One minus one plus one minus one - Numberphile
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- Опубликовано: 24 июн 2013
- Discussing the brain-bending Grandi's Series and Thomson's Lamp - featuring Dr James Grime.
More links & stuff in full description below ↓↓↓
A little bit of extra footage from the very end of this interview at: • Grandi's Series (littl... (on Brady's own channel)
See our Zeno's Paradox video at • Zeno's Paradox - Numbe...
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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.
"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
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...
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*
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
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 love how much this dude genuinely loves math; he couldn’t hide it if he tried
yeah
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
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.
_"And I'm hoping it will cause a little bit of debate on the comments, 'cause I know RUclips 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 RUclips remind of people who try to divide a number by zero.
+AluhutSmasherTV . tell me about it .
“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
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 🎯!
I love this guys nerdy excitement in the beginning of the video!
"RUclips 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.
It's onf.
xD
NO, it's ofn
Acutally, it's onf only so ofn.
like the anti-ice system
No, it's nof!
Love his confidence as he writes with a sharpie 🌞
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
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
"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
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
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'
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.
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 😀
It’s funny how he’s doing calculus while desperately trying to avoid saying calculus terms cuz then everyone would go “aaaah calculus”
I love the incorporation of supertasks in this (the lamp problem, an infinite number of calculations within a finite amount of time)
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
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
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
Is it one or zero?
Yes.
Superpositioning
Zero-one duality.
a>-1
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
*James:* ”After 2 minutes (and being turned on and off infinitely many times), is the light on or off?”
*Me:* ”It’s broken 🙃.”
this video is great because it makes me think about how there is possibly an infinite amount of infinites between everything and yet we are able to break past that infinity if we just look past it.
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.
An infinite amount of mathematicians walk into a bar. The first orders 1 beer. The second orders 1/2 of a beer. The next orders a 1/4 of a beer. This goes on forever. The bartender then pours 2 beers.
He then says "You mathematicians just don't know your limits..."
MediNIeN you just won the internet for today
ugh dugh What is the answer. Its the sum of the harmonic serie I know - but there was no options.
MediNIeN Or he would have died before the mathematicians had finished their order.
edit: of old age*
nine87able It's not the sum of the harmonic series. The harmonic series is 1 + 1/2 + 1/3 + 1/4 +..., which doesn't have a finite value. It's the sum of an infinite geometric series.
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
My answer to the "what do you think at the end" - the light will be off, because the switch will have broken from the wear and tear of being flipped infinitely many times in two minutes
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
RUclips 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
If I imagine a superhero movie where a mathematician is a villain, this guy should undoubtedly be the villain!
Gold rat man?
Three mathematicians walk into a bar and the bartender says "do you all want a beer?". The first mathematician says "I don't know", the second one says "I don't know", and the third one says "yes".
One of my favourite math jokes.
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!
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
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
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.
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'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?
Theorem of Bounded Periodic Divergent Series of Integers:
The sum of a bounded and periodic divergent series of integers is equal to the average of its partial sums over any period.
Example: Sum of Grandi's series (which is bounded and periodic of period 2): (1+0)/2 = 1/2
His enthusiasm won my heart.
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...
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
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. ❤️
Hello, I would like to know if mathematicians have explored the abstractums of crochet, knitting or other handcrafts? I find crocheting being such a meditative medium for how a yarn can be formed into intricate patterns in lace for example.
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.
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.
Love this man so much
"He was a monk, a mathematician.. one of THESE types" lol
I'd rather watch these then sit through fucking math class...
U
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.
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.
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
The answer can't be 1/2. You only have 1s and 0s. regardless of how you work the problem.
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...
I love how excited he is for these maths lessons. I was always drooling over on my desk when the class started😂
10:45 so does that mean, it is 1/2 because it‘s neither on or off? Because we would be pressing the light in such a speed that we cannot consider it on and off?
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).
"Cause I know RUclips is the home of rational and informed debate..."
Lol'd.
The light is on quantum superposicion!
At infinity sets, order does matter. It's not commutative. Meaning a + b might not be the same if you switch it to b + a. Then the associative and the distributive properties will follow.
I understood everything in this video perfectly, until you said '1'.
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!
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.
The half would be the state at the end which would be half brightness as it would function like pwm does
3:38 Things get serious in the nineteenth century...
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?
The light switch would be so quick from on to off/off to on - that the time it takes the photons to spread in the room is exactly two times slower until the next switch happens, so basically the light bulb gets enough time to release only half of the photons by the time it switches again(same for the opposite)
I wonder if someone else wrote this
If you did the Thomson's Lamp experiment, you would break the light bulb; So it would make sense that the answer is 1/2, you just broke the lights.
