined , doesn't it?
So if you turn a light on and off really fast .. you just invented a dimmer switch and set the light to half brightness
You might want to read about pulse width modulation of LEDs.
That’s what I said!!
Thank you, came here to say this. Thanks to arduino for me knowing about it.
Turning light on is 1 and switching off is 0
Well yes, but after 2 minutes you don't touch the switch anymore. And if you stop time at any given moment, the light is even on or off. This is a mathematical perfect question, the answer shouldn't be what a human being can process in a few milliseconds close to the end of the super process.
I know this is an old video, but if anyone is interested, the idea of taking the average partial sums is known as "Cesaro summation". And it's just one way of "summing" divergent series.
Thanks for using "quotation marks". The Cesàro "sum" is obtained via an averaging process that doesn't usually produce the same "value", unless the series is strictly convergent (which it isn't, in this example). The quote marks around "value" is a reminder that the so-called "value" of the original series is not any number. There are also fancier ways to produce an associated value, but when the original series is not strictly convergent then none of them produces an actual direct sum of the original series. For example, one method starts with the series in Dirichlet form, uses analytic continuation which changes the values over a complex half-plane, then evaluates the zeta function at a point in that half-plane, which produces a finite value but no longer represents the original series.
Another answer could be -oo (infinity). If you use the parentheses correctly: 1-(1+1)-(1+1)-(1+1)....etc. This will eventually result in -infinity or -oo
This includes the parentheses without having to add more to the equation. In his example where he added parentheses: 1(-1+1)(-1+1)(-1+1)....etc. You would actually be multiplying 1*(-1+1)*(-1+1)*(-1+1)....etc. So, whilst he says this will result in an answer or 1, it will actually result in an answer of 0. No matter how you place the parentheses, you can never get an answer of "1" without adding more "+" to the equation.
@@MrMR-sk8jm I think I just experienced a mathematical stroke
@@MrMR-sk8jm I believe you are confusing the underlying mathematical reality with an artefact of shorthand notation.
*smirks in Ramanujan and Euler*
When you try to balance the light switch between on and off
That's only the position of the switch not the state of the light.
Ive tried to do that before. You start to get this electrical popping sound. I don't know that happens if you hold it there for a long time.
Gotta love dimmer switches.. this mathematical equation is just setting the mood 😀
I love how much this dude genuinely loves math; he couldn’t hide it if he tried
yeah
I've never seen mathematics explained in such an exciting way. If I only have had this guy teaching me mathematics 50 years ago.
This fellow is a "real" teacher. Real teachers are unfortunately rare.
Yes I agree it's exciting. But it's wrong. Don't believe this. 1 - 1 + 1 - 1... diverges. It's not 'equal' to any number. If you want, check out Sequences and series chapter in calculus.
@@rohankumar-lh7ys Yes, I must say it sounded little weird...but I will check out what you recommended. Thanks!
@@pte1808 You are welcome! If you don't have time to buy a book and read it, you can read articles about 'Convergence and Divergence of Series' on the internet. Few examples will help you understand, according to me.
How is it half with 1s 0s
I love this guys nerdy excitement in the beginning of the video!
"TH-cam is the home of rational and informed debate." HAHAHA
Yeah baby yeah!
James Keefer Same xD You just can't not love it!
MATH RULES!!
Stop calling it "nerdy"....It's pure excitement.
Okay, for the light experiment, (granted I am not a mathematician and have surely made a mistake) but assuming you could even switch a light on and off, and that the electricity could even reach the light in time to be considered on and off, the maximum number of times you could turn it on and off would be just about 51 times, where the 51st time would equal 4.2*10^-44 seconds away from the goal (which is just under the planck time length). The 51st time would be in the on position, in this case.
1/2 is just 5 years old me balancing the switch
Stolen comment
@@ithaca2076 and no one cares
@@j.hawkins8779 FBI does, I guess
@@stv3qbhxjnmmqbw835 but still, no one cares
Someone explain pls tnx
_"And I'm hoping it will cause a little bit of debate on the comments, 'cause I know TH-cam is the home of rational and informed debate. So I look forward to that."_
That's exactly my kind of humour. ^^
Haha, that part slipped by me!
finally somebody who shares my opinion . ☺ lol
Well if youtuber arguments are neither rational or logical than they are fallacies.
Most arguments people make on TH-cam remind of people who try to divide a number by zero.
+AluhutSmasherTV . tell me about it .
3:38 Things get serious in the nineteenth century...
"Grandi... was a monk, he was a mathematician, he was one one of those types" - lol!
😂😂😂
I don't get it
He was a monk with a few levels in wizard.
If we repeat that infinetely, he must be halfway to a monk or to a mathematician
@@mk_rexx ahh yes, a monkematician
I'm not a mathematician, but a lot of these questions just seem to be treating infinity as a really big number. If you add 1+½+¼+⅛, then by infinity you reach 2. Well, you don't reach infinity at all, by definition, so you're always a little short of 2. The sum 1-1+1-1... is ½. No, it's either 1 or 0 if you ever reach a point where you can answer the question, depending on whether the last number is 1 or -1. If the sum is infinite, then the answer keeps switching between 1 and 0 forever. If you switch a light on and off progressively faster for 2 minutes, is it on or off at the 2 minute mark? If you're moving the switch at infinite speed then it's probably broken and you may not need a light at that point anyway.
A lot of the paradoxes leave me thinking either "I don't think that's how reality works" or "I don't think that's how infinity works".
Years late, but to clarify your concern: infinity when used as a limit, _is_ the limit, there's no beyond it. It's the absolute limit of time one - instinctively - wants to think extends beyond.
