Reminds me of when I have too much milk in my cereal so I add cereal, then I have too much cereal so I add milk. The amount of cereal and milk I use has no limit, but the mix gets closer and closer to what I want. Sadly I got arrested for trying to explain to the cashier that purchasing an infinite amount of milk and cereal means the store owes me 1/12 of dollar.
@@jeffersonpaesdefariasfilho well the infinity symbol is a bit deceptive here. While they are all the same, when you add anything to infinity, it's still infinity but just a little more infinity. So the infinities in your equation aren't necessarily the same, but are "different" by the added amount
Kudos on the (sort of) correct use of 'yhe'. A real thorn in the side of linguists... My intuition is that it isn't surprising that an equation with the term 'infinity' in it can produce whatever result we want it to, because 'infinity' has no mathematical significance to begin with. (I hope this 'hard-core finitist' position isn't too shocking without a prior warning. Or even an abbot warning)...
@Supreme ideology Basically a sequence converges to some number if it can get as close to a number as it wants but also stays that close and doesn't frequently stray. So you can always take a rearrangement of the sum and make it stay super close to some value by shoving the later terms down the line.
Yeah it makes you realize you should not expect associativity to work. Associativity is captured by what you put in the summation symbol which is why "..." notation is ambiguous.
@@TheBeakertubethe ... notation is always short for a limit of some sort. So 0.9999... is just shorthand for sum n=[0,inf] 9/10^n 1+1/2+1/4... is just shorthand for sum n=[0,inf] 1/2^n The ... notation is just always easier to write down, but it depends on the intuition of the reader to interpret what comes next. You can come up with any number of sequences that start the same way as a given finite sequence, so it's not a good formal way of writing down a limit.
anthony showalter because infinity isn't included in any proper number set. It is all numbers. 0/0 is undefined because you can't divide by zero. Stupid question.
anthony showalter actually it was stupid. The reason 0/0 doesn't have an answer is because it's not a real problem you can have. We as a people are not able to define what the answer would be. The answer to any question could be (n) if you define n correctly. Also why do you type like that? It's still a dumb question and you're not proving anything. Also, you are able to define infinity and negative infinity as different "numbers", so the whole n shit you're going on about is just weird.
That's because a limit tending toward infinity never actually brings in the concept of infinity or uses it as an expression or anything. Instead, you just say that no matter how big a number someone can mention, you can always find a delta such that so long as you're closer than it on your input, your output will always be a bigger number.
Well done. That was the simplest and easiest to follow explanation I've yet heard. Riemann and Ramanujan are more worthy than most to be called "rock stars". It's criminal that Ramanujan passed away at such a young age.
The takeaway is that if you ignore the definition of “infinity” and just rearrange terms as you wish you can always get the result you want. This feels morally questionable
Another way to look at it which maintains our current understanding of infinity is this Inf + x = inf Therefore X = inf - inf You can use the same technique of rearrangement to change the sum to any real number (switch the sets for negatives)
Paradoxes are just misunderstandings. The solution for this one is something we all already know: that infinity is not a number, though we continually make the mistake of thinking of the concept as that of a really big number. "Infinite" means without end. So the clearest meaning of "infinity minus infinity" is "to add to zero forever and subtract from zero forever." It says nothing about the rate of addition or subtraction nor about what quantities are to be added/subtracted in what order. As long as both the addition and the subtraction keep going on forever. You see, a time element has been introduced. But the phrase "infinity minus infinity" says nothing about that time except that both the addition and the subtraction must go on forever. It does not say when to begin. Thus, if you start say, adding first, and add till you get to Pi, and then start adding and subtracting in equal measure, forever, you'll stay at Pi forever. If you alternate adding and subtracting but in smaller and smaller increments, as in this video demonstration, you'll hover around Pi and converge upon it. And you can do this of course, with any number. When looked at this way, as in terms of "infinite" correctly signifying 'without end,' rather that *in*correctly signifying 'a really big number,' the Paradox goes away. Suddenly it is all quite clear and obvious, easy to visualize, and containing no contradiction. What confuses people, ironically is that "infinity" is in the linguistic form of a noun, rather than an adverb. Correct this, in your mind, by always correcting the term "infinity" to "infinitely." That is the trick to solving, most likely, all "infinity" paradoxes. Think of it this way, if you want more on that: You can go running, or, you can go for "a run." If you run for 1 mile in a direction and then for 1/2 mile in the other direction, you don't end up where you started, and this is no mystery, obviously, because you didn't "run" the same distance in both directions. But if we use the noun form, and say you went for "a run" of 1 mile in one direction and then "a run" of 1/2 mile in the other direction it opens a possibility for confusion. Because whatever distance we suppose "a run" signifies, we have shown that "a run" minus "a run" does not equal zero. We have *assumed* that "a run" designates a fixed distance, when it does not. Likewise, "Infinity," a noun, implies to our minds a fixed length, or distance, or quantity. A certain number. But it is not that. Whether we can imagine it or not, we suppose that 'that number' that is infinity minus that number aught to equal zero. We are, thus, astonished when it does not. But hopefully we can see now, that, that is because we are falsely imagining "infinity" to be a number, a fixed length, when it is not. It simply signifies the quality of being without end.
Have you ever written a book or read one that could speak about maths the same way you do? I'd love to read something related to this sort of abstract perspective on this field.
Yes, It opened the mind. Infinity is a numeric set that brings together all numbers in its individual being like this: ...-3, -2, -1, 0, 1, 2, 3... Infinity is not this: ...(-3)+(-2)+(-1)±0+1+2+3... That's a amount of individual numbers with no relation of addiction between themselves, constructing a really so big number with a infinite number of digits.
EXACTLY. The whole premise of this video was a bait and switch. Mathematicians treating infinity as a number, rather than an unending sequence: guess I won't be subscribing here...
The letter P is pronounced as the "Pai" in "Pain", while the mathematical character "π" is pronounced as how the English pronounce the letter "P", while in English they pronounce the "π" as "Pie
You can ask him to solve the whole test if instead of asking "answer to question 1, answer to question 2 ...." You interpret the whole test as a problem and ask him "the n-tuple wich solves the test"
Recently a few people have been contributing translations and subtitles to various videos. If you are thinking of doing the same could you please let me know. (RUclips does not notify me of any subtitles waiting for my approval) Also I'd like to acknowledge any contribution like this in the description of the videos :) (added 17 June) Thank you very much Zacháry Dorris for contributing English subtitles for this video and Rodrigo Naranjo for contributing Spanish subtitles!
That's great. This whole channel is a one-man-show and adding any subtitles just takes up too much of my time. Any help in this respect is very much appreciated :)
There is definitely a bit of a connection here. If you want to find out about the details check out the "Steinitz's theorem" on this page en.wikipedia.org/wiki/Riemann_series_theorem#Steinitz.27s_theorem
Thank you very much offering. I actually speak Russian reasonably well. Would be great to have Russian subtitles :) Not sure whether you've ever tried to put together any subtitles. I have and it really does take forever, so don't feel bad if it all gets a bit too much :)
When we are discussing the Riemann R. Theorem, it is essential to point out that this is about divergent or conditionally convergent series. NOT on absolutely convergent series. Absolutely convergent series (the kind we usually deal with everyday) we can rearrange to our hearts content. Just don't forget that! (Otherwise, excellent video. Keep up the fantastic work!)
For what we are interested here the technical terms "conditionally convergent" and "absolutely convergent" don't add any extra value to what is captured much better by focussing straightaway on what the positive and negative terms add up to :)
It is important to mention that this is a property of conditionally convergent series, but that it is NOT a property of absolutely convergent series. For instance, if we write pi as a series of fractions of powers of 10, we can't rearrange the fractions to get anything other than pi.
Well, I do talk about absolutely convergent series at the end (those for which both the positive terms and the negative terms add to finite numbers) and I do say that you can reshuffle them any way you like without changing the sum. Maybe have another look :)
Soo... I was watching anime, then accidentally clicked on RUclips. And now it's 1.52am and am still here. On this time, am learning math. All hail The Algorithm, first of his name, and replacer of what used to be the old king - The Dark Side of RUclips. Haven't seen that guy since like 2016...
What people often don't realize: Infinity is not necessarily the same as infinity. Thus, infinity minus infinity can be any real number including infinity. Similarly, infinity times zero can also be any real number including infinity.
@@bosongod2830 I think you did not get the point: Infinity is not a single determined number. Therefore, arithmetics involving infinity often does not turn out the way one is acustomed to. Therefore, a rule like "x * 0 = 0 for all x" does not hold good when involving infinity. Example 1: x/x^2 with x -> infinity leads to 0. This equation is the same as "infinity * 0 = 0" when you apply the limits to the numerator and denominator seperatly. However: Example 2: 8 * x/x with x -> infinity leads to 8. This equation comes down to "infinity * 0 = 8" when you apply the limits to the numerator and denominator seperatly. Example 3: x^2/x with x -> infinity leads to infinity and therefore "infinity * 0 = infinity".
@@bosongod2830 consider x and 1/x. As x→∞, 1/x→0 So x•(1/x)→∞•0 But x•(1/x)=1 So ∞•0=1 in this case. More generally, use x and a/x to let ∞•0=a for any a.
The alternating harmonic series is not absolutely convergent so the Riemann Rearrangement Theorem shows that it can be rearranged to converge to any real number you want.
This video takes me back to my Cal II class. Infinity series are so much fun and interesting! I remember the day my professor blew my mind when I first heard of this concept. This made me want to go back and brush up on infinity series. Loved the video
Hi there. I am not a mathematician but only have a PhD in computational physics. And I usually have a very pragmatic view on things. That’s probably why I spotted a HUGE difference between generating ln(2) or Pi with that series: To generate Pi, you need to know Pi already BEFOREHAND (to decide when to switch back and forth between negative and positive summands). However, the algorithm for the summation sequence to arrive at ln(2) (or ln(2)/2 for that matter) does not need any info on the value of ln(2). You could actually FIND the value of ln(2) with that series, even if you never knew it before. The INFORMATION about ln(2) IS CONTAINED in the series. In contrast to that, there is NO information about Pi in that series (since you have to explicitly put it into the summation sequence to construct it). So, the real question mathematician should ask about such series is not ‘What does it converge to?’ but: WHAT INFORMATION DOES IT CONTAIN? And then you might get sensible (maybe even unique?) answers: In the above example that answer is ln(2). Even if you re-arrange the summation (with an algorithm not using info about the end result already in the construction) you get things like ln(2)/2. But that’s not new information since, if you know ln(2) you also know half of it (or any value which differs from ln(2) only by a rational number - be it a factor or a summand). Well… one could even go further and say: No sir, you can NOT construct Pi with that series but only an approximation of Pi. Since you need the end-result in the construction of your summation-order, you would need knowledge of Pi to the last digit to really construct Pi EXACTLY. And well… we all know, nobody will ever have that knowledge since there is no last digit. The same holds for any irrational number, which you would need to know already beforehand to construct the summation sequence. That means, contrary to common believe of the last couple of centuries: You can NOT construct any number with such series. You can construct only rational numbers (since you know them exactly to the last digit should you need them in the summation-sequencing-algorithm) and those irrational numbers (maybe only one per series?!?) whose information is actually CONTAINED in the series (ln(2) in the above example). PS: You can also contact me in my native language, which is the same as yours :-)
All good points, but here are a few things for you to consider. First, have a look at the section "Getting an arbitrary sum" on the wiki page dedicated to the Riemann rearrangement theorem en.wikipedia.org/wiki/Riemann_series_theorem Second, for people like myself to distinguish between a number and its various representations is very important. So, there is pi, the ratio of the circumference of a circle and its diameter and there is the representation of this number as a decimal number. What I do know from the theory of real numbers is that there is a unique way to express pi as a decimal number. What I don't know are the individual digits in the sense that there does not seem to be any "simple" pattern to the digits. On the other hand, I can write down an algorithms that will generate any of those digits and in a sense this algorithm is really just another type of pattern. Same thing with what I do in this video. What I show is the existence of a rearrangement of the series that will add up to pi as well as an explicit algorithm that will get you this rearrangement. In terms of practicalities truncated representations are usually all we need anyway and in terms of exact sums existence is fine in the first instance. Of course, simple patterns are great and you also get them in abundance in terms of infinite series summing to whatever we like from things like Mclauring series :)
The Riemann rearrangement theorem is true, no doubt. But the theorem says: For any real number, there EXISTS a permutation such that the series equals that number. I would never deny that. What I say is: Yes, that permutation exists, but you can never actually CONSTRUCT it. I have chosen my wording carefully, you see ;-) Contrary to ln(2), you cannot describe a permutation such that the alternating harmonic series approaches Pi to ARBITRARY precision. Why? Because you don’t know Pi to arbitrary precision and you EXPLICITLY need that knowledge to construct the permutation. Let’s say you have approached Pi with that sum to the best precision known to mankind. Then you cannot go further since you don’t know how many positive terms you have to add next before you switch to negative terms again (or vice versa). Sure, it CAN be done, since the Riemann rearrangement theorem tells you such a permutation exists. But you cannot actually DO it until you obtain knowledge about a more precise Pi FROM SOMEWHERE ELSE (i.e. from running the algorithm for Pi you mentioned a little bit longer). The information cannot be extracted from the series ITSELF. You have to input it from the outside. It’s a bit like representing Pi (the real thing I mean, i.e. the ratio of the circumference of a circle and its diameter) as a decimal (or binary or hexadecimal or whatever) number: It can be done to arbitrary precision, but that doesn’t mean the information about Pi is already contained IN THE NUMBER SYSTEM ITSELF. It’s not. You have to know Pi to the desired precision FROM SOMEWHERE ELSE (the Pi algorithm) before you can write it as a decimal number, i.e. before you know the PERMUTATIONS of the digits to construct it with the number system. This is exactly the same as with the alternating harmonic series: You have to know Pi to the desired precision from somewhere else before you now the permutations of the summands to construct it. It’s even the same as this: the English language is enough to represent Shakespeare’s Hamlet. But that doesn’t mean, the information about Hamlet is already contained in the language ITSELF. Shakespeare had to input that information from somewhere else (from his fantasy happening in his brain) to know which permutation of letters to use to construct Hamlet. See the difference to ln(2)? The algorithm for the permutation contains no reference whatsoever to the end-result that permutation leads to. It just leads there “automatically”. The permutations to use are all clear right from the start, no matter how far out you want to go. Therefore, you can construct ln(2) up to arbitrary precision. You do NOT need to know the end-result beforehand to construct the needed permutations. The information about ln(2) is contained IN THE SERIES ITSELF. Information about Pi is not. That’s my whole point here: For these kind of infinite series it is not so useful to ask “Where does the series converge to?” but it makes much more sense to ask: “What information is contained in the series ITSELF?” If one could define - with mathematical rigor - a new kind of limit as the answer to this question, we might be getting somewhere.
