In the End, It All Adds Up to -1/12 (1+2+3+4+ ··· ∞ = -1/12)

This was pretty cool to watch, even if I was lost before half way.

saneguy
So you have shown that programming is not mathematics.

So you have shown that you have no idea how a computer works, or understand that a computer is a glorified calculator. The name "computer" actually comes from the word "compute" which means "calculate". A computer's sole purpose is to solve arithmetic and or logical operations, and it does so by means of programming. Programming itself is a form of Discrete Mathematics... en.wikipedia.org...

The code I wrote in my last post is the equivalent to typing the equation (1+2+3+4+5...) in a calculator forever towards infinity. It logically proves the value of "sum" will never be negative... it's logically impossible for it to be.

The variable "x" is only INCREMENTED, never decremented, by a single whole number.

The variable "sum" is only INCREMENTED, never decremented, by the value "x".

The above facts prove "sum" will never be negative, and will never be a fraction. It will always be a positive whole number. The value of "sum" will never be -0.0833333333... which is a decimal representation of -1/12.

reply to post by WeAre0ne


Just because a computer is a calculator does not mean it's capable of doing all types of mathematical calculations. You can make it increment the number over and over again but it'll never give you the true answer to 1+2+3+4...∞.

Whether you think the answer is -1/12 or infinity, it wont give you either of those answers will it? It doesn't understand such abstract equations which continue for infinity, there is no way to encapsulate an infinite series into a simple set of calculations.

If you stop the program at any point it'll just give you a large number, which is wrong. In fact it can't even tell you the correct result for 4/3, it'll just give you a floating point number which is an approximation of the real answer, ignoring the fact that the 3 repeats for infinity.

They're right but the video has been dumbed down to the point it makes no sense (i.e treating a divergent series as a convergent one and not explaining the reasons for S - S2 etc, no Dirichlet conditions..).

The Ramanujan summation and Riemann Zeta function gives a far clearer proof of this, with the former being quite easy for someone with basic maths to follow (RZ takes a couple of years to work up to).

There's a decen follow up here that tackles the problems in the video - www.nottingham.ac.uk...


Strange how many people seem to think they know better than professors just because they can't follow a proof. Though not really surprising given how the education system teaches Maths completely wrong until Degree level.


The beauty of Mathematics is you can teach yourself - instead of dismissing this, work through the equations and examples yourself to find out how this is the answer.

EDIT: Hats off to ChaoticOrder and Arbitrageur for providing good, clear descriptions of what's going on here.

ChaoticOrder
You can make it increment the number over and over again but it'll never give you the true answer to 1+2+3+4...∞.

Whether you think the answer is -1/12 or infinity, it wont give you either of those answers will it? It doesn't understand such abstract equations which continue for infinity, there is no way to encapsulate an infinite series into a simple set of calculations.

There is no true answer to 1+2+3+4...∞. By definition it is a never ending equation, or never ending computation. Infinity is not a number, it is a concept of unending continuation. The word Infinity derives from the Latin word infinitas which means "the state of being without finish". So this entire topic is based on a flawed idea to begin with, that there is an answer to the equation.

The only thing we can do is observe and understand the computation itself. We can understand the computation via computer code, or by physically watching someone type it in a calculator forever, or other means. By observing the computation in action we can form logical conclusions of what the equation will or will not equate to if you stopped the computation at any point (which would not be the answer).

For example: If we watch an immortal guy type in a calculator 1+2+3+4+5... we can conclude he will NEVER press the minus button, only the addition button. He will never add a negative number. He will never add a decimal. And he will never press the equals button. We then can conclude, that if we stop the computation at any point (even though we shouldn't stop) the answer will never be a negative fraction, only a positive whole number.

The same with the computer code I wrote. while(true) is a never ending infinite loop, like an immortal guy doing an action for all time (infinity). sum is the current value of the computation in action. x++ is like the number the guy will type in the calculator. += is the addition button on a calculator. All there in one logical absolute form. Log the value of "sum" during the computation to view it, and you will see, it will never be a negative fraction, ever.

Luckily this equation is very simple, it is just adding all possible numbers together. With that we can make several conclusions... But numbers are infinite. So trying to find the true answer to the equation in question is similar to asking, "What is the last number??". There is no last number...

We have to be happy and accept the fact that there is no answer, but we know what the answer can and can't be if there was an answer.