That's to say : if your sum limit _is_ infinity, then by definition you can and _do_ achieve that limit in your answer, however far away that is. if your final sum was a "little short "of an infinite series, then you'd be a little short of X, but if your limit is infinity, then and only then do you reach X, and because we state the limit _as_ infinity, we're allowed to do that :)
Sometimes, such a result is said more in the frame of _"as we approach infinity, the answer approaches X"_ but mathematically, this means the same thing as _"at infinity, the answer actually = X"_ - because, if it _didn't_ mean that, then we'd never be able to calculate anything that contained infinity in the problem, and that would be a problem, and it would really upset physicists :))
@@IngieKerr months later too, but what he was trying to say is that infinity might make sense on math but on real physics, such as turning a light switch on and off, there is no infinity. Everything must happen in a finite amount of time, quantity and space if that makes sense.
Anyway, what I’m trying to say is that limits are just mathematical concepts and can’t be taken to reality.
@@gONSOTEThe gaps are getting smaller..
Anyway to anyone thinking the same, we use the concept of infinity and limits to break through the limitations of our dimensional reality(spacetime), which allows us to accept and understand things that otherwise wouldn't make any sense like, that the electron has a radius of zero, and no extent, yet we know it exist, interact, and can measure it's effect.
What I'm trying to say is it's an Oxymoron trying to tie infinity, limits and other Applicable Mathematical concepts to our reality(spacetime), But that's not what Mathematicians does when explaining things to us, they give us a "similar" situation in our reality, to grasp the concept not get an answer. Hence the Light switch or that guy from the textbook that bought 109,876,542 Apples from the market.
@@gONSOTE I would say limits very much exist in reality. The number e has never had its value determined exactly because every method of producing it requires infinitely many steps (it would also require infinite space to store all of its digits), with a formula for the value of e being (1 + 1/n)^n with n approaching infinity and another being the infinite sum of 1 + 1 + 1/(2!) + 1/(3!) + 1/(4!) + ...
The universe, on the other hand, doesn't have to calculate the value of e. e is a number that simply exists inherently within the physics of the universe (well, maybe. There's some argument that the physical laws might be probabilistic in nature, which could mean that even the universe itself is simply approximating numbers like pi and e).
Still, there's plenty of practical applications of the idea of 1 + 1/2 + 1/4 + 1/8 + ... = 2, in engineering, for example, 1.99 will potentially be approximated as 2, so if you have a few dozen terms of that infinite sum in some engineering problem, you will approximate the sum as 2.
Even the 1 - 1 + 1 - 1 + ... = 1/2 idea has practical applications. If objects like light bulbs oscillate between two states fast enough, the outcome may be nearly indistinguishable from a static system halfway between the two states. Fluorescent lights in the U.S. work like this; they in fact flicker at high speed due to the alternating voltage of the electrical grid. Some lamps also can very their brightness by flickering at various intervals. This is because our eyes essentially do an exponential moving average of the terms of the sequence, 1, -1, 1, -1, ... which is a bit different than the sequence of averages of partial sums, but the outcome is much the same. A value approximating 1/2.
Edit: Finally, the unique property of a limit like 2 for 1 + 1/2 + 1/4 + 1/8 + ... is that 2 is the smallest number that you can say with certainty will not be surpass by any partial sum of those terms. If you told me 1.99, I could find the first few dozen terms of 1 + 1/2 + ... that surpass 1.99. 1.999? I can do that too. 1.9999? Same deal. I can even tell you how many terms it will take without having to actually add everything up. 2 is the very smallest number that I could never find any quantity of terms of 1 + 1/2 + ... that will sum past that number of 2.
The light switch is a bit of a paradox, but the taking the limit of 1/n to be 0 is definitely real. Pretty much all of calculus relies on infinity and taking infinitely large sums of infinitely small numbers
after 2 minutes the light will be changing between on and of "infinitely fast" or having an infinite amount of switching in no time , so what would actually happen is that you would see the half of light intensity
+ahmed Imam No, why would you give the on and off states equal weight? The average intensity right before you reach 2 minutes is 2/3, due to the asymmetrical duty cycle.
+Doug Gwyn Hmm, I didn't know , Iám just thinking mathematically not physically
+ahmed Imam well thats exactly what physicist do
+Eirik Hovind of course , I meant I was trying to apply the arithmetic average to light
+Doug Gwyn Technically not, we're concerned with the state of the lamp at the instant the 2 minutes are up, not with the average amount of light over the course of the 2 minutes. This leads back to a half, as the light just as the timer ticks to 2 minutes it will be alternating infinitely fast.
It is both one and zero because there is no set end, and depending on which point you break it off it could be either... and 1/2 is right between the two. I like that one.
The question is flawed. Averages as a process can bend reality to a result that is impossible. This can be easily seen as the average number of people being a rational result. For example, demographics find the gender of a group of people. You then pull a random person from that group and ask the question of what is the gender of this individual?
The data is split into fractions and these fractions represent probabilities. You cannot say for certain the gender of that individual, but you can definitively state the probability. The same logic applies to the question of whether the light is on or off.
That said, the 0.5 result is not the representation of a literal result. Since it relied on averages, it innately becomes a probability. That's why it can be accepted as a result to the Grandi's Series. The result of Grandi's Series is a probabilistic question, so it makes sense for it to have a probability as a result.
The equal sign is like humanity's way of forcing the numbers that represent a reality to conform to our hypotheticals. Without it, the numbers are basically acting like a liquid that evens out with whatever other numbers are in the pool until they settle in the middle.
Indeed. In many cases, probabilities are used to used to find values one can expect when the probabilities being compared have the same units.
Grandi's Series is one case of this since approaching infinity would lead to 50% of the domain being a value of 0 and the other 50% being a value of 1.
To be fair though, in these theoretical scenarios that holds to their premise and never changes their process, equal signs can be used just fine. That is pretty much what mathematical induction is after all.