If there is no limit or precision applied to Pi in this case, would that make the answer of inf mins inf sum up to infinity? (im a layman btw, just nerding out on the internet).
@@HansPeterDeutsch_hpd this made so much sense to me. Most of the math seems like a feedback loop, in which we input incomplete info and get incomplete output and we keep doing it...
Yes, there is definitely a difference in the algorithm's "complexity" resp. the "information contained" between "take every second term" and "know the desired end result, then wait until the sum has passed the threshold". My intuition would like very much to make that a clear-cut difference, but the terms of computability theory or even complexity theory are not "fine-grained" enough to capture it, I'm afraid. And I don't see any other options immediately... Loosely related fun fact: The set of algorithms to construct a permutation re-ordering the sequence is countable of course (as the set of all algorithms is), while the set of real numbers of course is not. Nevertheless you can construct a permutation of the sequence converging to each given real number. The magic lies in *given* I suppose, meaning that you need an algorithm to construct the number before you can construct your permutation. Guess that is the difference you and I were looking for after all: For ln(2) the algorithm is contained/trivial, for pi you have to put it in externally and for uncountably many others there is none.
Its not just that I think. The whole method of adding and rearranging sequences like that just doesn't seem to prove anything. You can do something similar to this and prove that 1+1=0 which is obviously wrong.
Thank you for sharing this. It reminds me of quantum mechanics where you can have all outcomes or no outcome, or just a single outcome, or anywhere in-between. I think this a mathematical framework that needs a lot more study as it relates to probability.
Infinity-Infinity is an indeterminate form and we use limits to remove this form!!. So we can't determine the exact value of these indeterminate forms like.. 0/0,infinity - infinity,etc
Maths with infinity series like the ones in this talk are actually hypnotism at play 😊. Think of it, the bracketing process makes you to believe it is legit, but then u get drowned under the sleight of MIND! The obvious secret to break the conclusion is to call the contradiction & detect which part is false. That requires talent of course, as much as the math mesmerizer!
Perhaps "crank" is too great of a term to say. He is a published mathematician and certainly isn't a bad teacher. But many of attitudes are very crankish. People like Wildberger want only the math that "resembles" reality, while somehow not grasping the fact that a positive integer itself is an abstract concept that doesn't exist in reality either. To take existence proofs, distances (which he abandons for quadratures), infinite sets, and etc. from mathematicians GREATLY cripples the subject. If we then continue this line of destruction farther than even Wildberger himself and proceed to remove everything from mathematics that doesn't exist in the real world then there would be no math left from the slaughter.
Yeah, "crank" is too strong because he's never put forth any crank results, at least none that I've seen. You say his attitudes are crankish, but it isn't self-evident to me that being a constructivist or a finitist is a crank attitude, especially seeing as how the axiom of infinity is an axiom, not a theorem of set theory. I don't agree with Wildberger on most issues and I have no problem with real numbers as he does, I just don't think he is an example of a crank. Nice examples of math cranks include those denying Cantor's diagonal argument or that 1/3 is not 0.333... (I've actually debated a crank on the latter).
Mathologer, I have an argument at my end. I'm saying infinity IS used in mathematics and my friend says it ISN'T, that it has no place in maths. Of course we both understand that infinity isn't a number and all the rest, and I know about square root of -1 (is that imaginary number i? Which is really useful in maths and just as silly as infinity? Isn't zero itself in a similar category?) too. Riemann's paradox seems to be just one of many examples in which infinity is used (in the practical sense?) in mathematics. Please do a video explaining how the layman's concept of infinity differs from or is the same as the mathematician's!
@Tiffany Wart Probably no. I have from another maths/physics friend an opening statement like 'Infinity is ubiquitous in calculus and quantum' but in the end he agreed with something like 'Infinity is not used in equations that have practical application.' Then we talked about zero and square root of -1 and for me at least the matter is not well resolved.
I isn't actually silly, it's an actual number that can also be written as 1;90° as it's still a number just in a different direction. Welch labs has a great video on it.
Order / position matters in an infinite series. The first issue arises in the Pi example as well. You can go to town with brackets and breaking up terms as long as they're calculated at the same point in time as their parent term.
Let's stop for a moment and apreciate the monumental amounts of patience and kindness that the author of video has for those arrogant morons from comment section, who think they know better about mathematics than someone who spend most of his lifetime learning and teaching about it.
If I may, it is not faith more than it is common sense. Arguments of authority are only invalid when the said authority isn't specialized in the subject. I would trust a mathematician about math more than I would trust someone who doesn't have evidence backing what he's saying in the field of mathematics. You may argue that mathematical theorems are independent of people. I agree but the great thing with math is that when you use intermediate results to solve an exercice, you don't have to always look at the demonstration, because you know that people who are better than you have already done it. In that sense, mathematical results are simply tools you take from your toolbox. I don't know how hammers are made but I do know how to use one. This way of thinking is far more productive than reviewing every math theorem. I can guarantee that not a single mathematician has done that because their job is to build (discover ?) new results. It is a possibility that a mathematician has gone wrong when demonstrating something and nobody has spotted the mistake, but I'd recommend you read about Ockham's razor.
I'd love to see a follow up video on the generalizations you mentioned at the end. From Wikipedia: "given a converging series of vectors in a finite-dimensional real vector space E, the set of sums of converging rearranged series is an affine subspace of E" Super interesting.
Challenge accepted at 6:49. Your series, - 1/2-1/4-1/6-... is just - 1/2 times the harmonic series, so - 1/2-1/4-1/6-... will also diverge. Then your other series 1+1/3+1/5+..., we can use the direct comparison test. If the harmonic series diverges, then 1/2 times the harmonic series will also diverge, that is, 1/2+1/4+1/6+... will diverge. If you compare 1+1/3+1/5+... to 1/2+1/4+1/6+... term-by-term, the series 1+1/3+1/5+... will be larger than a series already known to diverge. Perhaps there's an even simpler argument?
I didn't use a comparison for the series with the negative terms, just an observation that it's -1/2 times the harmonic series. I don't see why limit comparison is easier compared to direct comparison here. Certainly it gets the job done, but it's more machinery (limits) than needed.
Of course I can use observations. Here's the argument in more detail, starting with - 1/2-1/4-1/6-... Hopefully you see that if I take 1+1/2+1/3+1/4+..., multiply by - 1/2, I get - 1/2-1/4-1/6-... We already know that the harmonic series diverges and I claim that multiplying a divergent series by - 1/2 produces another divergent series. I'll prove that statement by contradiction. Suppose - 1/2-1/4-1/6-... converges to some finite number, call it A. Multiplying - 1/2-1/4-1/6-... by -2 implies that the harmonic series converges to -2A, a contradiction. In the case of the series, 1+1/3+1/5+..., I started out by claiming that taking 1/2 of the harmonic series produces another divergent series 1/2+1/4+1/6+... (I can prove that by a similar proof by contradiction, as above). Now look term-by-term: 1>1/2, 1/3>1/4, 1/5>1/6, etc. The series 1+1/3+1/5+... diverges by direct comparison with the divergent series 1/2+1/4+1/6+... The integral test works but I think it's like going hunting with a bazooka.
There's an intuition that since the harmonic series diverges, multiplying it by some nonzero number can only produce another divergent series. That is, multiplying infinity by a nonzero constant produces another infinite number. That's why I figured that recognizing - 1/2-1/4-1/6-... is - 1/2 times the harmonic series suffices. The same concept can be used to justify the divergence of 1/2+1/4+1/6+... which I then compared to the second series in question, 1+1/3+1/5+... As I said, one could use other methods like limit comparison or the integral test but one has to have prior experience with calculating limits and integrals for those to be intuitive.
+Bryan Leist You merely state my methodology is not valid and that the argument that I presented above does not hold. Care to tell us why? Also care to tell us why I cannot manipulate infinity (even though I don't need to manipulate any infinities in my proof)? I know how to sum an infinite series, don't you worry.
The issue is clear, the problem is that the series isn't absolute convergent, so the rearrangement trick won't work. You can ONLY do that if the series is absolutely convergent.
If the series is infinite, how do you just decide to use 400 of one and 2 of the other? That's not rearranging, you've removed things. I guess you could just subtract the remaining numbers from each one, but now your formula would be ∞-∞-∞-∞=π
There's an infinite number of terms, so ALL of the terms are being used, just... eventually. A lot of positive terms are used between each negative term, but there's INFINITELY many terms. So, all positive numbers AND all negative numbers will get used. The negative numbers just, take longer.
You're right Aaron; to mich on a side, so less in the other! If the quantum of plus is the same of the minus quatum from zero to infinity. Solid logic...
As a non maths guy, this sounds like essentially he's demonstrated you can't reliably apply the commutative property to infinite series. It only works with a subset of all infinite series. Am I understanding this right or have I completely missed the boat here?
You are absolutely correct. The "infinity - infinity" way of posing the question is just a way to make the concept more approachable and show the viewer how math can sometimes be counter-intuitive.
I really appreciate the fact that you point out the errors. Lots of people doing math on RUclips using all the properties in all kinds of algebraic structures and make others think 1=2...
the major problem i see is that "infinity" is not a number, but a concept.. and there are plenty of forms of infinity. there is an infinity of whole numbers, an infinify of fractions betweem two single whole numbers... an infinity of numbers that end in 4... and so on... so yeah...
ndgambella nope. infinity, mathematically speaking, is a series of numbers... one that never ends... infinity is not "a number" anymore than "A" is a number... (that is not to say that A and infinity are the same thing either, just that they are not numbers)
That second one is incorrect. The answer to "subtract all the numbers" is -∞, not ∞. Although I think I take your point. ∞ is the set of all positive numbers. A thing cannot be a number if it is a category to which all numbers belong. It is not a number, but it is its own thing. Hrm.
Well, maybe, yes? I dont think so. Numbers are information, math is something that is working with information, to make it easier and to communicate it, but infinite is an information which can not be solved, It could be a number to say, but It contains infinite information, that means it can not be solved, It has no ending.