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
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EPIC
EPIC FOR THE WIN
I REALLY LOVE YOUR PORTOLE REFERENCE
I AM SUCH A GAMER
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nop e calm...
I know answer.
If its 1/2= light bulb is off, but everything is on fire because of fuses :)
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.
that example wouldnt actually be fair cuz it takes a minimum amount of time to "turn" the light either on or off
so to that be the case, it should be an instant to turn the light on/off so that it couldnt be devided by 2 in any way
but, considering the turn is instant, how about this question: "how much time would the light be up and how much would it be off?"
Until 1:32 it is a subtle mistake but he starts 1 on in the numberline of the sum eg 1-1+2-2+3-3+4-4+5-5=0
(1-1)+(2-2)+(3-3)+(4-4)+(5-5)=0 in those instances each new number introduced is counteracted by its predecessor but when bracket moved 1 on that no longer happens and there will remain those values
1(-1+1)+(-1+1)+(-1+1)+(-1+1)=1
Wich isnt changed by the actual brackets but by the amount of numbers in the sequence
thats not the mistake, the mistake is that he assumed that the series in convergent while it is infact divergent therefore there is no infinite sum.
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.
I have a sneaking suspicion that one day the maths that are required to enable us to travel faster than the speed of light will be based on a collection of Numberphile videos.
except that Mr . the-man-who-took-his-picture-with-his-tongue-out or Einstein,,, neglects this
@@santoshmishra121 who knows maybe in a decade or 2 someone would neglects mr.Took-a-picture-with-tounge-out-man theory
@@rafigoghimarfirman3480 by violating CPT symmetry
@@ammyvl1 and replace it with another one :p
wait is that actually possible?
Numberphile - for Ramanujan's birthday please do a video with all your usual folks and ask them one thing they would ask him if he were alive today! Would be very interesting to see. And it's tragic he died at 33.
I can think of something else that oscillates between states and exists as an average (until the system is resolved by observation): quantum superposition. If you graphic the Grandi Series in 2-D with one axis as progress (time or steps, whatever), and interpolate the plotted dots, you get a zigzag sort of triangle wave. Now take the average (or median, if you prefer the statistical route) of that and you get a line that starts at 1 and tends quickly towards 1/2.
Try any other balanced offset and find the same answer. eg, instead of 1-S, something ½-S or ⅝-S, etc, and get the same outcome. (Note that an offset using positive addition only results in identical terms on either side and is not useful.)
What's interesting is if you consider the Grandi Series as the (2-D) real part of a complex mapping that's missing the imaginary part, and what that would graph to.
-"after 3 minutes, is the light on or off?"
-Yes.
Actually there was no specification for the state at or past the limit point.
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
how would bedmas affect the answer? if you did addition first, would the number be negative?
I’m fairly sure the light would be broken
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
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?
The flaw is in subtracting S from 1 at one end of the equation, while you cannot do the same to the other side of the equation because it has infinity in it. I think the guy who invented this, while I have a lot of respect to him, he fooled many mathematicians.
The Lamp problem is usually misapplied to Grandi's Series...the question of Thompson's Lamp is not "what is the sum of the entire series", the question is "what is the last term of the series". It doesn't make any physical sense, even for a finite number of terms, to say you are "adding together" all of the physical switch flips. It only makes sense to average them (you are determining what the switch is doing on average) or what the last switch is (you are asking for the immediate state of the switch at a specific point in time).
Because you are interpreting the question "what is the last term in an infinite series" as a physical analogy, it has to be answered with physics and we say that there are hard limits on how quickly and how many times the switch can be flipped due to the quantization of spacetime and energy/matter.
Values of divergent sums like this have lots of applications in physical reality, but the question "what is the last thing in an infinite list" does not.
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
I say ask Schrodinger's cat.
+FOLLOWTHEROAD Came to the comments for this.
and what about schroedinger's mice who've got no idea whether they've been caught or not?
Ray Kent I think they also might know
+FOLLOWTHEROAD It died long time ago even before somebody opened the box
Maybe it is all uncertain to me.
I freakin' love getting math lessons from Brick Top.
When you get three different answers to the same "equation", maybe it's not an equation, and "equals" is the wrong sign. ;) Fun video, thank you!
I think the answer is that the light burned out because you flipped the switch too many times. J/K
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.
It means I was drinking water from a glass and vomiting it back in the glass and do this forever, suddenly u entered and asked me is glass empty? "Well I m just drinking so it is half filled half empty".
I think that the lights are flicking so fast that you can not choose a single moment where the light is on or off. In one moment, the lights are both on and off, leading to them appearing on only halfway. After all, the flickering would be infinitely faster than the speed of light.