Відео

1:32 if my friend already has some debts and I am removing it then it must result in zero. How it is positive??? 😅😂😂

The following three articles explain that every number is divisible by zero. In doing so, they refute the claims in this video. I recommend reading these articles if you want to learn division by zero from different perspectives. 1.Division by zero in the light of the five fundamental principles - Beş temel ilkenin ışığında sıfıra bölme 2.A study to prove that the denominator can be zero in fractional numbers - Kesirli sayılarda paydanın sıfır olabileceğini kanıtlamaya yönelik bir çalışma 3.The problems created by zero in the division operation, their reasons and an attempt at a solution - Sıfırın bölme işleminde oluşturduğu; sorunlar, nedenleri ve çözüme yönelik bir deneme çalışması

According to Boolos' original formulation, Random doesn't answer Yes or No randomly, but rather tells the truth or lies randomly, so the original puzzle can be solved in two questions!

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And yes. I'm not racist

First!

I'm about level 4

20N2024 12:40

OK, you had me at Levels 1 through 4. Level 5 made me cry.

From the perspective of quantum mechanics and ancient spiritual wisdom, nothing is everything. so 0 times anything can technically become it. That's what I believe.

Youre about 10 years too late on this video

Love example of group with clock

The world where 1+2+3+4+&c = -1/12 is the same where 1+1=0, and it's easier to prove this than to prove that equality. Not talking about modular arithmetic, just regular high school maths.

Please do not treat a number as a limit

1+2+3+.... is infinity not a negative fraction.....

I never passed algebra.

e 13:22

This is the best explanation of the -1/12 I’ve seen although some of it is above my intelligence! It’s good to see the example of 1+1+1+1…… This may be a dumb question but I’ll ask it anyway! We have all seen the ‘How to Disprove Pythagoras’ puzzle. The one where you put a a load of steps along the hypotenuse of a right angle triangle’ it doesn’t matter how many steps you put in the sum is always the opposite + the adjacent! E.G. if it is a 1,1,sqrt(2) triangle even with millions of steps the length of the hypotenuse will be 2!

I basically use the level one explanation verbatim when I introduce e to my math students. Since it directly involves money, it is more interesting to most of them and it's a handy way to explain compounding interest at the same time. Two birds with one stone 😂

yet another proof that: "repeat this infinitely many times" is not well defined. My favorite example: you have a bag with infinitely many billiard balls (numbered 1,2,3,4 ... etc). Step: remove 1 ball from the bag. Repeat the step infinitely many times. How many balls are left? a) zero -- since you take the next numbered ball each time b) infinitely many -- you only take the even numbers. c) seven -- you only take balls numbered 8 and above. d) anything else. All correct.

{1n+1n ➖ }+{2n+2n ➖ }+{3n+3n ➖ }+{4n+4n ➖ }+{5n+5n ➖ }+{6n+6n ➖ }+{7n+7n ➖ } {2n^2+4n^2+6n^2+8n^2+10n^2+12n^2+14n^2}={56n^14➖ 1}/12=56n^14/12=48n^14 2^24n^14 2^12n^7^7 2^6n^3^4^3^4 2^2^3n^3^2^2^3^2^2 1^1^3n^1^1^1^1^1^2 3n^2 (n ➖ 3pin+2).

When mathematicians wander in to philosophy.

Even as a 1st year math student I understood less than 20% of what he said

To verify if it's true, put dollar signs on each side!

An easier method. lets say 0.9999999.....=x so 10x=9.9999999...... 10x - x = 9x 9.99999999........- 0.999999999....... = 9 9x=9 x=1

Cool. Btw, does it have anything to do with regularisations of sums of functions, in which these are regarded as distributions? (You integrate them against some test functions with good properties, which allows you, say, to integrate by parts and then derivatives of the series converge better, than you can integrate by parts back)

Best explanation I’ve seen. Regularization is hugely important.

The microwave example is actually very good. Noone would deny, that it takes half the max energy on the food, whatever happens between zero and infinity!

1+2+3+4+...+ = infinity. It's unbounded. You can't group something that doesn't converge to make it meaningful.

It is wrong to say 3 + 4 = 2 unless we also say (mod 5) at the end, informing the reader of a non-standard context. So, I think 1 + 2 + 3 + ... = -1/12 is missing some notation to provide this context. Maybe as simple as stating the function maps N -> Q (though the zeta function gets there via complex variables)

off is not minus one, it is plus zero

I don't think it's amazing some function be analytically extended and it has some value (-1/12). It only looks amazing when you put it next to a divergent sum and unfairly use an equals sign.

It could be also -1/8 as shown in the blackpenredpen's video 1+2+3+4+5+6+7+8+9+10+11+...=S 1+(2+3+4)+(5+6+7)+(8+9+10)+...=S 1+9+18+27+...=S 1+9(1+2+3+4+...)=S 1+9S=S 8S=-1 S=-1/8 In general the moment someone calls that the sum is equal to whatever, he has to define what that "equal" means.