MMOplayeerr infinity is not a number. Its a concept, an idea.. It is "forever"... Forever is not a number, its not a time or a distance but rather the concept of a number, time, or distance that is unending. Infinity holds no value on its own, but rather shows us that values will forever grow and expand.
It's a flaw in our mathematics at the edges. 1 divided by 0 is also infinity, so by this logic, pi = 1/0. You can play all sorts of games with maths at the margins.
Dividing by zero is undefinable. We can say the quotient of a Division tends towards + or - infinity as the denominator aproaches (gets really close to) 0 but we only aproache it! Try to zero it (which hopefully you won't do again) and you've made one of the greatest and most common among students mistakes in Math. That's by the way how you end up with nonsenses such as 1=2. edit: IF you are curious enough, check out the video of this excellent teacher to see why. ruclips.net/video/J2z5uzqxJNU/видео.html
Yeah m8.... 1/0 =/= infinity. Check it using limits. Yes 1 divided by an infintismally small number number does approach infinity. But what about 1 divided by an infintismally small Negative number? That is just as valid to represent being close to zero but you end up with NEGATIVE infinity. So in 2D space 1/0 = Negative AND Positive infinity. You should see the map for more dimensions as you approach zero.
The absolute value of 1/0 is infinity. 1/0 itself is simultaneously infinity, negative infinity, i times infinity... basically, if you could draw a circle with infinite radius on the complex number plane, 1/0 would be every point on the circumference of that circle.
Bertrand Russell: "Anything can be deduced from a contradiction." A voice from the back of the room: "Zero equals one. Therefore, you are the Pope." BR: "All right. 0 = 1. Hence, 1 = 2. The Pope and I are two. Therefore, the Pope and I are one."
The thing is: When you break math, the universe is broken. The mathematical constructs that hold our universe together will decrease in power and they will break, as the multiverse, in which we are living, will free all other universes and there will be a huge collaps that would lead to an explosion with results in a dead multiverse, aka ur dead m8. As you can see, if you or someone else really broke math, there can only be one possible solution for us to save us from the big problem: Meth. Hah
NeonGamerLP but the multiverse hasn’t collapsed though has it? This makes the xenoverse, a verse containing multiverses, collapse which causes the same thing, creating a paradox. This eventually leads to the omniverse collapsing and now we’ve officially broken psychics, creating another infinite loop. Meanwhile we’re all still alive. Let’s do a recap. First the multiverse collapsed and then the xenoverse collapsed. I guess the zeroth thing that happened was us, making infinity minus infinity equal zero. Unfortunately we have already defined this as Pi, breaking math again which doesn’t change anything since math has already broke several times before.
I think the thing that makes the rearrangement anomaly in infinite series is that the rearrangement operations themselves are infinite - the catch is in the 'and so on...' phrase
Infinity isn't a number; it's more of an abstract concept like love or beauty. And just as you can't subtract love from beauty, you can't subtract infinity from infinity. Only numbers can be subtracted. The math isn't broken, but one's understanding of how it is applied can be.
I disagree. These are infinite *series* of numbers, which add up to infinity. Besides, math is a concept designed by humans to help interpret the universe. Maths simply tries to understand the universe. Infinity is a very useful concept when dealing with things such as the infinite nature of the universe in space and time. Besides, math is what mathematicians make it, not what ordinary people would define as 'making sense'. Negative numbers used to be thought of as absurd, but the concept comes in handy especially if you just think of it as 'opposite'. Antimatter is basically negative matter, if you think about it. Sure, math is about concepts, but about concepts that come in handy no matter where you are in the universe. 1 + 1 will always equal 2 no matter what. Sure, maybe different ways of communicating this could be used but the concept remains the same. Math is also scalable, unlike the concepts of 'love' and 'beauty'. Sure, they are applicable to all humans and sexual life forms, but do not matter in the realms of atoms or galaxies. Maths matters there however, and everywhere else. It and infinity are universal concept, and such are much more applicable in many situations.
@@ataready8810 You're certainly free to disagree, so long as you're not disagreeable. But actually you do agree with me somewhat when you wrote : " Infinity is a very useful CONCEPT." Mathematicians no longer say that divergent infinite series add up to infinity, instead they say that no number N, can be chosen which exceeds the limit to a divergent series. In the past, top mathematicians did use "infinity" but this results in problems, as Cantor inadvertently demonstrated. Infinite sets lead to paradoxes, or actually, antinomies. particularly in establishing a logical foundation for mathematics. For the last century or so, starting with Karl Weierstrass, mathematicians have used precisely defined concepts involving limits and continuity rather than infinity. If a mathematician does talk about infinity, it is because he is speaking informally as an aid to understanding, using a concept that we intuitively understand but which is not mathematically rigorous.
there's a very quick inductive proof right just one arithmetic step away from the thumbnail. pi = inf - inf => pi + inf = inf. If we assume the negative infinity which gets move to the left side is truly infinity, then it would seem at first glance to be an inequality. But when we consider what infinity is, *something* that is larger than any number, then if we try to add any value to it, it cannot get any larger. It is already larger than any number. Thus, you can add or subtract any number from it, and it will still be a "whole" infinity since if any numerical quantity changed it, it could not possibly have been infinite to begin with. Maybe it's not inductive. Maybe it's circular. And hand waving. ;)
Since infinity is a concept and not a number, you don't have circular reasoning nor hand waving. It's all in the definition of the concept of infinity.
Stupid question, but I am curious to know, none-the-less. If pi=Infinity-Infinity, & the Area of a circle is pi•r^2, then Area of a circle is (Infinity-Infinity)•r^2. Is this a true statement? If not, why?
@adam spears in maths, infinity is not a number but an idea which represents incalculability. For example all the incremental values between 1 & 2 are infinity. Does this also mean infinity < 2?? :)
Many have already answered reasonably, but here's my take: That statement is false because of the asymmetry of the equality operation when operating on infinities. "Symmetry" regarding equality means that x = y iff y = x. The symmetry is found for many (most?) sets of scalars (e.g. integers, reals), but for infinities it does not apply. Mathologer convincingly demonstrated that ∞ - ∞ = π, in a sense. However, by the asymmetry of equality, ∞ - ∞ = π does not necessarily imply that π = ∞ - ∞, so we can't substitute it in for π in the circular area formula etc. Not sure if totally correct, but it's my 2¢.
All this stuff about sums of infinite series fascinate me since I studied it at University. Moreover I really appreciate the way you communicate and the graphical/visual explanations. These contents are challenging but thanks to you encouraging and inspiring teaching the concepts involved become more understandable.
In the first example the regrouping combines each positive element from the series with 2 negative elements from the series. The density ratio of negative to positive terms is 2:1 regardless how many terms are used which gives a result of half what it should be. I wonder if there is a way of rigorously taking into account the distribution of the elements used in the rearrangement pattern?
When you start comparing terms of an infinite series with a condensed version of itself, where you have combined multiple terms into one, you are no longer comparing equivalent infinite series. You are comparing a subset to the whole, so saying 1= 2 is really saying, part of everything is less than everything. I don't see how changing the terms of the series can change the sum, but comparing terms to the unchanged series would be different. Any mathematician knows that not all infinities are equal. The sum of an infinite series, where the terms of one series are of a larger magnitude, or you add up a different number of terms, may both result in infinity, but to say they are the same infinity is nonsensical. No paradoxes here, just fun math manipulations to confuse the non-mathematically minded.
So what step of the process is invalid? You haven't pointed out any invalid steps, you've just said you're comparing a subset to the whole. But the process of getting from the whole to the subset must have an invalid step in there. The invalid step can only be rearranging the terms, since all of the other steps can be proven to be valid. For example, if you have a convergent series, adding parentheses to group terms together does not change the sum. The sequence of partial sums of the newly grouped series is a subsequence of the sequence of partial sums of the original series. It's a general fact that if you have a sequence which converges to L, then every subsequence must also converge to L. So this shows that the grouping part (which is what gets you from all of the terms of the original down to only some of the terms of the original) is a completely valid step. The general reasoning that Mathologer used to get a sum of pi also does not involve the step of grouping terms together, so the claim that he is just "comparing the subset to the whole" is incorrect.
@@MuffinsAPlenty - 7:26 How can you honestly say the he doesn't group things to get pi? He groups the first 76 positive terms in the series, then 1 negative term, then 126 more positive terms, then 1 negative term, etc ... He is basically summing all of the positive terms, with very few of the negative terms, which is why it sums to a number (3.14...) greater than the original summation (0.69314...). The invalid step is absolutely rearranging the terms. The original series, where they alternate positive/negative, includes equal numbers of each, but summing to pi includes far more positives than negatives. You are comparing two completely different infinities as if they are the same thing, which they are not. (all of the terms) 0.69314... = 3.14... (the subset with more positive terms than negative) We know the series has equal numbers of positive and negative terms, yet the summation to get pi is overloaded with positive terms. You even state that convergent series, which this is, sum to the same number regardless of grouping, so how does one get pi from a convergent series the sums to ln2?
Perhaps we're using terminology a bit differently here. When I say "grouping" I mean putting parentheses around terms. By reordering, I mean terms go in different positions than they originally were in. For finite sums, regrouping doesn't affect the sum: a+b+c+d = a+(b+(c+d)) = ((a+b)+c)+d = (a+b)+(c+d), etc. But for infinite series, "grouping" can affect the sum. For example, consider Grandi's series: 1−1+1−1+1−1+... If you group terms like this: (1−1)+(1−1)+(1−1)+..., you get 0+0+0+... = 0 But if you group terms like this: 1+(−1+1)+(−1+1)+(−1+1)+..., you get 1+0+0+0+... = 1 But Grandi's series is divergent because the sequence of partial sums doesn't converge to anything; it just keeps alternating between 1 and 0. If you have a convergent series, then adding in parentheses (which is what I mean by "grouping") cannot affect the sum of the series. For finite sums, reordering also doesn't affect the sum: a+b+c+d = a+c+d+b = b+c+a+d, etc. Reordering can affect the sum of an infinite series, like this video points out. If you keep adding positive terms until you get above pi, and then add negative times until you get below pi, then keep adding positive terms to terms until you get above pi again, etc. that's reordering the terms, because you're changing the order in which the terms appear. What's really weird is that reordering _can_ change the sum of even some _convergent_ series! But reordering is not comparing a subset to a whole. Every term from the original series must appear as a term in the new series exactly the same amount of times it was a term in the original series. The strange thing about infinity is that you can take terms at different rates and still get them all in there. And that's what allows the sum to change.
By adding or substracting infinite series of numbers, you can get anything you want. You can even add an infinite serie of positive numbers and get -1/12...
Be careful with that!! This kind of things can be done only with particular series. There exists series that converge to a unique number (or to infitity), so you can not generalize the Riemann theorem to any infinite serie of numbers. And the -1/12 thing is false :O
Sorry, chum. A series with all positive terms will always produce positive partial sums and that sequence, being positive, cannot converge to anything negative.
+Gilles Soulet You can add (or multiply) infinities and be totally right. It's only subtraction (or quotient) that doesn't work. The disjoint union of two infinite sets is necessarily infinite. Therefore infinity plus infinity is infinity. Cartesian product of two infinite sets is necessarily infinite so infinity times infinity is infinity. On the other hand, if you subtract an infinite subset from an infinite set you get something that may be finite (of any size, including 0) or infinite. That's why it's called indefinite. For example if you subtract the even numbers from the integers you are still left with an infinite set: the odd numbers. But if you subtract the nonzero integers, you are left with a finite set - the singleton {0} or if you subtract everything you are left with the empty set.
duffypratt Infinity is not a real number, but all numbers, like words, are made up. We get to define terms in math in a way that is convenient and useful to us.
Herr Mr. Polster, I am very enthusiastic about your lectures. You have a very thorough, but also a very funny/joyful way to explain the problems and solutions. Thank you! Greetings/Grüße Dr. Sebastian Kühnert (working in the field "Functional Time Series Analysis")
I did something similar with integrals to get ln(2). The integral from 0 to ∞ of 1 = ∞ The integral from 0 to ∞ of tanh(x) = ∞ Subtract the second from the first and make them one integral to get the integral from 0 to ∞ of 1 - tanh(x) = ∞ - ∞ = ln(2) Can you write this with series?
Anybody else, at some point in the video, have their eyes glaze over and the sound "blurr"? Videos like this are good reminders for me that I'm never gonna be a mathematician or a physicist.
Can you make a video about Zeta function plz?! I would love to understand the problem that Rieman left and why it is so hard for Mathematicians to solve!!