The big issue, when translating it into reality, is that maintaining the soundness of the theory due to how difficult it is to prove a process will never change. A consistent process is important to eliminate negligible values like the trailing -1 or +1 at the end of Grandi's Series or the remainder of 0.3333... when evaluating 1/3 in decimal.
holycrapitsachicken that's exactly what I was thinking👍🏼
But quantum physics is reality.... particles do happen simultaneously in both ways, and maybe it just means that the lamp will reach a superposition state, when the speed of turning on and off is approaching infinite.
That's right Lin. Superposition can certainly occur. However, if it was in superposition, you would not call it on nor off. You would call it some kind of mix between the two, something new derived from the combination of the two.
You know what's interesting actually. In terms of lights turning on and off very quickly, there is a very real-world application of this for lights in electronics. This is called *pulse-width modulation* in which at high enough frequencies of turning a light on and off can change the *brightness* of a light.
Perhaps in the real world, the 0.5 refers to the superposition state of on and off being at half-brightness. Half of the time it's on, half of the time it's off. So our eyes perceive it as half of the brightness it's supposed to be when it's on.
Love his confidence as he writes with a sharpie 🌞
The answer" You discovered quantum mechanics and the light is in a superposition of on and off. :)
I cant find another comment about quantum. I had the same thought initially!
Sounds like Schrodinger's shenanigans
Obviously he's explaining Schrodinger's Cat but with math!
But strangely if you look at the bulb the bulb closes (or opens)
until you look at it
I swear if it's -1/12
That’s 1+2+3+4+5+6+7+8...
@@hectorsk9490 watch the video on Reinmann's Hypothesis
It is right if you prolonge the Zeta function, but wrong if you use simple math and convergent rules it is wrong, that's what most people don't understand.
@@DK-xw5hp The value of the zeta function for -1 is -1/12, not 1+2+3+4...
The zeta function does exist for negative values, the -1/12 thingy is just an analitic extension.
@@eternalio5885 1/1^-1 + 1/2^-1 + 1/3^-1... = 1+2+3+4...
shrödingers light switch
semi awesomatic Now spell him correctly.
I was literally just about to comment this.
@@dothemaths1256 Ok, Ill do the maths
x+x(xx)+x = 1752?!
x=12
boom
+Do the Maths r/woooooooosh
@@MrR4nD0mDUd3 I was figuratively about to do the same...
“This junk is getting smaller and smaller and smaller”... i know that feeling all too well :(
It's just cold water, don't worry, it will come back
I was in the pool!
like a frightened turtle
My parents never let me turn the light on and off very quickly. Today I learned why.
just so you dont accidentally answer an unsolved mathematical question
@@teemuaho4807 or so you don't lose your mind
Why?
@@teemuaho4807 Yup. No other reason. Heaven forfend InstaSound do the math!
Trust me, after 2 minutes, the light broke.
Vsauce?
I'm sure you tried IRL
After two minutes, you are having a seizure from a light turning on and off infinitely many times
+tomfoolyaface in 2 minutes
TH-cam strikes again. Where am I? I'm up in 6 hours for work and I'm here watching a random mathematical equation /show/mind boggle. Love it.
Callum Rowley relatable
I swear, that in personal journal writing I wrote and 'started seeing' something like 0:23-0:43 once but everything from 0:43+ is where it's 'starts becoming it's own thing'--the nice feeling of validation when you find words to an experience might sort of apply here now that I know of 'Grandi series'
THE SOLUTION:
the lamp is broken.
+babnasyes industries No. The lamp SWITCH is broken.
+chamcham123 No, He kicked the lamp, its broken now.
As long as you don't change the color of the wall the lamp's on, the alien will turn 90 in purple years...
x Lunar xD
+N3cr0m0rph biotch * don't do math
Physicists answer: A broken lamp.
Penguin Design yep
Actually, it would most likely be that it is like dealing with quantum entanglement. 2 atoms connected, both opposing spins, yet they are both spinning one way and the other at the same time until observed.
I would think that would be the engineer's answer
Thomson’s lamp? Is it on or off?
Yes.
If a series doesn’t converge it diverges... that’s just a rule.
This series is alternating between 0 and 1 forever and therefore never converges so it is divergent.
My opinion (which definitely can be wrong 100%) is that when you try to do math like that with a divergent series, it’s like dividing by 0, you can make anything you want happen.
That seems like a logical explanation for why you can get multiple answers, but again, just my opinion.
except it doesnt truly diverge, when a sum diverges it grows indefinitely, there is no growth in this sum only the constant and equal switching of the values (in my opinion of course).
it doesn't matter whether it's convergent or divergent, he's trying to find the sum not the limit. he even said it has no limit
@@lafyouthclub281 an infinite sum, is itself a series that has a limit.
And its limit doesn't exist
@@fareedabifarraj483 I see the series as a way to show the number 1/2. Imagine 1+1, that's equal to 2. That's one way to show the number 2. And so 1-1+1-1+1-1... series, because it's defined being infinite, instead of seeing it literally as a divergent series, I see it tending to 1/2 because it doesn't want to stick to 0 or 1 values. Think of it as two 'sweethungry' guys that want to eat a single pie and decide to cut it in two, so that both get equal amount.
Plus, because there are an infinite amount of -1s and +1s, it can equal 1-1 or 0+1 with how the series works, BUT it's exactly how the series works that the result is not 0 or 1, so it's essentially forced to be a half. Again, that's the way I see it.
the way i see it, he is trying to give the series a value that is meaningful and useful in calculations . you can’t have a square root of -1 but we can say it is = i so that we can have a meaningful value attached to it for calculations. in the same way that we obviously can’t have the square root of a negative number so it is undefined yet we assign it a value of i so that we can use it in calculations, we can assign a meaningful value to this series that we obviously can’t have a value for so that we can use in calculations.
In school i hate maths, but this channel always blows my mind.
Well it’s spreading false information since this is a divergent series lol
You've mathematically described the concept of PWM (pulse width modulation - for lights that would be flashing the light so fast that it looks like it's on continuously at a lower brightness).