It can't be negative, it can't be a fraction. Period.

ChaoticOrder
Just because a computer is a calculator does not mean it's capable of doing all types of mathematical calculations.

B.T.W. I never claimed a computer can do all types of mathematical calculations. But it can do almost all of them if you are an experienced programmer and mathematician. Luckily this equation is simple addition.

ChaoticOrder
You can make it increment the number over and over again but it'll never give you the true answer to 1+2+3+4...∞.

In a manner of speaking, it will give you the true answer to 1+2+3+4...∞. You just have to leave the computer running for infinity. Which is forever.

ChaoticOrder
In fact it can't even tell you the correct result for 4/3, it'll just give you a floating point number which is an approximation of the real answer, ignoring the fact that the 3 repeats for infinity.

It doesn't ignore the fact that 3 repeats for infinity. If you wanted it to, it will compute the result of 4/3 to infinite decimal places, but none of us have time for that. And that is not a limitation of computers, its a limitation of the decimal system.

WeAre0ne

ChaoticOrder
You can make it increment the number over and over again but it'll never give you the true answer to 1+2+3+4...∞.

In a manner of speaking, it will give you the true answer to 1+2+3+4...∞. You just have to leave the computer running for infinity. Which is forever.

These are infinite series though - it's far quicker to calculate by hand and plot graphs based on when n tends to infinity. Hence to convergence from a divergent series.

WeAre0ne
In a manner of speaking, it will give you the true answer to 1+2+3+4...∞. You just have to leave the computer running for infinity. Which is forever.

Hmmm, I'm not sure I can wait that long for the answer. I'm not sure there is an answer, since it's an abstract concept. But yeah no matter how long you let the program run it's never going to end up with a negative number as long as it's summing positive numbers.

On the other hand if you ask the program to fit the curve of the sums and find the y-axis intercept of the curve (when n=0) it should come up with -1/12.

bastion
These are infinite series though - it's far quicker to calculate by hand and plot graphs based on when n tends to infinity. Hence to convergence from a divergent series.

The graphs seem to be a good way to look a this, however, note that the Ramanujan sum and the zeta regularized sums appear on the graphs I posted on page 3 when n tends to zero, not when n tends to infinity. When n tends to infinity the graph tends to infinity, so you can visualize all three sums at the same time.

reply to post by WeAre0ne

It doesn't ignore the fact that 3 repeats for infinity. If you wanted it to, it will compute the result of 4/3 to infinite decimal places, but none of us have time for that.

Even if you could leave a computer running forever it wouldn't have the ability to do that calculation because it has a finite amount of memory. You cannot store an infinite series in a computer because it would require an infinite amount of memory. So it does ignore the fact it repeats for infinity because it has no other option but to ignore it and store a finite amount of precision.

Consider this:

x = 0.999...
so
10x = 9.999...

Thus:

10x - x = 9.999... - 0.999... = 9
so
9x = 9

Thus:

x / 9 = 9 / 9 = 1

Hence:

x = 1

And there are many other ways to prove that 0.999... = 1 if you don't believe that example. However that is not something you can mathematically prove with a computer, because you have no way of encapsulating an infinitely repeating decimal such as 0.999...

Consider this code:

float x = 1/3;
float y = x * 3;

In many programming languages (not all) you would get something other than 1 as the value for y because x doesn't have infinite precision. x would have a value of 0.333333 but it wouldn't have an infinite number of 3's, so when it does x * 3 it doesn't have the necessary precision required to realize that the answer should be 1, it simply thinks the answer should be 0.333333+0.333333+0.333333, which is 0.999999. But of course we know that 3 multiplied by x should be 1, so it's giving us the wrong answer. This stems from the fact that it cannot handle infinite series, and the languages which do produce 1 are simply guessing that you want 1 when in reality you might have explicitly set x as 0.333333 and you actually want y to be 0.999999, so it messes up your calculations in the opposite way.

1+2+3+4+5+6... = -1/12
if it works for you, I don't care... calculate with it.


but I will not, let my tell you why I see it different

S1 = 1-1+1-1+1-1+1...
what is S1 here... I see it as a structure, forget the numbers for a moment,
S1 is this whole infinite ( 1-1+1-1+1-1... ) structure,even if you average it to 1/2

S2 = 1-2+3-4+5-6+7-8...
what is S2 here... I see a totally different structure here even if you average it to 1/4

S = 1+2+3+4+5+6+7...
another structure... the whole thing

this one is an apple this other a pear
My grandpa told me not to add or subtract apples and pears, but I can add or subtract the number of fruits after all

but as I said, if you want to add 1 apple to 1 pear and get 1 orange, got for it!
you mathematics/physicists are some extraordinary guys, for you, you can even travel a negative velocity in some 10th dimension...