Yes, usually, whenever it is possible to make sense of infinity minus infinity you find that the result can be anything you want. Still this "whatever you want" can carry a lot of meaning as in the case of infinite series :)
Marty & Friends - Thanks again! This is a great example of how seemingly complicated and/or mysterious (or mind-bogglingly inscrutable) maths and numeric logic can be explained with elegantly yet deceptively simple examples. > + Why deceptive, i.e., what's hidden or unobserved or unproven? Of course, Marty didn't need to repeat the rules, axioms, and well-known principles of arithmetic required for the results. However, what was never shown or explained (even by Riemann, among all other super-stars of maths) is why. + In other words, what are the deeper enabling principles of maths and numbers, and their potentials? Also, how can we discover them, if that's even possible? We can also wonder why it might matter, especially if the super-stars didn't care, or we could wonder what else the answers could tell us about maths, numbers, symbols, and reality (etc.). + First, it may seem obvious that--to exist and "work"--the logical operational principles of maths require enabling principles of form, structure, functionality and, maybe, relativity (enabling the relations, functions, etc.). How we find those deeper principles requires observation and some study. For example, we see the forms of numeric symbols and the operational symbols (+, --, =., /, etc.), and we can see the functional results of their relations and some of the reciprocity. Yet, clearly, even the greatest geometers and giants of maths failed to see the virtual structural nature of numbers and maths and why they do what they do. Why that matters is revealed by a) all the failures to solve the RH problem (among many others), b) the long failure to understand the reason, c) the inability and ability to recognize insufficient proofs (of RH, etc.), and d) over 500 other theorems and results related to RH and R.'s zeta function (etc.), even without proof of RH. Of course, virtually all of Marty's videos give us potentially infinite opportunities for seeing and discovering the nature of reality, maths, geometry, trig, numeric logic (etc.), and their intrinsic relativity. + What the enabling meta-logical principles, the quest, and its results can tell us is too vast for a comment, but it's the basis of 2 papers @ (ORCID.org/0000-0001-5029-7074).
@@benterrell9139 Pure Bullshit. There is no common notation. ln(x) can be written down by log(x). Others just use log(x) not with base e but with base 10, which is in common notation called lg(x). So there's a common notation for ln(x) and lg(x), but not for log(x). You just have to use your brain to get the context whenever someone writes log(x).
I think it depends a little on the field you're studying, but in general ln (the natural logarithm) occurs so much more often than log (base 10 logarithm) that people often call ln "log". In physics you basically never see base 10 logs, so we always just refer to ln as log. It's usually clear from context (there's no reason you would see a logarithm with some other base here, for example).
This also works with 1-1+1-1+1-1+1-… and it’s much faster to see, and you can even make it sum to infinity very quickly. What kind of sequences that diverge like that can you do this?
That infinite series isn't convergent so rearranging by bracketing is invalid. If you do and try to compute a sum, you find crazy results, like: It's obviously 0 as it cancels out, but it's 1 as 1 - 0 is 1. Adding all the positive terms together and negative ones leads you to a meaningless infinity - infinity. It's easy to see any finite series length n will add to 1 or 0 and one n+1 will always flip to the other value. That remains true as n -> infinity.
@@raphaelcardoso7927 As erroneously rearranging gives (infinity - infinity) then any number is equally as valid in the invalid calculation, because ¼ + infinity is infinity when you allow rearranging. Divergent sequences have sums for finite number of terms (1 or 0 in this case) but not infinite terms as it's either a non-converging 1 or 0 never tending to a single value. In other words an integer like 10 is no less wrong than π, because you permit invalid operations.
The null properties of addition and multiplication are exceptions to the inverse properties. In general, if you add a number to its opposite, you get 0, and if you multiply a number by its reciprocal, you get 1. But a null addend such as infinity has no opposite; and a null factor such as 0 has no reciprocal.
yeah but at one point the sum is finite and it reads 3.1415... it was ambiguous whether the sum was exactly equal to pi or only the first few digits were the same. that's why i was confused
Is it not because pi is actually what the series tends to as the number of terms tends towards infinity? Cutting the series off with finitely many summands would lead to a rational number, but we have infinitely many terms so it's okay.
Math isn't broken. Commutativity over addition e.g. a + b = b + a etc., was only defined for a finite number of numbers. In fact commutation still works with absolutely convergent series such as the geometric series.
Maybe because the series is defined from it's chosen endpoint back toward zero, so of course rearranging the terms ruins the sum, since you end at a different interval, and worse still, close to the larger fractions where it counts as much as a half.
I think you still have to get into the swing of things here. Nothing gets "ruined", nothing is "worse", etc. I'd say relax a bit. There is this standard definition for what the sum of an infinite series is supposed to be that I talk about at the beginning. Everything else is just about being logical and following our nose (and marvelling about what we find along the way :)
I don't really know what you are talking about. I was trying to reaffirm your point but obviously you've misunderstood what I meant and (somehow) taken it as an insult. I wasn't even addressing you. Don't tell a first time viewer to 'relax', and don't condescend.
Also, I don't know jack about math, so whatever you said is probably true. My comment was me trying to get my head around it, but I'll try that elsewhere in future.
I've just replayed the video because I couldn't really remember what it was, and I realise now English is your second language, so I'll take that into account, hopefully so do you.
No, problem at all and I was definitely not trying to be condescending. Try to look at it from my perspective. I've been teaching this stuff for the better part of my life and I've never had any complaints from my students that I am not expressing myself clearly (on the contrary). At the same time any video on anything to do with infinity always attracts a huge number of cranks who start off their comments by saying "You are lying!", "This is BS!", "Infinity is not a number!", "Mathematician don't know what they are doing!" etc. I really get very annoyed by this and I think it is important to set the record straight to make sure that this comment section is not swamped with such nonsense. Also it is very hard to tell when a comment with negative connotations is left by a crank or just somebody who is honestly struggling to understand what is going on.
Reminds me of when I have too much milk in my cereal so I add cereal, then I have too much cereal so I add milk. The amount of cereal and milk I use has no limit, but the mix gets closer and closer to what I want.
Sadly I got arrested for trying to explain to the cashier that purchasing an infinite amount of milk and cereal means the store owes me 1/12 of dollar.
Underrated comment.
I will interchangeably like and dislike your comment until you owe me one like.
-1/12
This comment was better than the video. Bravo.
@@parikshitmusic510 yeah that's why it says *owes*
This guy looks and sounds familiar, so I looked at his "about me" and found out he was my maths lecturer.
nice lmao, you just found out your lecture is a youtuber.
He was the lecture? He was the entire lecture itself?
for real?
welp there goes ur uni's location
@@mexomenti Not the one I go to now.
height of pizza=a
radius of pizza=z
volume of pizza= Pi*z*z*a
That's about the smartest equation I ever saw. Kudos.
This could be rewritten as Pi*z^2*a
what about hamburguer and hot dogs?
@@Yottifferent You just made it into a simpler equation, but it's still the same equation.
@@Yottifferent woosh
It's pretty simple to prove
π + ∞ = ∞
therefore ∞ − ∞ = π
Brilliant
What about when i say
1 + ∞ = ∞
1 = ∞ - ∞
Its also right
π+∞=∞
π+∞-∞=∞-∞
If you cancel the infinity in one side, you will have to cancel in the other, so:
π=0
Something isnt working here
@@jeffersonpaesdefariasfilho well the infinity symbol is a bit deceptive here. While they are all the same, when you add anything to infinity, it's still infinity but just a little more infinity. So the infinities in your equation aren't necessarily the same, but are "different" by the added amount
@@jeffersonpaesdefariasfilho 4+2-2=2-2
=> 4=0
The euqation is wrong you cant set two different numbers equal since they arent equal
I’m a simple man. I see something I don’t understand, I click.
Max Zatlin ya
That's a good thing. Slowly but surely you'll start understanding.
@@pietrotettamanti7239 yes, infinity is no longer infonite and can now be measured
@Max Zatlin me too.
@@ileryon4019
Nope
"Maths is wrong"
*collective gasps of every human ever since the dawn of time*
"Or you made a mistake"
*Collective sigh of relief*
Mine is opposite to dat
wouldnt be the first time math was wrong...
its just a man made tool. We adjust and refine it as we move forward.
Kudos on the (sort of) correct use of 'yhe'. A real thorn in the side of linguists... My intuition is that it isn't surprising that an equation with the term 'infinity' in it can produce whatever result we want it to, because 'infinity' has no mathematical significance to begin with. (I hope this 'hard-core finitist' position isn't too shocking without a prior warning. Or even an abbot warning)...
The mistake is applying mathematical functions to things that are not really numbers: GIGO, in other words.
godel's inompletness theorem
well...
inf + 42 = inf
therefore
42 = inf - inf
ur god
1234bliblablau inf-inf= any rational/irrational number
Mind=blown
wtf no that would go to 0=0
1234bliblablau maths is broken
you can't just 42 to the infinite, the infinite is not a real number. That is not an operation you are allowed to do.
This makes alot of sense when you think about the definition of a limit
Great boi!
@Supreme ideology Basically a sequence converges to some number if it can get as close to a number as it wants but also stays that close and doesn't frequently stray. So you can always take a rearrangement of the sum and make it stay super close to some value by shoving the later terms down the line.
Yeah it makes you realize you should not expect associativity to work. Associativity is captured by what you put in the summation symbol which is why "..." notation is ambiguous.
@@hybmnzz2658 what's your alternative then?
@@TheBeakertubethe ... notation is always short for a limit of some sort. So 0.9999... is just shorthand for sum n=[0,inf] 9/10^n
1+1/2+1/4... is just shorthand for sum n=[0,inf] 1/2^n
The ... notation is just always easier to write down, but it depends on the intuition of the reader to interpret what comes next.
You can come up with any number of sequences that start the same way as a given finite sequence, so it's not a good formal way of writing down a limit.
LIFE HACK:
so if you dont know the awnser to a mathematical question, just awnser with (∞-∞) and you'll always be right!
Kalkaanuslag ahahaha
anthony showalter yes you can? What? 1+1=2, infinity minus infinity can equal 2.
anthony showalter because infinity isn't included in any proper number set. It is all numbers. 0/0 is undefined because you can't divide by zero. Stupid question.
anthony showalter actually it was stupid. The reason 0/0 doesn't have an answer is because it's not a real problem you can have. We as a people are not able to define what the answer would be. The answer to any question could be (n) if you define n correctly. Also why do you type like that? It's still a dumb question and you're not proving anything. Also, you are able to define infinity and negative infinity as different "numbers", so the whole n shit you're going on about is just weird.
Michael Steshenko how am I writing nonsense? 0/0 is undefined.
When 1=2 math is having a really bad day
ruclips.net/video/hI9CaQD7P6I/видео.html lol
but surely in steps 4-5 you divide by 0, no?
FAT cat yay I can do smart stuff
@FAT cat u can cancel b & b iff b not equal to 0 , but you have done it which is blunder.
But still I play these things in my younger ages.
@Floofy shibeshibe how did you get line 5 a+b=b?
Not sure if paradox or just being naughty with definitions of infinity.
The latter.
That's because a limit tending toward infinity never actually brings in the concept of infinity or uses it as an expression or anything. Instead, you just say that no matter how big a number someone can mention, you can always find a delta such that so long as you're closer than it on your input, your output will always be a bigger number.
kinda hurts ur brain huh
Not really, you just have to know there's more than one type of infinity.
Nicholas Heathfield infinity is not a number and cannot be subtracted 😂
Well done. That was the simplest and easiest to follow explanation I've yet heard. Riemann and Ramanujan are more worthy than most to be called "rock stars". It's criminal that Ramanujan passed away at such a young age.
Can't even imagine what our math world would be if Ramanujan lived at least to his middle age.
The takeaway is that if you ignore the definition of “infinity” and just rearrange terms as you wish you can always get the result you want.
This feels morally questionable
Another way to look at it which maintains our current understanding of infinity is this
Inf + x = inf
Therefore
X = inf - inf
You can use the same technique of rearrangement to change the sum to any real number (switch the sets for negatives)
Michael deBidart It doesn’t converge absolutely
Myanmar. ..p. 9:00: com
You summed it up succinctly. This is the area of math that I find useless, and therefore infuriating.
@@masterlangtau useless and therefore funny*
Paradoxes are just misunderstandings. The solution for this one is something we all already know: that infinity is not a number, though we continually make the mistake of thinking of the concept as that of a really big number.