That's what I thaught about aswell. So it would literally be half on wich would equal to 1/2 as the answer to the sum
I understood everything in this video perfectly, until you said '1'.
There is no answer. This is exactly why infinity does not work as a quantity. Infinite series are good for convergences, but that does not mean there is a literal value of infinity where the convergence will be met.
The minute you declare you have found the answer, I'll just go on 1 number further than you looking for it.
I thought James' discussion of the lamp was going to describe PWM (pulse-width modulation) whereby you imitate an analog value with digital values. If you only have digital values (eg voltages) like 0v and 5v, you can "simulate" analog values like 2.5v by rapidly turning it on and off as fast as you can for an equal amount of time. You can simulate a value like 1.25v by turning it on for one time unit and off for three. Of course, that's not really an explanation for infinite series, but it's related.
It's off because you can't afford electricity anymore.
but you've used half as much electricity
and what kinda of pleb cant afford to keep a light on for half of 2 minutes
that means it has 1/2 brightness.
Great Answer!
ChocolateFlavoredRamen its off cause you burned it
The answer can't be 1/2. You only have 1s and 0s. regardless of how you work the problem.
I'm scared to watch these videos.
What is your name!
His name is Hugh Mungus.
harley395 Hugh Mungus what?
Goratchthemule Its Hugh Mungus. Thats his name.
harley395 is that a sexual harassment?
3:59 "If we pick a nice infinite sum, cause there are nice infinite sums and there are bad infinite sums"
Nice infinite sums being convergent and bad being divergent. Convergent ones have a result of summing them, divergent ones either diverge and we don't do math with them because they have no result. That includes our "One minus one plus one minus one"
Edit: Look up "Divergence tests" Today to see how to determine if Your sum is divergent
The light is off. Because after infinite switches on and off the light bulb's fuse would go out. Therefore the answer to 1-1+1-1+1.... is 0. Because of fuses.
yea, the cheese is real :D
whoulde said the same tho
AzraeltheRightous Perhaps; but the cake is a lie.
daemonpoet
AHAHAHAHAHAHAHAHA
EPIC
EPIC FOR THE WIN
I REALLY LOVE YOUR PORTOLE REFERENCE
I AM SUCH A GAMER
AHA
A HA
HA
nop e calm...
I know answer.
If its 1/2= light bulb is off, but everything is on fire because of fuses :)
is this the solution to what came first - the egg or the chicken?
It was half chicken and half egg?
a stillborn
the chicken-and-egg thing is a non-question. It stems from the problem that there is no strict definition as to where the chicken's ancestors became chicken and not some other animal. The demarcation of species is a human concept, and arbitrary. A chicken just is, nothing came first. A real philosophical question to ponder would be where the universe came from, and that is a real unanswerable question in my view.
Gabriel Cavendish I like your explanation...
+Gabriel Cavendish
The egg came first because an egg some other animal laid had a mutation that caused a new species: the chicken. Obviously some other animal wasn’t just walking along and then magically turned into a chicken, that’s not how evolution works.
@@vibaj16 that egg came from a chicken though :) some bird did not give birth to a chicken. That's also not how evolution works
The light on/off question is actually used in discrete control systems. Circuits can only technically have 2 states--on and off. Sometimes, though, you need to produce an "analog" signal with a varying amplitude. But how do you do that if you only have 2 states? Basically what a digital controller will do is just flip that particular circuit on and off at an extraordinary rate, and the weighted average of the "on"s and "off"s ends up being your attenuated amplitude.
So let's say within one second, the flips are such that the circuit is on 80% of the time and off 20% of the time. This will equate to an amplitude attenuated to 80% of your maximum amplitude. In the instance of the specific situation Dr. Grime was talking about, you would get a 50% attenuated amplitude--or the 1/2 you would get from the infinite sum!
This is the best answer to Dr Grime's last question. It sounds like pulse width modulation to me.
breakbumper Yes. :)
I could argue that if you had an incandescent lamp, the result after two minutes would be on but dimmed by half. The power input would fluctuate so fast that the result would be an attenuation of the power input by half, hence the bulb would be half as bright. I thought about this when he was explaining the final part.
***** That's pretty much exactly what I said. On a bit of a tangent, though, you wouldn't need to attenuate an incandescent bulb like that because you can actually give it a true analog signal. But it would still work if you had it hooked up to a digital circuit :)
Somebody give this guy an award for the most intelligent TH-cam comment ever.
I love the incorporation of supertasks in this (the lamp problem, an infinite number of calculations within a finite amount of time)
As you aproach the 2 minute mark, you are getting to a fraction of time of which you have the light on or off. When you have reached the 2 minute mark, you have turned the light on and off an infinite number of times. As infinity is not a number the answer is it is not defined, as strange as it may sound. We can see this with the series as well. The sum is not defined
Yes exactly. You could not have turned the light on and off an infinite number of times in 2 minutes because there is a smallest amount of time it takes to turn the light on and off. You cannot keep doing it faster and faster - in the real world.
By this logic all infinite sums and limits are invalid.
@@thischannelhasnocontent8629 no by his logic only divergent infinite series (a series whose partial sums do not tend to a value) are undefined, such as the one in the video. Convergent infinite sums still approach a some limit and have a value.
@@ryanduggan6738 you can use the alternating series test or the geometric series test to prove that this series is divergent - therefore it does not have a value by definition
Physics answer: superposition
shrodingers lamp
give me Nobel plz
_ I _
Physics could get 1/2 even if lamp would switch completely randomly. Mathematics would get undefined answer with each level of mediation in that case.
*Schrodinger
*Schrödinger
So is the lamp both on and off, or neither on or off?
Yes is not an acceptable answer.
Guys stop trying to change his mind. HE'S SMARTER THAN YOU! Accept it. This is accepted as are the other infinite sums this channel covers.
I’ve seen other expert mathematicians who criticize some of this channel, including this one. According to them, the “sum” is nonsense because it just oscillates between zero and one. That’s the whole answer.