Personally, I find the result that 1-1+1-1.....=1/2 makes sense. It's like saying if you jump from 0 to 1 forever thats equivalent to 1/2. It's kind of a cool way to deal with infinity.

saneguy
Personally, I find the result that 1-1+1-1.....=1/2 makes sense. It's like saying if you jump from 0 to 1 forever thats equivalent to 1/2. It's kind of a cool way to deal with infinity.

sure the average is 1/2 if you think of it as a whole, frequency 0 to 1 to 0 to 1

In colloquial language average usually refers to the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However, the word "average" can be used to refer to the median, the mode, or some other central or typical value. In statistics, these are all known as measures of central tendency. Thus the concept of an average can be extended in various ways in mathematics, but in those contexts it is usually referred to as a mean (for example the mean of a function)

1-1+1-1+1-1... = 1/2
so this is one structure or central tendency


1-2+3-4+5-6... = 1/4 however is a different average therefore a different central tendency

as I said in my preview posting
apples and pears

Miniscuzz
It was all basic Algebra even a 5th grader could understand.

"Basic algebra a 5th grader can understand"....

If you have 1 apple, then add 2 apples, then add 3 apples, then add 4 apples and you keep adding apples in this pattern forever and ever and ever; how many apples do you have in the end?


-1/12 of an apple is the 5th graders basic response?????

Really?

I don't know why anyone would think it could be a negative number. Basic math tells you when you add any positive number to another positive number, you get a positive number . . forever, and ever, and ever.

This mathematical induction trick with moving numbers to the right is a joke. The trick was shifting the numbers over, and putting 0 in the first spot. That's where he went on a tangent. You don't need long strings of numbers to see 1+2+3 . . +100 and so on will not bring you to a negative number. It's common sense you'll get a larger and larger number with each addition number you add.

saneguy
Personally, I find the result that 1-1+1-1.....=1/2 makes sense. It's like saying if you jump from 0 to 1 forever thats equivalent to 1/2. It's kind of a cool way to deal with infinity.

It's wrong. You can't just average something out and say "Yup, there it is". Infinity is Infinity because it never ends and can't be averaged by just picking an alternating group of numbers.

OptimusCrime
...This mathematical induction trick with moving numbers to the right is a joke...

I wouldn't call it a "joke". The math used to come up with the answer "-1/12" is NOT invalid, and the concept used is in fact useful (and valid) in many other math applications.

However, having said that, it is obvious that the answer to adding a string of whole integers will NEVER result in a negative and/or non-whole number, no matter how many integers are included. Therefore, the application of this math concept is not very valid "in the real world" in this case (in the case of trying to calculate the sum of whole positive integers).

The answer people are looking for in the real world is not based on an average, but on a real straightforward sum. The answer of "-1/12" is a function of the math itself, NOT a function of what really happens when I add whole things together.

I do understand that this answer is in large part due to the fact that we are taliking about "infinity" here, and infinity is inherently almost impossible for mathematics to truly define. However, the limitations cuased by the inability to define "infinity" is not my problem, but math's problem. I simply know that I will never ever ever get -1/12 apples if I start piling them in a bushel using an infinite divergent series.

There are several people posting here who do not understand advanced mathematics. It is quite humorous to see people criticizing the work of some of the best mathematicians in history.

reply to post by Soylent Green Is People

The answer people are looking for in the real world is not based on an average, but on a real straightforward sum. The answer of "-1/12" is a function of the math itself, NOT a function of what really happens when I add whole things together.

YES, but if you watch the vids thy claim it is -1/12

saneguy
There are several people posting here who do not understand advanced mathematics. It is quite humorous to see people criticizing the work of some of the best mathematicians in history.

and here we go again...

because somebody said something it does not have to be truth, even if he was doing nothing else than mathematics his whole life.
It also does not have to be truth if 1000 other people repeat it and say so !

once and for all 1+2+3+4... is not -1/12 like suggested in those clips

reply to post by KrzYma


That video is exceptional thank you for the contribution.