"Infinite" means without end. So the clearest meaning of "infinity minus infinity" is "to add to zero forever and subtract from zero forever." It says nothing about the rate of addition or subtraction nor about what quantities are to be added/subtracted in what order. As long as both the addition and the subtraction keep going on forever.
You see, a time element has been introduced. But the phrase "infinity minus infinity" says nothing about that time except that both the addition and the subtraction must go on forever. It does not say when to begin.
Thus, if you start say, adding first, and add till you get to Pi, and then start adding and subtracting in equal measure, forever, you'll stay at Pi forever. If you alternate adding and subtracting but in smaller and smaller increments, as in this video demonstration, you'll hover around Pi and converge upon it.
And you can do this of course, with any number.
When looked at this way, as in terms of "infinite" correctly signifying 'without end,' rather that *in*correctly signifying 'a really big number,' the Paradox goes away. Suddenly it is all quite clear and obvious, easy to visualize, and containing no contradiction.
What confuses people, ironically is that "infinity" is in the linguistic form of a noun, rather than an adverb. Correct this, in your mind, by always correcting the term "infinity" to "infinitely." That is the trick to solving, most likely, all "infinity" paradoxes.
Think of it this way, if you want more on that: You can go running, or, you can go for "a run." If you run for 1 mile in a direction and then for 1/2 mile in the other direction, you don't end up where you started, and this is no mystery, obviously, because you didn't "run" the same distance in both directions. But if we use the noun form, and say you went for "a run" of 1 mile in one direction and then "a run" of 1/2 mile in the other direction it opens a possibility for confusion. Because whatever distance we suppose "a run" signifies, we have shown that "a run" minus "a run" does not equal zero.
We have *assumed* that "a run" designates a fixed distance, when it does not.
Likewise, "Infinity," a noun, implies to our minds a fixed length, or distance, or quantity. A certain number. But it is not that. Whether we can imagine it or not, we suppose that 'that number' that is infinity minus that number aught to equal zero. We are, thus, astonished when it does not.
But hopefully we can see now, that, that is because we are falsely imagining "infinity" to be a number, a fixed length, when it is not. It simply signifies the quality of being without end.
Yes how infinity going larger or the way to diverge is different, there is no way to tell
Have you ever written a book or read one that could speak about maths the same way you do? I'd love to read something related to this sort of abstract perspective on this field.
I am happy to read you man I am enough of these youtuber that mislead people to make view.
Yes, It opened the mind. Infinity is a numeric set that brings together all numbers in its individual being like this:
...-3, -2, -1, 0, 1, 2, 3...
Infinity is not this:
...(-3)+(-2)+(-1)±0+1+2+3...
That's a amount of individual numbers with no relation of addiction between themselves, constructing a really so big number with a infinite number of digits.
EXACTLY.
The whole premise of this video was a bait and switch.
Mathematicians treating infinity as a number, rather than an unending sequence: guess I won't be subscribing here...
Its kinda refreshing hearing german words and names pronounced correctly :P
Word!
If they pronounce Pi correctly, then how would they distinguish it from the Latin letter "P" as it is pronounced in the English language?
In Dutch we pronounce pi like the letter P, except we pronounce the letter P like Pé
Sub Nano Give me an example of an English word that each of those letters rhymes with.
The letter P is pronounced as the "Pai" in "Pain", while the mathematical character "π" is pronounced as how the English pronounce the letter "P", while in English they pronounce the "π" as "Pie
1:25
I wish you suddenly appeared like this in my mathematics exam...
_Your math genie is at your command, you have two problems left_
You can ask him to solve the whole test if instead of asking "answer to question 1, answer to question 2 ...." You interpret the whole test as a problem and ask him "the n-tuple wich solves the test"
laughed at this :D
Recently a few people have been contributing translations and subtitles to various videos. If you are thinking of doing the same could you please let me know. (RUclips does not notify me of any subtitles waiting for my approval) Also I'd like to acknowledge any contribution like this in the description of the videos :)
(added 17 June) Thank you very much Zacháry Dorris for contributing English subtitles for this video and Rodrigo Naranjo for contributing Spanish subtitles!
You should definitely start with adding English subtitles as others can use that as a template so all subtitles derive from the same original source
That's great. This whole channel is a one-man-show and adding any subtitles just takes up too much of my time. Any help in this respect is very much appreciated :)
So it's like a linear system. It can only have 1 or 0 or an infinite amount of solutions.
There is definitely a bit of a connection here. If you want to find out about the details check out the "Steinitz's theorem"
on this page en.wikipedia.org/wiki/Riemann_series_theorem#Steinitz.27s_theorem
Thank you very much offering. I actually speak Russian reasonably well. Would be great to have Russian subtitles :) Not sure whether you've ever tried to put together any subtitles. I have and it really does take forever, so don't feel bad if it all gets a bit too much :)
This is a heck of a lot simpler than I thought it would be.
And yes, we do all want Pi. Thank you, Zach (captioner), for pointing that out.
When we are discussing the Riemann R. Theorem, it is essential to point out that this is about divergent or conditionally convergent series. NOT on absolutely convergent series. Absolutely convergent series (the kind we usually deal with everyday) we can rearrange to our hearts content.
Just don't forget that!
(Otherwise, excellent video. Keep up the fantastic work!)
For what we are interested here the technical terms "conditionally convergent" and "absolutely convergent" don't add any extra value to what is captured much better by focussing straightaway on what the positive and negative terms add up to :)
It is important to mention that this is a property of conditionally convergent series, but that it is NOT a property of absolutely convergent series. For instance, if we write pi as a series of fractions of powers of 10, we can't rearrange the fractions to get anything other than pi.
Well, I do talk about absolutely convergent series at the end (those for which both the positive terms and the negative terms add to finite numbers) and I do say that you can reshuffle them any way you like without changing the sum. Maybe have another look :)
I beg your pardon. I see you did that at the end. You are right.
Well, keep up the hard work. You have some great videos.
All under control then :)
Soo... I was watching anime, then accidentally clicked on RUclips. And now it's 1.52am and am still here. On this time, am learning math.
All hail The Algorithm, first of his name, and replacer of what used to be the old king - The Dark Side of RUclips. Haven't seen that guy since like 2016...
This just happened to me
@@neilbrons529 what anime were u watching
Anime sux
∞ - ∞ =
(∞ - ∞)
Hahahaha
(0-0)
uWu
Chernobyl boobs, I like it
Like it
Before that video, I was confused. Thanks to this video, now everything cleared out : I am even more confused.
This is me. I watch all his videos and feel like I need advanced math just to understand meme math
I like your serious approach to series. I was tempted to put in a very bad pun, but I refrained this time.
Why so series?
I'm glad you did, Terry.
Infinite series? Do go on...
What people often don't realize: Infinity is not necessarily the same as infinity. Thus, infinity minus infinity can be any real number including infinity. Similarly, infinity times zero can also be any real number including infinity.
How infinity times zero can be any number? It should be zero although it's a non determinant form
@@bosongod2830 I think you did not get the point: Infinity is not a single determined number. Therefore, arithmetics involving infinity often does not turn out the way one is acustomed to. Therefore, a rule like "x * 0 = 0 for all x" does not hold good when involving infinity.
Example 1: x/x^2 with x -> infinity leads to 0. This equation is the same as "infinity * 0 = 0" when you apply the limits to the numerator and denominator seperatly. However:
Example 2: 8 * x/x with x -> infinity leads to 8. This equation comes down to "infinity * 0 = 8" when you apply the limits to the numerator and denominator seperatly.
Example 3: x^2/x with x -> infinity leads to infinity and therefore "infinity * 0 = infinity".
@@bosongod2830 consider x and 1/x.
As x→∞, 1/x→0
So x•(1/x)→∞•0
But x•(1/x)=1
So ∞•0=1 in this case.
More generally, use x and a/x to let ∞•0=a for any a.
The alternating harmonic series is not absolutely convergent so the Riemann Rearrangement Theorem shows that it can be rearranged to converge to any real number you want.
Finally someone else that knows this theorem!
This video takes me back to my Cal II class. Infinity series are so much fun and interesting! I remember the day my professor blew my mind when I first heard of this concept. This made me want to go back and brush up on infinity series. Loved the video
Great, wasn't sure how well this one would fly. Anyway, I really love this stuff :)
Hi there. I am not a mathematician but only have a PhD in computational physics. And I usually have a very pragmatic view on things. That’s probably why I spotted a HUGE difference between generating ln(2) or Pi with that series: To generate Pi, you need to know Pi already BEFOREHAND (to decide when to switch back and forth between negative and positive summands). However, the algorithm for the summation sequence to arrive at ln(2) (or ln(2)/2 for that matter) does not need any info on the value of ln(2). You could actually FIND the value of ln(2) with that series, even if you never knew it before. The INFORMATION about ln(2) IS CONTAINED in the series. In contrast to that, there is NO information about Pi in that series (since you have to explicitly put it into the summation sequence to construct it). So, the real question mathematician should ask about such series is not ‘What does it converge to?’ but: WHAT INFORMATION DOES IT CONTAIN? And then you might get sensible (maybe even unique?) answers: In the above example that answer is ln(2). Even if you re-arrange the summation (with an algorithm not using info about the end result already in the construction) you get things like ln(2)/2. But that’s not new information since, if you know ln(2) you also know half of it (or any value which differs from ln(2) only by a rational number - be it a factor or a summand).
Well… one could even go further and say: No sir, you can NOT construct Pi with that series but only an approximation of Pi. Since you need the end-result in the construction of your summation-order, you would need knowledge of Pi to the last digit to really construct Pi EXACTLY. And well… we all know, nobody will ever have that knowledge since there is no last digit. The same holds for any irrational number, which you would need to know already beforehand to construct the summation sequence. That means, contrary to common believe of the last couple of centuries: You can NOT construct any number with such series. You can construct only rational numbers (since you know them exactly to the last digit should you need them in the summation-sequencing-algorithm) and those irrational numbers (maybe only one per series?!?) whose information is actually CONTAINED in the series (ln(2) in the above example).
PS: You can also contact me in my native language, which is the same as yours :-)
All good points, but here are a few things for you to consider. First, have a look at the section "Getting an arbitrary sum" on the wiki page dedicated to the Riemann rearrangement theorem en.wikipedia.org/wiki/Riemann_series_theorem
Second, for people like myself to distinguish between a number and its various representations is very important. So, there is pi, the ratio of the circumference of a circle and its diameter and there is the representation of this number as a decimal number. What I do know from the theory of real numbers is that there is a unique way to express pi as a decimal number. What I don't know are the individual digits in the sense that there does not seem to be any "simple" pattern to the digits. On the other hand, I can write down an algorithms that will generate any of those digits and in a sense this algorithm is really just another type of pattern. Same thing with what I do in this video. What I show is the existence of a rearrangement of the series that will add up to pi as well as an explicit algorithm that will get you this rearrangement.
In terms of practicalities truncated representations are usually all we need anyway and in terms of exact sums existence is fine in the first instance. Of course, simple patterns are great and you also get them in abundance in terms of infinite series summing to whatever we like from things like Mclauring series :)
The Riemann rearrangement theorem is true, no doubt. But the theorem says: For any real number, there EXISTS a permutation such that the series equals that number. I would never deny that. What I say is: Yes, that permutation exists, but you can never actually CONSTRUCT it. I have chosen my wording carefully, you see ;-)
Contrary to ln(2), you cannot describe a permutation such that the alternating harmonic series approaches Pi to ARBITRARY precision. Why? Because you don’t know Pi to arbitrary precision and you EXPLICITLY need that knowledge to construct the permutation. Let’s say you have approached Pi with that sum to the best precision known to mankind. Then you cannot go further since you don’t know how many positive terms you have to add next before you switch to negative terms again (or vice versa). Sure, it CAN be done, since the Riemann rearrangement theorem tells you such a permutation exists. But you cannot actually DO it until you obtain knowledge about a more precise Pi FROM SOMEWHERE ELSE (i.e. from running the algorithm for Pi you mentioned a little bit longer). The information cannot be extracted from the series ITSELF. You have to input it from the outside.
It’s a bit like representing Pi (the real thing I mean, i.e. the ratio of the circumference of a circle and its diameter) as a decimal (or binary or hexadecimal or whatever) number: It can be done to arbitrary precision, but that doesn’t mean the information about Pi is already contained IN THE NUMBER SYSTEM ITSELF. It’s not. You have to know Pi to the desired precision FROM SOMEWHERE ELSE (the Pi algorithm) before you can write it as a decimal number, i.e. before you know the PERMUTATIONS of the digits to construct it with the number system. This is exactly the same as with the alternating harmonic series: You have to know Pi to the desired precision from somewhere else before you now the permutations of the summands to construct it. It’s even the same as this: the English language is enough to represent Shakespeare’s Hamlet. But that doesn’t mean, the information about Hamlet is already contained in the language ITSELF. Shakespeare had to input that information from somewhere else (from his fantasy happening in his brain) to know which permutation of letters to use to construct Hamlet.
