Or to put it more directly 1+2+3+4+5... = -1/12

Innocuous enough but this little equation has got some sections of the internet buzzing leading to derision from parts of the maths community and defence from physicists.

Here is the Youtube clip that kicked the debate off.

http://www.youtube.com/watch?feature=player_embedded&v=w-I6XTVZXww

A very fragmented education has left me with less than satisfactory mathematical foundations but I am still intrigued by the subject. The two in the video above do dumb things down for us plebs and that is where most of the criticism has been directed.

But in many ways the furore forces an examination of reality vs abstract and I have been having enormous fun debating others on the notion so I thought I would bring it here for others to enjoy if so inclined.

Great, facinating, thanks, Bye.

Dear Steele, (Hi),

I'll be reading the posts on this discussion
with great interest. And I hope to learn something
from doing that. However, as maths was never
my strong suit - I can't contribute anyting of value.
It's way over my head. I'll be admiring those who
can. I only wish I could.

To BEGIN WITH..infinity..isnt a number
http://www.google.com.au/search?q=infinity+is+not+a+number&i

and if it was a number..it would be a whole..NUMBER
Not a fraction..[let alone..a minus fraction..[the only WAY..YOUR RESULT COULD BE CORRECT..is by deceit..and by confounding/BAFFLING IGNORANT'S WITH BULLs hit]

adding infinite numbers WOULD RESULT IN A bigger but whole number
bUT WHY BOTHER?..9.999.99999.999.6 out of 9.9999999.9999 couldn't care LESS.

http://www.google.com.au/search?q=quantum+fractions&
dammm i wernt going to post nuthin..to..no one

ANYHOW work backwards..from positive infinity..TO NEGATive infinity
the answer is zero..eitHER WAY...few people study math..because..the theory and imaginary numbers..IS often AS Absurd..as their warminG-models

SteeleRedux, I can see why 1 - 1 + 1 - 1 + 1 – 1…… might be thought to = ½. But I'm not sure that this conclusion is right, as the real answer could be either 1 or 0 but never ½. So ½ is simply the average of the correct answers, but is not a correct answer in itself.

The conclusion that 1! = -1/12 is based on this sort of thinking, which I'm afraid does not inspire my confidence!

So, what are the key points to the debate that you have been having elsewhere?

Dear Steele,

I've been thinking about infinity.
And I had always assumed it was an abstract concept.
With no beginning and no end.
Therefore I would have assumed that no amount
of numbers could explain it. Numbers are a creation of
man, and infinity existed before our time.

Dear spindoc,

My pleasure.

Dear OUG,

Thank you, you never know, you could be right.

Dear Foxy and Ludwig,

Here is one of the blogs which is thrashing the topic around at the moment.
http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/

I quote;

“And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn't true. From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.”

Now for the really interesting crux of it. The physicists are happy bending mathematical rules to suit what they find in their real world experiments. The mathematicians on the other hand scream blue murder when those rules are not adhered to to the letter. Kind of like Christ vs the Pharisees.

So while the mathematicians are carrying on like pork chops the physicists are finding their calculations are being verified in the real world.

Cont...

Cont...

Yet the calculation is not new. Srinivasa Ramanujan who “was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.”
http://en.wikipedia.org/wiki/Srinivasa_Ramanujan

His mathematics was driven by intuition rather than formal training very much along the lines of Einstein's physics.

““During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. However, the mathematical mainstream has been rather slow in absorbing some of his major discoveries”.”

He concluded in what is known as the Ramanujan Summation concerning the Sum of Divergent Series that the sum of 1+2+3+4... = -1/12
http://en.wikipedia.org/wiki/Ramanujan_summation

Admittedly Ramanujan had concerns how his equation would be received. In a Letter to Hardy he wrote;

"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …".
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

Well it is now a basic part of string theory.

Here is another video exploring the 1-1+1-1 more closely.
http://www.youtube.com/watch?v=PCu_BNNI5x4

All great stuff.

Well maths?
As Homer Simpson may say my brain is hurting.
Did find it interesting.