Well regarding the lamp, if it would turn on and off INFINITELY fast, then it would only make sense that it would create an apparition of a dim light let's say- a not fully bright light as compared to the one when it's fully turned on (the one representing 1) since it would create a singular image due to its speed. If you think about it even without knowledge of Grandi's Series, it would makes sense thus also making it a supporting proof for Grandi's Series itself since a dim light, in fact, is a light level in the middle of an on and off state of the lamp (representing 1/2).
The average intensity of the lamp over the entire interval is 2/3, not 1/2. If you start averaging at some moment other than the beginning of the interval (with the toggling still done exactly as specified) then the average from there to the 2-minute mark will be different, somewhere in the range [1/3,2/3] depending on when you start averaging. The fact is that the state of the idealized lamp at precisely 2 minutes is indeterminate, not 0 nor 1 nor 1/2 nor 2/3, etc.
i think the answer is that u break the switch lol
@@rohangeorge712 Same. I don’t think the lamp can take the abuse, either.
Well for this to work, we have to assume 1+1-1+1-1+... is even a number at all. Why should we, when it isn't even convergent?
I think the 1/2 solution is close to a probability solution. Since the sum is virtually infinite, we cannot discuss where it will end. Therefore, it has a half chance to end in 0, and a half chance to end in 1, thus the answer is 1/2.
I think the way you explain it lines up with the partial sums
An infinite sum does not "end", i.e. you never "end" adding it up. This one simply does not converge (like the guy said, there is no limit). The fact that the average sum converges is interesting though.
That's very interesting. It's like when you roll a 6 sided dice, the average will be 3.5 even though the dice never rolls that value.
its acctualy 3
(1+2+3+4+5+6)/6 = 3.5
That's fun and all with the exception that there is no such thing as infinite, its just an abstraction
The first sum (1-1+1-1+1…) will depend entirely on where you stop the sum, the result will loop between 0 or 1, the concept of infinite is not applicable here
A reminder of the golden rules to be adhered to when dealing with divergent series:
1) Do not use brackets.
2) Do not remove any zero (unless you have proven that the divergent series is stable).
3) Do not shuffle around more than a finite number of terms.
Not adhering to these rules yields incorrect sums.
I enjoy how you always correct them in the comments
@@NorthRoyalton Just trying to switch the light on...
Math: Screw this, I'm not doing this anymore
*hands lamp over to Quantum Mechanics*
Quantum Mechanics: |1⟩ + |0⟩
Math: *facepalm*
I actually facepalmed before I read "facepalm", I must be a nerd.
Schrodinger’s cat
Exactly...A rational answer ....
Pretty sure flipping a light switch on and off an infinite number of times in the span of two minutes would completely destroy the switch. So yeah, the light would probably be off by the end of it.
shark puppet Hey, get your real world solutions out of our allegorical abstractions!
I know this is old. But how about looking it like this:
You are trying to quantify something which is infinite using our finite methods. It's almost like a quantum bit. It's both 1 and 0 until you observe it, then it falls back to finite state and it's either 1 or 0. It's the same principle here. If you stop the series at infinity in a finite method it will be 0 or 1 depending on when you stopped the series.
I think this series might actually be great for explaining physical properties of quantum bits. :)
I was thinking the exact same thing :3
+arekusanda1 i was looking for this :D exactly what i thought :D and man this might actually be the best comment section on youtube, he doesn`t need to use it sarcastic in his video :D
After a night of thought I came to this same conclusion. Now at 2 minutes the switch is both on and off, but what is happening at 3 minutes? In theory we'very gone past infinity!?!??
+Some Guy. Pretty sure going further than infinity is a larger infinity making it the same problem. I could be wrong though. I doubt I can properly comprehend a lot of the stuff on this channel.
+arekusanda1 an* ,and you have the smartest solution out there,i like how you're thinking :D
This thing is used in digital circuits to create analogs outputs, and it's called PWM. If I keep switching on and off a pin with a fixed frequency (and that's the assumption), what I get is a square wave with a duty cycle of 50% and therefore the voltage value which is seen from the external world is half of the supply voltage, that is, 1/2. That's my physic interpretation of the value this series converges at.
As described, the "frequency" (duration of 1 cycle) is not constant. Depending on where in a cycle you start, the average could be anywhere from 1/3 to 2/3.
Yes you are right, I was referring to the first half of the video, when they don't speak about the frequency. In the real world their example in which I keep increasing the switching frequency is not feasible, since there is a limit on how fast a micro processor can go. It was an interesting thought experiment, though. Thanks for your answer =)
Yep, this is where the real world tell the math world that it doesn't care what math thinks should be. If you are turning it on and off rapidly enough then it would literally be both 1/2 on and 1/2 off. There is no paradox there. The problem is that math assumes there is only a 1 or a zero, but that assumption doesn't work in practical applications like light switches. You can definitely have a half-state in light switching and math can't claim otherwise. And at the end of 2 minutes you have equal chance of either state being the end, thus 50/50.
The duty cycle as described is variable, with an overall average of 2/3, not 1/2.
If only they tackled the zeta function with this approach in mind emphasizing that "the limit of a series is not always the same as its partial sum (and therefore not really the same as analytic continuation)". More honest presentation.
Yeah
Damn you, Numberphile and especially you James! I'm at work, plus I have a lot of studying to do for my exams and I'm just sitting here watching your videos O_O
Maybe if you were to infinitely turn a lamp on and off for 2 minutes, the light bulb would be so worn that it only produces half of its original potential.
Google wanted me to change my username so I changed it to a description of why I changed my username Your name is perfect
Love this answer
Comment & like only because of your name! lol
Google wanted me to change my username so I changed it to a description of why I changed my username
Nice name fam
Daily Updates on Michael Jacksons Health Condition if you'r e only turning it off until 2 minites you're not doing it infinitely?