See the difference to ln(2)? The algorithm for the permutation contains no reference whatsoever to the end-result that permutation leads to. It just leads there “automatically”. The permutations to use are all clear right from the start, no matter how far out you want to go. Therefore, you can construct ln(2) up to arbitrary precision. You do NOT need to know the end-result beforehand to construct the needed permutations. The information about ln(2) is contained IN THE SERIES ITSELF. Information about Pi is not.
That’s my whole point here: For these kind of infinite series it is not so useful to ask “Where does the series converge to?” but it makes much more sense to ask: “What information is contained in the series ITSELF?” If one could define - with mathematical rigor - a new kind of limit as the answer to this question, we might be getting somewhere.
If there is no limit or precision applied to Pi in this case, would that make the answer of inf mins inf sum up to infinity? (im a layman btw, just nerding out on the internet).
@@HansPeterDeutsch_hpd this made so much sense to me. Most of the math seems like a feedback loop, in which we input incomplete info and get incomplete output and we keep doing it...
Yes, there is definitely a difference in the algorithm's "complexity" resp. the "information contained" between "take every second term" and "know the desired end result, then wait until the sum has passed the threshold". My intuition would like very much to make that a clear-cut difference, but the terms of computability theory or even complexity theory are not "fine-grained" enough to capture it, I'm afraid. And I don't see any other options immediately...
Loosely related fun fact: The set of algorithms to construct a permutation re-ordering the sequence is countable of course (as the set of all algorithms is), while the set of real numbers of course is not. Nevertheless you can construct a permutation of the sequence converging to each given real number. The magic lies in *given* I suppose, meaning that you need an algorithm to construct the number before you can construct your permutation.
Guess that is the difference you and I were looking for after all: For ln(2) the algorithm is contained/trivial, for pi you have to put it in externally and for uncountably many others there is none.
This was one of the best mathematics lessons I have listened to in my life. Thank you.
what doesn't equal infinity - infinity
me
zero?
Teddy Woodburn a higher infinity (like ∞^∞)
no. you cant simply subtract infinity with infinity. cause how is infinity defined? every math prof will tell you you cant do this :D
Kalkaanuslag but infinity to the power of infinity is just infinity right?
Me looking at pure math: "You know applied maths never treated me like this..."
I'm pretty sure you can't just apply standard arithmetic operations to undefined values like infinity.
Its not just that I think. The whole method of adding and rearranging sequences like that just doesn't seem to prove anything. You can do something similar to this and prove that 1+1=0 which is obviously wrong.
Actually those arithmetic operations are valid in the extended real number field, in which infinity is a special member
Some you can, like 1/infinity=0
bookashkin that doesn't make sense, since it would follow that 0*infinity = 1
There is no 1/infinity afaik, just like there's no 1/0
Thank you for sharing this. It reminds me of quantum mechanics where you can have all outcomes or no outcome, or just a single outcome, or anywhere in-between. I think this a mathematical framework that needs a lot more study as it relates to probability.
the Reimann-Zeta hypothesis is the Dark Matter problem for Mathematics
Its always the comments section that makes a video great😂😂😂
Infinity-Infinity is an indeterminate form and we use limits to remove this form!!. So we can't determine the exact value of these indeterminate forms like..
0/0,infinity - infinity,etc
Determine the value of indeterminate forms ? Sounds very contradicting to me.
It is as making a list of things that can not be placed in a list.
@@erikblaas5826 Not exactly determining the value but the value where it tends to be.
so, the thing is.....
I came with a question and am leaving with more questions
Strangely, that's how you know you're gaining knowledge, i.e the more you know, the more you know you don't know!
Maths with infinity series like the ones in this talk are actually hypnotism at play 😊. Think of it, the bracketing process makes you to believe it is legit, but then u get drowned under the sleight of MIND!
The obvious secret to break the conclusion is to call the contradiction & detect which part is false. That requires talent of course, as much as the math mesmerizer!
Number one math channel.
Wildberger is a crank though.
He's just a finitist. What specifically makes you think he's a crank?
+Sen Zen good to know! thanks
Perhaps "crank" is too great of a term to say. He is a published mathematician and certainly isn't a bad teacher. But many of attitudes are very crankish.
People like Wildberger want only the math that "resembles" reality, while somehow not grasping the fact that a positive integer itself is an abstract concept that doesn't exist in reality either.
To take existence proofs, distances (which he abandons for quadratures), infinite sets, and etc. from mathematicians GREATLY cripples the subject. If we then continue this line of destruction farther than even Wildberger himself and proceed to remove everything from mathematics that doesn't exist in the real world then there would be no math left from the slaughter.
Yeah, "crank" is too strong because he's never put forth any crank results, at least none that I've seen. You say his attitudes are crankish, but it isn't self-evident to me that being a constructivist or a finitist is a crank attitude, especially seeing as how the axiom of infinity is an axiom, not a theorem of set theory. I don't agree with Wildberger on most issues and I have no problem with real numbers as he does, I just don't think he is an example of a crank. Nice examples of math cranks include those denying Cantor's diagonal argument or that 1/3 is not 0.333... (I've actually debated a crank on the latter).
Mathologer, I have an argument at my end. I'm saying infinity IS used in mathematics and my friend says it ISN'T, that it has no place in maths. Of course we both understand that infinity isn't a number and all the rest, and I know about square root of -1 (is that imaginary number i? Which is really useful in maths and just as silly as infinity? Isn't zero itself in a similar category?) too. Riemann's paradox seems to be just one of many examples in which infinity is used (in the practical sense?) in mathematics. Please do a video explaining how the layman's concept of infinity differs from or is the same as the mathematician's!
@Tiffany Wart Probably no. I have from another maths/physics friend an opening statement like 'Infinity is ubiquitous in calculus and quantum' but in the end he agreed with something like 'Infinity is not used in equations that have practical application.' Then we talked about zero and square root of -1 and for me at least the matter is not well resolved.
I isn't actually silly, it's an actual number that can also be written as 1;90° as it's still a number just in a different direction. Welch labs has a great video on it.
@@rockstarvolkov4256 yes, i isn't more silly than -1. Better, both are relative notions because 1 is the negative of -1 and also the imaginary of -i.
2:20 Not really a paradox, just bad dealing with series that converge conditionally.
That's what the video is about.
Wow so smart
Order / position matters in an infinite series. The first issue arises in the Pi example as well.
You can go to town with brackets and breaking up terms as long as they're calculated at the same point in time as their parent term.
Let's stop for a moment and apreciate the monumental amounts of patience and kindness that the author of video has for those arrogant morons from comment section, who think they know better about mathematics than someone who spend most of his lifetime learning and teaching about it.
Glad you think so and thank you very much for saying so :)
Jacek Jarmasz so is maths based on faith? You're appealing to authority.
If I may, it is not faith more than it is common sense. Arguments of authority are only invalid when the said authority isn't specialized in the subject. I would trust a mathematician about math more than I would trust someone who doesn't have evidence backing what he's saying in the field of mathematics.
You may argue that mathematical theorems are independent of people. I agree but the great thing with math is that when you use intermediate results to solve an exercice, you don't have to always look at the demonstration, because you know that people who are better than you have already done it. In that sense, mathematical results are simply tools you take from your toolbox. I don't know how hammers are made but I do know how to use one. This way of thinking is far more productive than reviewing every math theorem. I can guarantee that not a single mathematician has done that because their job is to build (discover ?) new results. It is a possibility that a mathematician has gone wrong when demonstrating something and nobody has spotted the mistake, but I'd recommend you read about Ockham's razor.
You're the ONE
I'd love to see a follow up video on the generalizations you mentioned at the end.
From Wikipedia: "given a converging series of vectors in a finite-dimensional real vector space E, the set of sums of converging rearranged series is an affine subspace of E"
Super interesting.
Challenge accepted at 6:49. Your series, - 1/2-1/4-1/6-... is just - 1/2 times the harmonic series, so - 1/2-1/4-1/6-... will also diverge. Then your other series 1+1/3+1/5+..., we can use the direct comparison test. If the harmonic series diverges, then 1/2 times the harmonic series will also diverge, that is, 1/2+1/4+1/6+... will diverge. If you compare 1+1/3+1/5+... to 1/2+1/4+1/6+... term-by-term, the series 1+1/3+1/5+... will be larger than a series already known to diverge. Perhaps there's an even simpler argument?
I didn't use a comparison for the series with the negative terms, just an observation that it's -1/2 times the harmonic series. I don't see why limit comparison is easier compared to direct comparison here. Certainly it gets the job done, but it's more machinery (limits) than needed.
Of course I can use observations. Here's the argument in more detail, starting with - 1/2-1/4-1/6-... Hopefully you see that if I take 1+1/2+1/3+1/4+..., multiply by - 1/2, I get - 1/2-1/4-1/6-... We already know that the harmonic series diverges and I claim that multiplying a divergent series by - 1/2 produces another divergent series. I'll prove that statement by contradiction. Suppose - 1/2-1/4-1/6-... converges to some finite number, call it A. Multiplying - 1/2-1/4-1/6-... by -2 implies that the harmonic series converges to -2A, a contradiction.
In the case of the series, 1+1/3+1/5+..., I started out by claiming that taking 1/2 of the harmonic series produces another divergent series 1/2+1/4+1/6+... (I can prove that by a similar proof by contradiction, as above). Now look term-by-term: 1>1/2, 1/3>1/4, 1/5>1/6, etc. The series 1+1/3+1/5+... diverges by direct comparison with the divergent series 1/2+1/4+1/6+...
The integral test works but I think it's like going hunting with a bazooka.
There's an intuition that since the harmonic series diverges, multiplying it by some nonzero number can only produce another divergent series. That is, multiplying infinity by a nonzero constant produces another infinite number. That's why I figured that recognizing - 1/2-1/4-1/6-... is - 1/2 times the harmonic series suffices. The same concept can be used to justify the divergence of 1/2+1/4+1/6+... which I then compared to the second series in question, 1+1/3+1/5+...
As I said, one could use other methods like limit comparison or the integral test but one has to have prior experience with calculating limits and integrals for those to be intuitive.
No, the harmonic series 1+1/2+1/3+1/4+... diverges. Perhaps you're thinking of 1+1/2+1/4+1/8+... (a geometric series) which indeed converges to 2.
+Bryan Leist
You merely state my methodology is not valid and that the argument that I presented above does not hold. Care to tell us why? Also care to tell us why I cannot manipulate infinity (even though I don't need to manipulate any infinities in my proof)? I know how to sum an infinite series, don't you worry.
The issue is clear, the problem is that the series isn't absolute convergent, so the rearrangement trick won't work. You can ONLY do that if the series is absolutely convergent.
you can get infinity -infinity to equal anything
but thanks for proving it
It equals to my credit card Number :0
Yes, you can also make it equal to your credit card number :)
If the series is infinite, how do you just decide to use 400 of one and 2 of the other? That's not rearranging, you've removed things. I guess you could just subtract the remaining numbers from each one, but now your formula would be ∞-∞-∞-∞=π
There's an infinite number of terms, so ALL of the terms are being used, just... eventually. A lot of positive terms are used between each negative term, but there's INFINITELY many terms. So, all positive numbers AND all negative numbers will get used. The negative numbers just, take longer.
You're right Aaron; to mich on a side, so less in the other!
If the quantum of plus is the same of the minus quatum from zero to infinity.
Solid logic...
As a non maths guy, this sounds like essentially he's demonstrated you can't reliably apply the commutative property to infinite series. It only works with a subset of all infinite series.
Am I understanding this right or have I completely missed the boat here?
You are absolutely right!
The1stImmortal you sound like a maths guy to me, thumbs up
You are absolutely correct. The "infinity - infinity" way of posing the question is just a way to make the concept more approachable and show the viewer how math can sometimes be counter-intuitive.
he said exactly that in his video...