Of course, in nonstandard analysis, we can say that the standard part of the sum 1+2+3+... is -1/12, separating the standard and nonstandard parts. In some sense what we are saying is that the sum is "an infinite quantity, which we can separate out, minus 1/12." I once used the nonstandard identity 1 + 2 + 4 + 8 + .. = -1 (again this can be established by zeta regularization) in an argument trying to define (for a programming language) the results of bitwise Boolean operations in a system that admitted arbitrary-precision integers. I got there in part by pointing out that it holds (for ordinary binary arithmetic) in any finite word size. From there I could define what it would mean to do AND, OR and NOT among the bits of signed arbitrary precision numbers, using the identity ~x = -1 - x. That identity is meaningless in the reals, but for 2-adic numbers, it's extremely useful!

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Hey nice, you made a video about the -1/12 sum I suggested in a comment on one of your videos several months ago. I had recently watched a then new Numberphile video about a weighting system they made, based on some work that Terence Tao had done, that derived -1/12 from the traditional sum of 1+2+3+etc. using a cosine term in front of each summand. I'm not sure if this video was a response to my comment specifically, but I really enjoyed your take regardless. I've done a lot of applied math at my jobs and during graduate school, so I love seeing how theoretical math can oftentimes be useful in applied math and physics, oftentimes decades after it was deemed "useless."

The partial sum of the sequence is n(n+1)/2. If you allow the variable to be all real numbers instead of just non-negative integers, then the formula will give negative values between -1 and zero. Guess what the value of the integral of x(x+1)/2 between x = -1 and x = 0 is. Yep, -1/12!

It’s -1/12, you just need to relax what you mean by “sum” and “equals”. Which seems like cheating but we actually do it all the time.

Yall wanna know why humanity is stuck at a type 0 civilization level? It is because access to knowledge is often locked behind paywalls just like with "BRILLIANT", and learning is controlled by rigid structures. Imagine if education were free and stress-free, where people could learn at their own pace, focused on what genuinely interests them. This freedom would make learning enjoyable, which would foster curiosity, critical thinking, and a much broader skill set across society. Instead, we live in a system where nearly everything, including education, comes at a cost. In Norway, for example, students pay up to 70% of their income just to keep a roof over their heads-it's absurd. If education were truly worth the hassle, it would be accessible to all, without forcing students to jump through financial and structural hoops. With a society more empowered to learn and think critically, we could make significant strides toward becoming a more advanced civilization. All this because people need their greed to be satisfied instead of thinking big as a freaking race.

This is weird because 1+2+3+4+....., is not equal 0 +1+2+3+4+..... 😂😂😂

In the physics case there is one ingredient missing called "renormalization". This is the "throwing away" of the infinite part of the sum. The regularization mentioned here is more or less identifying which infinities occur. The formal treatment of infinities and when it's ok to throw them away is highly important in past and current theoretical physics and there is no "good" unique way to do it. Some version of this famously appears in classical magnetostatics (maybe even earlier) for the vector potential of an infinite wire. More recently this continues in quantum field theory and general relativity and why they don't "work" together as theories in some technical sense.

In a simulation, everything is possible - even something as incongruous as this.

Thanx for this amazing video about the fascinating correlation between the sum of all natural numbers and -1/12! Especially, all these graphical illustrations are impressive! Nevertheless, using the regular equal-sign "=" to describe this relation feels wrong to me, because "+" is (at least in my understanding) the math-symbol for just regular _arithmetic_ summation. Cesàro- and Abel summation, or - all the more - analytic continuation of Riemann's ζ-function is something very different. Therefore, I'd stick with 1+2+3+... ≠ -1/12 Am I wrong? 🙂👻

usefull to make view on yt.

There are different ways to calculate this sum - but all end up with -1/12. This is really remarkeble and amazing. Especially because it is completely counter -intuitive.

A Nice Depiction of Gibbs Peaks : ua-cam.com/users/shortsOGwN3Zb1sXQ?si=44YffkPiUdcCcG6j

Hardy's book "Divergent Series" plumbs the depths of regularization, Norlund, Abel, Cesaro means etc, Tauberian theory and much more.

3:45 Isn't the Cesaro sum of 1 - 2 + 3 - 4 + ... 0? The partial sums are 1, -1, 2, -2, 3, -3, 4, -4, so every even-positioned term is -1/2, and the others approach +1/2.

-1/12 is an obvious paradox. But all paradoxes are the result of a basic misunderstanding or a misuse of theory. I just ignore the so called proof. Math trickery.

I don't study math, but rather Latin and Ancient Greek. I love your channel. I've learnt a lot! Your voice being so calm is even ASMR for me hahahahahaha. Keep on the good work! 💪🏻