I think that since you're always in transition between the light on and the light off and as the speed at which you transition between on and off increases and the time of which it's either on or off shortens, it gets closer and closer to impossible to tell if the lights on or if the lights off. If this goes on for an infinite amount of time and what it actually looks like in the room is that the room will look dimly light, with a light intensity that's half way between on and off. Hence the 1/2 answer.
The light is pulse with modulated which with real lights is dimmed. It with be at half brightness with a 50% duty cycle. So I say the best fit is 1/2 or 0.5
This guy looks like a Harry Potter character. A crazy wizard.
He does kinda look like a Weasley lol
Yeah he looks like Fred/George
@Allan Bozz SERIOUSLY though!!! It’s probably just cuz he’s British... I wish Americans could be wizards ;(
That cause he is one obviously! How else do you find these things?!
Actually come to think of it, he does look like I always imagined Rincewind to look
Is it one or zero?
Yes.
Superpositioning
Zero-one duality.
a>-1
the square root of 69 is 8 something
-drake
Hahahaha
He's not wrong...
***** No one said he's wrong. It's just funny
Pennyinmouth 8.306623862918075
Cus I been tryna work it out all day
It’s funny how he’s doing calculus while desperately trying to avoid saying calculus terms cuz then everyone would go “aaaah calculus”
I would argue that you wouldn't even _get_ an answer, because to get an answer you would have to end. And infinity doesn't have an end.
i think you kinda misunderstand the meaning. Of course there is no "answer" to infinity. I think, even if they may phrase it different, or smth. its more like they are searching for a value. Because with Values you can work and with "infinities" not.
AzraeltheRightous Two things.
First, if you bothered to read my entire post i said the concept of (inf - inf) is fallacy. This has been proven so I'll not go over it.
Seond, if the series within the parenthesis is, indeed, infinitely long then by definition it will turn out to be (inf - inf). Thus, the first point holds true.
Simple rearrangement is key here. For a short example we show that (1 - 1 + 1 - 1) is also the same as (2 - 2) I simply re arranged the positive and negative ones. With me so far? So, by that measure we have an infinite number of positive ones so we can re arrane them into positive infinity. Likewise we have an infinite number of negative ones so we can rearrange them to get negatve ininity. So, the problem can be rearranged to look like (inf - inf) = S. So, 1 + S = 1 + (inf - inf)
Now, the reason it is so important to realize the falacy of inf - inf is because 1 + inf = inf so the above equation of 1 + (inf - inf) can be rewritten to be inf - inf due to th communitive property.
If you're talking about the lamp thing, he specifically mentioned Zeno's paradox. The infinite process WILL end after 2 minutes.
***** Where did Zeno's Paradox come from? That is a different video all together.
eric mcdowell What you say doesn't make a difference and I'll show you why.
The original problem is (1 - 1 + 1 - 1 + .... ) as an infinitely repeating pattern. Using the assosicative property we can rearrange the problem where are the positive ones are on the left and the negative ones on the right to give us :
(1 + 1 + 1 + ... + 1 - 1 - 1 - ... - 1) which will give us the formula (infinity - infinity). At this point it no longer matters what you put outside the parenthesis, except zero, because the formula will always revert to (infinity - infinity). Even if you raised the whole thing to the power of infinity you'd still get (infinity - infinity).
Thus, the paradox comes out to (infinity - infinity).
Folks who are having a hard time with this, consider this: you could reason that since no numbers multiplied together equal -1, then "what is the square root of -1" has no answer. However, pretending such an answer exists leads to all sorts of interesting results. Similarly, even though it is "clear" that this series "has no answer," pretending that it does leads to all sorts of interesting results. S = -1/12 is no more "wrong" than sqrt(-1) = i
The light has been pulse width modulated to half intensity.
But that's only true for 1=light/0=dark, not 1=light/-1=dark. "Dark" is not negative energy.
See it is for 1 and 0 as it is the two answers from the bracketing. The sum is flipping between the two. That's how I am seeing it anyway.
Converting the math into a phenomenon (light pulse) it becomes a time series, so you use the partial sums which is a series of 1s and 0s.
Light is off, the switch broke after first 10 million toggles.
Not if the switch broke with the light on...
The average of the partial sum works if you have limit because you eventually drown the earlier values into an infinite number of values that get infinitely close to the limit. But if you don't have a limit, that's just an average. It shouldn't be considered as the actual sum of the series.
I think the answer is that the light burned out because you flipped the switch too many times. J/K
"Cause I know TH-cam is the home of rational and informed debate..."
Lol'd.
It's all of the above.
+Efreeti Until it is observed with will imposed upon it.
It's A
+HD Candela you don't get it do you
Everything and nothing, Oam
+Efreeti
No. Actually he is doing it wrong if we follow the laws of mathematichs.
I do understand where he is going with this video but when he writes 1-1+1-1+1.... this actually means 1+(-1)+1+(-1)+1.... which doesn't have an answer. The answer alternates between 0 and 1.
This video confuses people that don't really know math.
On the off chance that you actually read this comment:
I saw a number called SSCG(3).
Could you do a video on that one, in your big number series? I heard it's larger than TREE(3), and yet I don't know how and why TREE(3) must be this immeasurably big (is it not a sort of lines and graphs after all?), and I know you worked on biggest number imaginable (Rayo's number I believe), although I didn't quite understand it. I struggle in English and my PhD was in Persian literature, so the last time that I HAD TO study (veeeeeeeeeery basic) maths, and in Persian, was about 16 years ago for university's entrance exam, and I didn't quite ace at it (14%. I know. All the shame is mine).
But I'm really fascinated by your big number series and your teaching skills, and I'd love to hear about that particular number or function (whatever that is) on your channel.
Lots of love from Iran for you, beautiful people and great scientists and outstanding teachers. ❤️
The way you proved the 1-S=S, 1+S=S is also possible to prove. Like this:
1+1-1+1-1+1-1......