I really appreciate the fact that you point out the errors. Lots of people doing math on RUclips using all the properties in all kinds of algebraic structures and make others think 1=2...
the major problem i see is that "infinity" is not a number, but a concept.. and there are plenty of forms of infinity. there is an infinity of whole numbers, an infinify of fractions betweem two single whole numbers... an infinity of numbers that end in 4... and so on... so yeah...
infinity is definitely a number, mathematically
ndgambella nope. infinity, mathematically speaking, is a series of numbers... one that never ends... infinity is not "a number" anymore than "A" is a number... (that is not to say that A and infinity are the same thing either, just that they are not numbers)
That second one is incorrect. The answer to "subtract all the numbers" is -∞, not ∞.
Although I think I take your point. ∞ is the set of all positive numbers. A thing cannot be a number if it is a category to which all numbers belong. It is not a number, but it is its own thing. Hrm.
Well, maybe, yes? I dont think so. Numbers are information, math is something that is working with information, to make it easier and to communicate it, but infinite is an information which can not be solved, It could be a number to say, but It contains infinite information, that means it can not be solved, It has no ending.
MMOplayeerr infinity is not a number. Its a concept, an idea.. It is "forever"... Forever is not a number, its not a time or a distance but rather the concept of a number, time, or distance that is unending. Infinity holds no value on its own, but rather shows us that values will forever grow and expand.
So this video shows us how fucked up it gets when you make 1 mistake
Except there is no mistake :)
Mathologer 3:17?
It's a flaw in our mathematics at the edges. 1 divided by 0 is also infinity, so by this logic, pi = 1/0. You can play all sorts of games with maths at the margins.
I dunno if I am doing a right math here but isn't it π=(1/0)-(1/0)
π≠1/0
hence...
1/0 ≠ (1/0)-(1/0)
coz ∞≠∞-∞
and 1/0≠ -(1/0)
Dividing by zero is undefinable. We can say the quotient of a Division tends towards + or - infinity as the denominator aproaches (gets really close to) 0 but we only aproache it! Try to zero it (which hopefully you won't do again) and you've made one of the greatest and most common among students mistakes in Math. That's by the way how you end up with nonsenses such as 1=2.
edit: IF you are curious enough, check out the video of this excellent teacher to see why.
ruclips.net/video/J2z5uzqxJNU/видео.html
Yeah m8.... 1/0 =/= infinity. Check it using limits.
Yes 1 divided by an infintismally small number number does approach infinity. But what about 1 divided by an infintismally small Negative number? That is just as valid to represent being close to zero but you end up with NEGATIVE infinity. So in 2D space 1/0 = Negative AND Positive infinity. You should see the map for more dimensions as you approach zero.
The absolute value of 1/0 is infinity. 1/0 itself is simultaneously infinity, negative infinity, i times infinity... basically, if you could draw a circle with infinite radius on the complex number plane, 1/0 would be every point on the circumference of that circle.
No this video is absolutely different from that concept wtf
You the first bloke who understands maths the same way i do well done 👍 totally straight forward and understandable simple put
Meanwhile im watching this instead of doing my math homework
then all of your answers are ∞-∞, it'll be always right
Question remains, math homework more or less difficult?
Even math is interesting if it’s a RUclips video
Meanwhile I’m watching this at 1 am when I have a test tomorrow
Bertrand Russell: "Anything can be deduced from a contradiction."
A voice from the back of the room: "Zero equals one. Therefore, you are the Pope."
BR: "All right. 0 = 1. Hence, 1 = 2. The Pope and I are two. Therefore, the Pope and I are one."
u could make it 0=2 too aswell as u can make it 0= infinity or whatever number u want.....
Math is broken...
Understandable have a great day.
RavenMast3r try to break numbers that doesn’t exist physically
Math machine broke
@@aeop not trying To .be smartass but it is possible
The thing is: When you break math, the universe is broken. The mathematical constructs that hold our universe together will decrease in power and they will break, as the multiverse, in which we are living, will free all other universes and there will be a huge collaps that would lead to an explosion with results in a dead multiverse, aka ur dead m8.
As you can see, if you or someone else really broke math, there can only be one possible solution for us to save us from the big problem:
Meth.
Hah
NeonGamerLP but the multiverse hasn’t collapsed though has it? This makes the xenoverse, a verse containing multiverses, collapse which causes the same thing, creating a paradox. This eventually leads to the omniverse collapsing and now we’ve officially broken psychics, creating another infinite loop. Meanwhile we’re all still alive. Let’s do a recap. First the multiverse collapsed and then the xenoverse collapsed. I guess the zeroth thing that happened was us, making infinity minus infinity equal zero. Unfortunately we have already defined this as Pi, breaking math again which doesn’t change anything since math has already broke several times before.
I think the thing that makes the rearrangement anomaly in infinite series is that the rearrangement operations themselves are infinite - the catch is in the 'and so on...' phrase
Brilliant video (and videos in general!) - thank you so much! I would LOVE to see some more analysis!
Coming up :)
+Mathologer are u a teacher
I am one of the math professors at Monash university in Australia :)
Infinity isn't a number; it's more of an abstract concept like love or beauty. And just as you can't subtract love from beauty, you can't subtract infinity from infinity. Only numbers can be subtracted. The math isn't broken, but one's understanding of how it is applied can be.
I disagree. These are infinite *series* of numbers, which add up to infinity. Besides, math is a concept designed by humans to help interpret the universe. Maths simply tries to understand the universe. Infinity is a very useful concept when dealing with things such as the infinite nature of the universe in space and time. Besides, math is what mathematicians make it, not what ordinary people would define as 'making sense'. Negative numbers used to be thought of as absurd, but the concept comes in handy especially if you just think of it as 'opposite'. Antimatter is basically negative matter, if you think about it. Sure, math is about concepts, but about concepts that come in handy no matter where you are in the universe. 1 + 1 will always equal 2 no matter what. Sure, maybe different ways of communicating this could be used but the concept remains the same. Math is also scalable, unlike the concepts of 'love' and 'beauty'. Sure, they are applicable to all humans and sexual life forms, but do not matter in the realms of atoms or galaxies. Maths matters there however, and everywhere else. It and infinity are universal concept, and such are much more applicable in many situations.
@@ataready8810 You're certainly free to disagree, so long as you're not disagreeable. But actually you do agree with me somewhat when you wrote : " Infinity is a very useful CONCEPT." Mathematicians no longer say that divergent infinite series add up to infinity, instead they say that no number N, can be chosen which exceeds the limit to a divergent series. In the past, top mathematicians did use "infinity" but this results in problems, as Cantor inadvertently demonstrated. Infinite sets lead to paradoxes, or actually, antinomies. particularly in establishing a logical foundation for mathematics. For the last century or so, starting with Karl Weierstrass, mathematicians have used precisely defined concepts involving limits and continuity rather than infinity. If a mathematician does talk about infinity, it is because he is speaking informally as an aid to understanding, using a concept that we intuitively understand but which is not mathematically rigorous.
The whole rearranging of terms thing seems way less confusing when one uses Sigma notation instead of x+x+x....
How are you gonna find the variable term?
The sum will still tend to infinity.
And we are left with infinity - infinity
Intuitively, the reason is that the positive sequence gets kinda stretched so it becomes "less dense".
there's a very quick inductive proof right just one arithmetic step away from the thumbnail. pi = inf - inf => pi + inf = inf. If we assume the negative infinity which gets move to the left side is truly infinity, then it would seem at first glance to be an inequality. But when we consider what infinity is, *something* that is larger than any number, then if we try to add any value to it, it cannot get any larger. It is already larger than any number. Thus, you can add or subtract any number from it, and it will still be a "whole" infinity since if any numerical quantity changed it, it could not possibly have been infinite to begin with. Maybe it's not inductive. Maybe it's circular. And hand waving. ;)
Since infinity is a concept and not a number, you don't have circular reasoning nor hand waving. It's all in the definition of the concept of infinity.
Inf - inf is one of known indeterminate forms.
You MUST know the original expression before calculating such a limit.
Why do we waste our time looking at how slight of hand math procedures end up amazing idle minds when the magician proves that 1 = 2 ?
"As long as they match to produce the desired sum, they can be anything."
Stupid question, but I am curious to know, none-the-less.
If pi=Infinity-Infinity,
& the Area of a circle is pi•r^2,
then Area of a circle is
(Infinity-Infinity)•r^2.
Is this a true statement?
If not, why?
@adam spears in maths, infinity is not a number but an idea which represents incalculability. For example all the incremental values between 1 & 2 are infinity. Does this also mean infinity < 2?? :)
@@derekfrost8991 Great explanation. Thanks!
@@derekfrost8991 there is only 1 & 2. No 0. Not in the real world.
@@robb1038 that wasn't even part of the explanation
They're talking bout 1 - 2 and all numbers between (decimals )
Many have already answered reasonably, but here's my take:
That statement is false because of the asymmetry of the equality operation when operating on infinities. "Symmetry" regarding equality means that x = y iff y = x. The symmetry is found for many (most?) sets of scalars (e.g. integers, reals), but for infinities it does not apply. Mathologer convincingly demonstrated that ∞ - ∞ = π, in a sense. However, by the asymmetry of equality, ∞ - ∞ = π does not necessarily imply that π = ∞ - ∞, so we can't substitute it in for π in the circular area formula etc.
Not sure if totally correct, but it's my 2¢.
All this stuff about sums of infinite series fascinate me since I studied it at University. Moreover I really appreciate the way you communicate and the graphical/visual explanations. These contents are challenging but thanks to you encouraging and inspiring teaching the concepts involved become more understandable.
In the first example the regrouping combines each positive element from the series with 2 negative elements from the series. The density ratio of negative to positive terms is 2:1 regardless how many terms are used which gives a result of half what it should be.
I wonder if there is a way of rigorously taking into account the distribution of the elements used in the rearrangement pattern?
After +24 years of engineering mathematics, I’m lost after watching this video.
When you start comparing terms of an infinite series with a condensed version of itself, where you have combined multiple terms into one, you are no longer comparing equivalent infinite series.
You are comparing a subset to the whole, so saying 1= 2 is really saying, part of everything is less than everything.
I don't see how changing the terms of the series can change the sum, but comparing terms to the unchanged series would be different.
Any mathematician knows that not all infinities are equal. The sum of an infinite series, where the terms of one series are of a larger magnitude, or you add up a different number of terms, may both result in infinity, but to say they are the same infinity is nonsensical. No paradoxes here, just fun math manipulations to confuse the non-mathematically minded.
So what step of the process is invalid? You haven't pointed out any invalid steps, you've just said you're comparing a subset to the whole. But the process of getting from the whole to the subset must have an invalid step in there. The invalid step can only be rearranging the terms, since all of the other steps can be proven to be valid.
For example, if you have a convergent series, adding parentheses to group terms together does not change the sum. The sequence of partial sums of the newly grouped series is a subsequence of the sequence of partial sums of the original series. It's a general fact that if you have a sequence which converges to L, then every subsequence must also converge to L. So this shows that the grouping part (which is what gets you from all of the terms of the original down to only some of the terms of the original) is a completely valid step.
The general reasoning that Mathologer used to get a sum of pi also does not involve the step of grouping terms together, so the claim that he is just "comparing the subset to the whole" is incorrect.
@@MuffinsAPlenty - 7:26 How can you honestly say the he doesn't group things to get pi?
He groups the first 76 positive terms in the series, then 1 negative term, then 126 more positive terms, then 1 negative term, etc ...
He is basically summing all of the positive terms, with very few of the negative terms, which is why it sums to a number (3.14...) greater than the original summation (0.69314...).
The invalid step is absolutely rearranging the terms. The original series, where they alternate positive/negative, includes equal numbers of each, but summing to pi includes far more positives than negatives. You are comparing two completely different infinities as if they are the same thing, which they are not.
(all of the terms) 0.69314... = 3.14... (the subset with more positive terms than negative)
We know the series has equal numbers of positive and negative terms, yet the summation to get pi is overloaded with positive terms.
You even state that convergent series, which this is, sum to the same number regardless of grouping, so how does one get pi from a convergent series the sums to ln2?
Perhaps we're using terminology a bit differently here. When I say "grouping" I mean putting parentheses around terms. By reordering, I mean terms go in different positions than they originally were in.
For finite sums, regrouping doesn't affect the sum: a+b+c+d = a+(b+(c+d)) = ((a+b)+c)+d = (a+b)+(c+d), etc.
But for infinite series, "grouping" can affect the sum.
For example, consider Grandi's series: 1−1+1−1+1−1+...
If you group terms like this: (1−1)+(1−1)+(1−1)+..., you get 0+0+0+... = 0
But if you group terms like this: 1+(−1+1)+(−1+1)+(−1+1)+..., you get 1+0+0+0+... = 1
But Grandi's series is divergent because the sequence of partial sums doesn't converge to anything; it just keeps alternating between 1 and 0.
