Nihil novi sub sole. This result was already known since the 18th century, but I've only heard about it last week: `1+2+3+4+\cdots=-1/12`
The infinite sum of all the natural numbers is somehow equal to negative one twelfth. I happened upon this crazy science on Youtube:
I tried to make sense out of this. I wanted a graphical representation of this infinite sum. If the result is not infinity, there should be a way to visualize the limit, right?! Well, I failed to find one. The only thing I could do was to show that the growth is logarithmic: $$\begin{align} A[n]&=n, \forall n\in\mathbb{N}_{>0} \\ B[n]&= A[n]+\cases{0 & \text{if } n = 1 \cr B[n-1] & \text{if } n > 1}, \forall n\in\mathbb{N}_{>0} \\ C[n]&=\frac{A[n]}{B[n-1]}, \forall n\in\mathbb{N}_{>1} \\ \\ A&=1,2,\ 3,\ 4,\ \ 5,\cdots \\ B&=1,3,\ 6,\ 10,15,\cdots \\ C&=\ \ \ \frac{2}{1},\frac{2}{2},\frac{2}{3},\frac{2}{4},\cdots \end{align} $$ Adding the `n`-th integer (thus `n`) to the sum means adding `2/{n-1}` time(s) the current sum to the sum. In other words, the sum increases by a factor of `{n+1}/{n-1}` at each step. When `n` tends towards infinity, this factor approaches `1`. The infinity we're adding is nothing but an epsilon...
Graphically, if we plot `B[n]` against `n`, the gradient of the curve (it's not actually a curve, just a set of points, but I guess the notion of gradient/slope still makes sense in this discrete context?) is equal to `n` (or `n-1`, or `n-1/2`? Not sure. Not important.) at each step. The gradient is ever increasing, so the series seems to increase exponentially. Absolutely no sign of a plateau; the slope gets more and more "vertical", forever. No limit. Yet our analysis above shows that the growth slows down. Well. The growth rate does decrease... but a smaller growth factor (which is by definition always greater than `1`) multiplied by an ever-bigger base yields an ever-bigger result. Oh, and not a trace of `-1/12`!
I was hooked. I youtube'd some more. The following video vulgarizes related mathematical concepts such as Cesàro mean, Cesàro sum, Riemann zeta function:
John Baez elaborates on the relevance of this equivalence in quantum mechanics, and how this result yields that string theory is 26-dimensional:
I am not very good at maths, nor knowledgeable in physics. Yet I find these explanations surprisingly clear and simple. It does makes sense... And at the same time it does not make sense. How can we can get a negative result by summing only positive numbers? As a programmer, my first thought was that we ran into a glitch in the Matrix. More likely, this is a clue about how the universe is really shaped. There is no reason for human mathematics to accurately represent a universe largely unknown to humans. If there are indeed 26 dimensions, why would our numbers only have two dimensions, "real" and "imaginary"?... [Edit 2016-10-25: I've just read about hypercomplex numbers] Who knows, a great discovery might be for tomorrow. That's exciting! It makes me want to brush up on my calculus.
I tried to make sense out of this. I wanted a graphical representation of this infinite sum. If the result is not infinity, there should be a way to visualize the limit, right?! Well, I failed to find one. The only thing I could do was to show that the growth is logarithmic: $$\begin{align} A[n]&=n, \forall n\in\mathbb{N}_{>0} \\ B[n]&= A[n]+\cases{0 & \text{if } n = 1 \cr B[n-1] & \text{if } n > 1}, \forall n\in\mathbb{N}_{>0} \\ C[n]&=\frac{A[n]}{B[n-1]}, \forall n\in\mathbb{N}_{>1} \\ \\ A&=1,2,\ 3,\ 4,\ \ 5,\cdots \\ B&=1,3,\ 6,\ 10,15,\cdots \\ C&=\ \ \ \frac{2}{1},\frac{2}{2},\frac{2}{3},\frac{2}{4},\cdots \end{align} $$ Adding the `n`-th integer (thus `n`) to the sum means adding `2/{n-1}` time(s) the current sum to the sum. In other words, the sum increases by a factor of `{n+1}/{n-1}` at each step. When `n` tends towards infinity, this factor approaches `1`. The infinity we're adding is nothing but an epsilon...
Graphically, if we plot `B[n]` against `n`, the gradient of the curve (it's not actually a curve, just a set of points, but I guess the notion of gradient/slope still makes sense in this discrete context?) is equal to `n` (or `n-1`, or `n-1/2`? Not sure. Not important.) at each step. The gradient is ever increasing, so the series seems to increase exponentially. Absolutely no sign of a plateau; the slope gets more and more "vertical", forever. No limit. Yet our analysis above shows that the growth slows down. Well. The growth rate does decrease... but a smaller growth factor (which is by definition always greater than `1`) multiplied by an ever-bigger base yields an ever-bigger result. Oh, and not a trace of `-1/12`!
I was hooked. I youtube'd some more. The following video vulgarizes related mathematical concepts such as Cesàro mean, Cesàro sum, Riemann zeta function:
John Baez elaborates on the relevance of this equivalence in quantum mechanics, and how this result yields that string theory is 26-dimensional:
I am not very good at maths, nor knowledgeable in physics. Yet I find these explanations surprisingly clear and simple. It does makes sense... And at the same time it does not make sense. How can we can get a negative result by summing only positive numbers? As a programmer, my first thought was that we ran into a glitch in the Matrix. More likely, this is a clue about how the universe is really shaped. There is no reason for human mathematics to accurately represent a universe largely unknown to humans. If there are indeed 26 dimensions, why would our numbers only have two dimensions, "real" and "imaginary"?... [Edit 2016-10-25: I've just read about hypercomplex numbers] Who knows, a great discovery might be for tomorrow. That's exciting! It makes me want to brush up on my calculus.
Mathologer's just posted a video that criticizes Numberphile's methodology: https://www.youtube.com/watch?v=YuIIjLr6vUA
ReplyDeleteIt seems that in the end, it's not really a correct sum. -1/12 has some meaning (see the end of the video), but we shouldn't think of it as the result of a plain old sum...