1+(1-1)+(1-1)+(1-1).......
Now just swap the elements inside brackets, which becomes
1-1+1-1+1-1+1........
So 1+S=S
Which means 1=0 isn't it.
Zulkarnine Mahmud hmm
Duhh
Zulkarnine Mahmud No you have an extra kne
wtf is going on
Nope. 1-S=S
I'd say that here instead of saying that the limit is 1, 0 or 1/2, we should say that at infinite (or at any step of the serie), we have 1/2=50% chance to have 1 and 1/2=50% chance to have 0, but 0% chance to have 1/2...
Or we could say : IF the serie converge to a unique value, the value must be 1/2
But at any given two consecutive step, we have two value 1 and 0, the serie is not convergent...
This doesn't make sense as this is an infinite series
Guys there is an easier method.
This sequence is a geometric progression with a common ratio of r=-1 and the starting term of a=1
A geometric sequence goes like a, ar, ar^2,ar^3.....
If you substitute the values of a and r, you get 1,1(-1),1(-1)^2...
Which is 1,-1,1 upto infinity. The sum upto infinity of a geometric progression is given by a/(1-r)
Substituting,
1/(1-(-1) = 1/2 and there you go. There's the answer.
@@kwertie133 You can't do this. In the geometric series, the convergence radius is 1. when you choose the value a=1, you are taking the value on the boundarie where the series no longer converges.
plot twist: there is no infinite series. no one really does a infinite sum, so it does not exist.
@@luiz_46487 yes it does. But the answer isn't in math. It's in science. The answer is a superposition where the light is both on and off. Or the answer being both on and off in seperate scenarios. Look up quantum superpostion and the theory of quantum immortality
Everytime i get too cold - 7:59
Haha
Tending towards half of its original size.
🤣 I swear math videos are the funniest ones
1×1÷1×...
1-^1√1^...
The only reason he got 1-S=S which leads to S=1/2 is because these are two differntly defined S, so in fact it's 1-S1=S0 which is easy if S1=1 and S0=0 or the other way too.
1-1=0 and 1-0=1
If you convince the audience that 1 is the same as 0, then you can derive whatever false you'd like out of that.
End of story.
That’s a sheep’s answer. The question is *why* do we have to have these definitions? Remember that these definitions we made up ourselves using axioms. Just because we’ve decided to say something is false, that doesn’t mean we can then use our own definition as the proof for the definition. What if we changed the rules so that it does = 1/2? The idea I’m getting at here is don’t just rote learn the textbook as gospel. Question why things make sense logically from first principles. That’s what Numberphile have done so beautifully here.
Actually the heating filament in the lamp has thermal capacity, so if you turn it on and off infinite times, in the end you will practically drive it on a 50% duty cycle pulse width modulated signal, thus the lamp will literally be halfway up. Also the coil has parasitic inductance which will average the high frequency signal out further
If you make a paradox, make a working one :P
You have a disproportionate amount of knowledge regarding lamps
What can I say... I like dimmers
Getting into it seemed like a bright idea but it's pretty complicated and at first you just go around in the dark.
It's reminscent of how a rectifier would work maybe, with the thermal capacity acting like the capacitor in the rectifier. If it behaves like that, I would expect it to work a bit like RMS voltages, where the brightness factor would be sqrt(2). But I am guessing
+Adam Weishaupt One might even go so far as to say he's enlightened about lamps
+Michael Bauers You don't need rectification if you do it with a solid state relay or triak (or even combined with a variable frequency drive), or let's talk about a DC supply and a fast switching transistor. Let save the hassle and go simple :D
the light is off because if the switch is not broken, the light bulb is for sure.
Light on or off at the end of 2 mins is like Schrödinger’s cat question
Mathematics describing reality (quantum mechanics) , but they don't know it yet.
The ansaer is superposition
Bruh I hate it when my quantum mechanics gets mixed with my math of infinite series
Yes! I agree! I think that this creeps into the Schrödinger territory where 1/2 represents the light being both on and off at the same time.
@@ML-5 But in reality its only a try to get a result wich isnt even possible to get...
It is 2024 and my mind still blows every time i see this
In answer to the question about the light, I tried to do the experiment myself just with drawing different types of lines in a sequence rather than using an actual lightbulb. The answer to the puzzle is this:
It is actually mathematically impossible to work it out. I tried to draw different lines at different intervals so: after 60 seconds, I would draw a straight line, after 30 seconds a curly line, after 15 seconds a straight line, after 7.5 seconds a curly line, after 3.75 seconds a straight line, after 1.875 seconds a curly line and so on. But, just as it is explained in the video, the number of seconds between each line drawing never quite reaches exactly two minutes, it is always a fraction of a second before no matter how far you go into the sequence.
Therefore, we will never find out what will happen after two minutes because that two minutes target will never actually be reached!
You could argue that you could draw a straight line at 0.9375 seconds and then two minutes would come and you could say: "Right, it must be a straight line and the light must be on then." But actually, you have missed loads more line-drawing at intervals such as 0.46875, 0.234375 and so on. Even if you can draw lines or flick switches at the speed of light, time will beat you and two minutes will come and you will never be able to keep going with the infinite series of switch-flicking as it goes onto infinity and never reaches two minutes. Consequently, we can never know.
You dont seem to know high school maths.
any number divided by infinity is 0 . 2-1/n=2-0.Also 0.999... = 1.
+lich erdor do you know high school maths? Infinity is a concept, not a number that can be used in a sum to produce rational logical results.
If you want to be correct about it:
for all n element of R:
Lim(a -> +inf) n/a = 0
With words:
For every rational number n out there; if you divide n by a with a going to infinity, the result will be 0. I think this is what he meant.
Correction : real not rational
You're basically asking if infinity is even or odd. I see no problem leaving this undefined.