If you have a convergent series, then adding in parentheses (which is what I mean by "grouping") cannot affect the sum of the series.
For finite sums, reordering also doesn't affect the sum: a+b+c+d = a+c+d+b = b+c+a+d, etc.
Reordering can affect the sum of an infinite series, like this video points out. If you keep adding positive terms until you get above pi, and then add negative times until you get below pi, then keep adding positive terms to terms until you get above pi again, etc. that's reordering the terms, because you're changing the order in which the terms appear.
What's really weird is that reordering _can_ change the sum of even some _convergent_ series!
But reordering is not comparing a subset to a whole. Every term from the original series must appear as a term in the new series exactly the same amount of times it was a term in the original series.
The strange thing about infinity is that you can take terms at different rates and still get them all in there. And that's what allows the sum to change.
By adding or substracting infinite series of numbers, you can get anything you want. You can even add an infinite serie of positive numbers and get -1/12...
Be careful with that!! This kind of things can be done only with particular series. There exists series that converge to a unique number (or to infitity), so you can not generalize the Riemann theorem to any infinite serie of numbers. And the -1/12 thing is false :O
Sorry, chum. A series with all positive terms will always produce positive partial sums and that sequence, being positive, cannot converge to anything negative.
low quality bait
+Gilles Soulet You can add (or multiply) infinities and be totally right. It's only subtraction (or quotient) that doesn't work. The disjoint union of two infinite sets is necessarily infinite. Therefore infinity plus infinity is infinity. Cartesian product of two infinite sets is necessarily infinite so infinity times infinity is infinity. On the other hand, if you subtract an infinite subset from an infinite set you get something that may be finite (of any size, including 0) or infinite. That's why it's called indefinite. For example if you subtract the even numbers from the integers you are still left with an infinite set: the odd numbers. But if you subtract the nonzero integers, you are left with a finite set - the singleton {0} or if you subtract everything you are left with the empty set.
+bookashkin physicists would seem to disagree with you on that.
The paradox is not all that complicated but people overthink it
Thank you so much for uploading this video. It is helping me get through the pandemic!
Uh???? + Lemniscate minus Lemniscate?....
so (+∞) - (∞) = ∅ ???
Yeah, So the solution is the outro to Bohemian Rhapsody!
Infinity is not a number. Thus, there is no definition for subtraction of infinity. You might as well say that tuna fish minus asphalt = pi.
duffypratt Infinity is not a real number, but all numbers, like words, are made up. We get to define terms in math in a way that is convenient and useful to us.
i like your shirt, had a little giggle when i saw what was happening.
I have a collection of almost 100 math themed t-shirts :)
Fantastic! i need to get a collection going!
Herr Mr. Polster,
I am very enthusiastic about your lectures. You have a very thorough, but also a very funny/joyful way to explain the problems and solutions.
Thank you!
Greetings/Grüße
Dr. Sebastian Kühnert (working in the field "Functional Time Series Analysis")
A little funfact: there are infinitely many ways to get pi, in this way. 🤯
Not mindblowing in the slightest
Amazing Editing on this one I appreacited it!
I did something similar with integrals to get ln(2).
The integral from 0 to ∞ of 1 = ∞
The integral from 0 to ∞ of tanh(x) = ∞
Subtract the second from the first and make them one integral to get the integral from 0 to ∞ of 1 - tanh(x) = ∞ - ∞ = ln(2)
Can you write this with series?
Anybody else, at some point in the video, have their eyes glaze over and the sound "blurr"?
Videos like this are good reminders for me that I'm never gonna be a mathematician or a physicist.
Can you make a video about Zeta function plz?! I would love to understand the problem that Rieman left and why it is so hard for Mathematicians to solve!!
Isn't infinity-infinity technically whatever you want
Yes, usually, whenever it is possible to make sense of infinity minus infinity you find that the result can be anything you want. Still this "whatever you want" can carry a lot of meaning as in the case of infinite series :)
pi is niether here nor there in this context, (anything) = [infinity] - [infinity] and not just pi exclusively.
That is absolutely correct but we are here on RUclips and I think this is a great RUclips title :)
0 = 0
2 * 0 = 5 * 0
2 = 5
this is short sense of all math paradoxes
just find zero in your equations
in a ring 0 is not regular
Not always
Marty & Friends - Thanks again! This is a great example of how seemingly complicated and/or mysterious (or mind-bogglingly inscrutable) maths and numeric logic can be explained with elegantly yet deceptively simple examples. >
+
Why deceptive, i.e., what's hidden or unobserved or unproven? Of course, Marty didn't need to repeat the rules, axioms, and well-known principles of arithmetic required for the results. However, what was never shown or explained (even by Riemann, among all other super-stars of maths) is why.
+
In other words, what are the deeper enabling principles of maths and numbers, and their potentials? Also, how can we discover them, if that's even possible? We can also wonder why it might matter, especially if the super-stars didn't care, or we could wonder what else the answers could tell us about maths, numbers, symbols, and reality (etc.).
+
First, it may seem obvious that--to exist and "work"--the logical operational principles of maths require enabling principles of form, structure, functionality and, maybe, relativity (enabling the relations, functions, etc.). How we find those deeper principles requires observation and some study. For example, we see the forms of numeric symbols and the operational symbols (+, --, =., /, etc.), and we can see the functional results of their relations and some of the reciprocity. Yet, clearly, even the greatest geometers and giants of maths failed to see the virtual structural nature of numbers and maths and why they do what they do. Why that matters is revealed by a) all the failures to solve the RH problem (among many others), b) the long failure to understand the reason, c) the inability and ability to recognize insufficient proofs (of RH, etc.), and d) over 500 other theorems and results related to RH and R.'s zeta function (etc.), even without proof of RH. Of course, virtually all of Marty's videos give us potentially infinite opportunities for seeing and discovering the nature of reality, maths, geometry, trig, numeric logic (etc.), and their intrinsic relativity.
+
What the enabling meta-logical principles, the quest, and its results can tell us is too vast for a comment, but it's the basis of 2 papers @ (ORCID.org/0000-0001-5029-7074).
Great video but can you stop saying log 2 and writing ln2 its confusing
But thats how it's written
@@Cuzjudd I think they meant there's a difference between ln(2) and log(2).
Ln(2) is log to the base e of 2, whereas log(2) is log to the base 10 of 2.
@@benterrell9139 Pure Bullshit.
There is no common notation. ln(x) can be written down by log(x).
Others just use log(x) not with base e but with base 10, which is in common notation called lg(x).
So there's a common notation for ln(x) and lg(x), but not for log(x).
You just have to use your brain to get the context whenever someone writes log(x).
I think it depends a little on the field you're studying, but in general ln (the natural logarithm) occurs so much more often than log (base 10 logarithm) that people often call ln "log". In physics you basically never see base 10 logs, so we always just refer to ln as log. It's usually clear from context (there's no reason you would see a logarithm with some other base here, for example).
Math: - I don’t feel so good...
This also works with 1-1+1-1+1-1+1-… and it’s much faster to see, and you can even make it sum to infinity very quickly. What kind of sequences that diverge like that can you do this?
You can't really get any number though, just integers
That infinite series isn't convergent so rearranging by bracketing is invalid.
If you do and try to compute a sum, you find crazy results, like:
It's obviously 0 as it cancels out,
but it's 1 as 1 - 0 is 1.
Adding all the positive terms together and negative ones leads you to a meaningless infinity - infinity.
It's easy to see any finite series length n will add to 1 or 0 and one n+1 will always flip to the other value. That remains true as n -> infinity.
@@raphaelcardoso7927 As erroneously rearranging gives (infinity - infinity) then any number is equally as valid in the invalid calculation, because ¼ + infinity is infinity when you allow rearranging.
Divergent sequences have sums for finite number of terms (1 or 0 in this case) but not infinite terms as it's either a non-converging 1 or 0 never tending to a single value.
In other words an integer like 10 is no less wrong than π, because you permit invalid operations.
Totally going to have to do the calculations for this myself just so I can wire in some more maths, this is epic, love number patterns.
Doesnt that mean that pi+inf = inf? you could say that about every number then
World :infinity
Россия: бесконечность
Казахи:спяший восемь?
Шрифты для того 🇹🇬
I have no idea what I wrote I don’t speak Russian can you say what it means
ッッ
World: infinity
Russia: infinity
Kazakh People: An 8 that sleeps
Infinity minus infinity equals zero.
*Change my mind*
∞≠∞ but as well ∞=∞
It does equal zero, correct ... and also every other number as well. If you have enough of nothing you get something! "We existing" is proof of that!
The null properties of addition and multiplication are exceptions to the inverse properties. In general, if you add a number to its opposite, you get 0, and if you multiply a number by its reciprocal, you get 1. But a null addend such as infinity has no opposite; and a null factor such as 0 has no reciprocal.
Hahahaha that's where my mind was going too... 0 and.... well?
Where do I go to add the infinite symbol to my key board? Ψ
Pizza!!!
My mind just keeps going round in circles on this one.
hold on at 6:41 the negative series and the positive series are going with convergent how can you say they add to infinity???
No, both the positive and the negative series are divergent (google harmonic series for a proof or check out my video on "Apu's paradox" :)
I see u are right, it looked for a second as a geometric progression with a common ratio of less than 1
Thanks I will btw
But they wouldn't end up at infinity... they would basically end up at one devided by infinity so technically zero.
kartik aalst EXACTLY! I WAS GOING TO SAY THE SAME.
Hi sir. Don't know why you reminders me breaking bad guy 😃
how can you express pi as a series of finite rationals? that would imply that pi is rational. and it's not
Untrue. It's an infinite series. There are more infinite series that give you pi. For example the Leibnitz Formula for pi.
you re using rational numbers but because you use infinitly many you end up with an irrational number
yeah but at one point the sum is finite and it reads 3.1415... it was ambiguous whether the sum was exactly equal to pi or only the first few digits were the same. that's why i was confused
Is it not because pi is actually what the series tends to as the number of terms tends towards infinity? Cutting the series off with finitely many summands would lead to a rational number, but we have infinitely many terms so it's okay.
Uh? The sum of 1/(2^n) including 2^0 is 2.
I barely have a grasp on grade 10 math (aka algebra) and I still watch these videos even though I don't know whats happening, what is wrong with me?
Maths is broken! That's what I'm telling my calc prof.
Math isn't broken. Commutativity over addition e.g. a + b = b + a etc., was only defined for a finite number of numbers. In fact commutation still works with absolutely convergent series such as the geometric series.
+Chris Seib I get it. I was just commenting on the funny notion of broken maths.
mathematicians have a weak sense of humor :P
Bad mathematicians*
Yes math seems broken. Because I cannot multiply or devide negative numbers...
Maybe because the series is defined from it's chosen endpoint back toward zero, so of course rearranging the terms ruins the sum, since you end at a different interval, and worse still, close to the larger fractions where it counts as much as a half.
I think you still have to get into the swing of things here. Nothing gets "ruined", nothing is "worse", etc. I'd say relax a bit. There is this standard definition for what the sum of an infinite series is supposed to be that I talk about at the beginning. Everything else is just about being logical and following our nose (and marvelling about what we find along the way :)
I don't really know what you are talking about. I was trying to reaffirm your point but obviously you've misunderstood what I meant and (somehow) taken it as an insult. I wasn't even addressing you. Don't tell a first time viewer to 'relax', and don't condescend.
Also, I don't know jack about math, so whatever you said is probably true. My comment was me trying to get my head around it, but I'll try that elsewhere in future.
I've just replayed the video because I couldn't really remember what it was, and I realise now English is your second language, so I'll take that into account, hopefully so do you.
No, problem at all and I was definitely not trying to be condescending. Try to look at it from my perspective. I've been teaching this stuff for the better part of my life and I've never had any complaints from my students that I am not expressing myself clearly (on the contrary). At the same time any video on anything to do with infinity always attracts a huge number of cranks who start off their comments by saying "You are lying!", "This is BS!", "Infinity is not a number!", "Mathematician don't know what they are doing!" etc.
I really get very annoyed by this and I think it is important to set the record straight to make sure that this comment section is not swamped with such nonsense. Also it is very hard to tell when a comment with negative connotations is left by a crank or just somebody who is honestly struggling to understand what is going on.
Let's make it 50/50 my frriend.
1-1+1-1+...= 0.5
Look up cesaro summation, that gets you to 0.5