There would be one other solution if we rearrange the terms of sum:
-1+1-1+1-1+1-1...= -1+(1-1)+(1-1)+(1-1)= -1+0+0+...= -1
i think of it this way:
because there is no last step, the + and - keep on going, i think that the number is being added and subtracted at the same time. So: 1-1=0; 0+1=1 and that keeps alternating and there is an equal amount of + and - in the equasion, so the answer is somwhere in the middle, or between (1-1) and (0+1), a half!
I'll do another one:
3+1-2+2-2+2-2+2...
3+1=4; 4-2=2; 2+2=4
between 4 and 2 is 3
so 3+1-2+2-2+2-2+2...=3
Okay.
1+1-1+1... does not have a sum and neither is it an infinite series because the mathematical operations performed are being nullified the very next step.
BUT! There exists a kind of series of this format:
x^0 + x^1 + x^2 + x^3 . . .and so on infinitely. If your "x" is 1 or anything more than 1, it diverges off to infinity (duh). If your x is 0, it causes the operation 0^0 which is not defined. If your x is -1, it produces this series in question (1+1-1+1-1+1-1...) thats becuz every even power of -1 is 1 and every odd power of -1 is -1. And anything less than -1 likewise diverges.
For values greater than -1 and less than 1, this sequence converges to a finite value. The value is equal to 1/(1-x). An example would be 0.5^0 + 0.5^1 + 0.5^2 and so on which produced the famous series: 1 plus a half plus a quarter plus an eighth and so on. We know it equals 2. Applying the formula gives 2 as well: 1/(1-.5) = 2.
Substituting x with 1 doesnt work as it causes division by 0.
We know that when x = (-1) this series diverges so it has no answer but What ramanujan & co did is substitute x = (-1) anyway and they get 1/(1- (-1)) which of course equals a half.
People would say well, how can they do it when it doesn't work for -1?
Okay. It may not work for -1 but it should work for values *very* close to -1. A tiny fraction more than -1. Like, say, -0.999999. In this case the infinite series where x = -0.999999 would be equal to 1/1-(-0.999999) which is 1/1.99999 which is crazily close to 0.5
So the closer and closer x gets to -1, the closer and closer the sum of all of its powers gets to 0.5.
For instance if your x was, say, -1 + (10^(-100)), then your x would look super similar to -1 and the infinite sum of all its powers would look super similar to 0.5. They would just converge very very very slowly.
0^0 is 1
@@streak9010 no, mathematically it is undefined. Look it up.
This is güüd
huh, so you put a lim inside a lim
@@G0dlol so lim[ε -> 0+] (lim [n -> ∞] (Σ[0, n] (-1 + ε)^n)) = 1/2. Yeah, I'm convinced.
"Will the light be on or off?" The light will be off. The lamp turned into a quantum particle with undetermined state and therefore the light is off. Got a problem?
if the state is undetermined then its not off
Noah haoN it is both simultaneously. Therefore it is *also* off.
U know water is getting liquid by some temperature and ... u dont know
***** Dude. That's the joke.
I find myself watching these videos and I understand about 20% of them, but I hope that some day someting stuck in my brain and I will use it for a pubquiz or something.
*James:* ”After 2 minutes (and being turned on and off infinitely many times), is the light on or off?”
*Me:* ”It’s broken 🙃.”
Since it's a natural number sequence imagine it's a wave function oscillating between 1 and -1.
Start the sequence from -1 and answer is -1/2. So light on or off depends on where you start measuring in this infinite sequence. Similar to wave function collapse with Heisenberg.
Is the probability to finish in 0 or 1
What do u mean by wave function
Wouldn't it be a perfect square wave if it was a wave function? The average would be zero.. I think I need more proof than the averages of partial sums to understand that it's 1/2.
Square wave
I'd rather watch these then sit through fucking math class...
U
Engineers answer:
Fix the electrical problem in the light circuit so it stops oscillating. Done ✔
I was thinking it would be better to switch it onto a 60VAC service from the 120VAC service after about 1:59.5 so you don't wear out the switch and there is no need to keep turning it on and off.
Increase the speed of the oscillation so no one can tell
All light bulbs has some oscillation frequency, and there's nothing you can do to stop it.
Actually the Engineers Answer was to build an electronic dimmer, which can make your light half as bright by quickly switching it on and off, and sell that to you. :-)
(As an engineer I actually expected that to be the conclusion to the "light" experiment.)
The error is in the question, as it's not a series converging to a limit but something better expressed as a function, so there is a set of sums depending on number of elements in the series. That is:
G(1) = 1, G(2) = 0, G(n) = 1 - G(n+1) = 1 - G(n-1) and abs(G(n) - G(n +/- 1)) = 1 ie the difference is always 1 no matter that n nears infinity, therefore the question is fallacious, the sum depends directly on n, where n is the number of elements written.
G(n) = 1 - (n+1) modulo 2 = n modulo 2
IOW when you write odd number of terms in the sequence, it's sum is 1 and 0 for even. The games with brackets then fall out as they simply reduce n:
G(n) = 1 - G(n-1) or 1 - G(n-1) = G(n)
-"after 3 minutes, is the light on or off?"
-Yes.
Actually there was no specification for the state at or past the limit point.
can i just say this is the 1st comment section with almost exclusively rational and informed debate
congrats every one
+flawlessgenius You ruined it. Congratulations
flawlessgenius no you can't actually
lucromel that's why his had his meeting with everyone else in the gas chamber, what a lovely person he was
no you can't
Well that light is like Schrodringer's cat, it's on and off at the same time :D
Schrodinger's lamp
Scribbity scrooby
Quantum theory and quantum mechanics ftw
Scribbity scrooby superpositions!!!! the true random
but if there are three possible values, wouldn't Schrodinger's light be on, off, & half on/off? All three values at the same time?
Yeah, that's what I thought, it's in a superposition and when we look it will be on 50% times. Maybe that's the solution.
Ever considered to define the sum to be undefined, like trying to divide by zero we can get various results therefore it has been define as undefined.