How did they arrive at 666 as reference to a certain character ?

cs/quote..<<..the physicists are finding their calculations are being verified in the real world.>>..

'in the real world close-enough'..is often good enough
so they make a best guess and tweak the numbers..but its not science fact..ITS THE BEST GUESS/HOPE FEAR expectation.. WITH SEXED UP NUMBERS

http://www.google.com.au/search?q=FRAUD+NUMBERS+climate+MODELing

THE REAl problem..is the requirement..of a the-suss
http://www.google.com.au/search?q=math+thesis+example&

after all there are not..infinite..'original thesis'..remaining
[NO THESIS NO Diploma][thus confound eM WITH BULL.]..

live by the number..die by numbers.
666=[6 TIMes 100]=[6 times 10]=[6 TIMES 1]
100=times OF derision../1O=DEGREES OF SPERATION/..1's=personum divinators.

BUT ITS A NUMBERING OF the end Sign..[by design]..[the sTAR OF DAVID]
SIX STRAIGHT lines..six intersecting POINTS..AND 6 ISOLATED Triangles
[surrounding a hex-o-GRAM*]

Good Morning Steele,

Thank You for providing another link and for
more detailed explanations. I was so intrigued
with all this that last night I looked up "Infinity"
in World Book encyclopedia. Here's what I found:

"Infinity is a term continuously used to refer to
a quantity or distance that is so large it cannot
be counted or measured in mathematics, the idea of
infinity forms an important part of set theory.

A set of objects or numbers is called "finite" if
the objects or numbers can be paired with the
positive "integers (whole numbers) less tan some
positive integer. For example, a set of playing cards of
one suit, which consists of 13 members, is finite. The
cards can be paired with the positive integers less than
14.

An infinite set is defined as one that is not finite.
Its members cannot be paired with the positive integers
less than some positive integer, because the set
continues without end. For example, the set of all positive
integers - 1,2,3,4, and so on - is infinite, as is the
set of all fractions. Both sets have an unlimited number
of members.

Infinite sets may be represented by placing three dots
after the last member noted. For example, the set of
even numbers above zero may be written 2,4,6 ...
There is a symbol which also represents infinity.

The idea of infinity has other applications in mathematics
in addition to set theory. In projective geometry, for
example, the "point at infinity" is defined as the
intersection of all parallel lines.

More information is available by looking up "Set theory"
(Finite sets and infinite sets).

As I stated earlier - it's all way over my head.

cs/MISS-quote..<<..the physicists are finding their calculations are being verified in the real world.>>..

NON-SENSe..one needs ask/define..which ones
economists?
http://www.infowars.com/how-junk-economists-help-the-rich-impoverish-the-working-class/

[even a dead..clock..is right twice A DAY.]

SEE THE FRAUD..REVEALED by..The ‘Pause’ of Global Warming
THAT..Risks Destroying The Reputation Of any model-bling/Science
http://wattsupwiththat.com/2014/01/26/the-pause-of-global-warming-risks-destroying-the-reputation-of-science/

…there has been no significant warming
over the most recent fifteen or so years…

In the light of all this, we have at least to consider the possibility that the scientific establishment behind the global warming issue has been drawn into the trap of seriously overstating the climate problem … in its effort to promote the cause.

It is a particularly nasty trap in the context of science, because it risks destroying, perhaps for centuries to come, the unique and hard-won reputation for honesty which is the basis of society’s respect for scientific endeavour…

HAWKING RE THINKS HIS BLackhole numbers
http://www.bobtuskin.com/2014/01/27/black-hole-theory-my-biggest-blunder-stephen-hawking-says/

MORE RE 666
BY..the fact-or..of 911
http://www.veteransnewsnow.com/2014/01/26/401954-israeli-militarism-predicated-on-911-deception/
http://theuglytruth.wordpress.com/2014/01/27/israeli-engineered-911-gateway-to-war-of-civilizations-between-the-west-and-the-muslim-world/

666..by the number of the thIRD..[the REICKE..numbers]
http://www.amazon.com/gp/product/1556520778/ref=ox_sc_sfl_title_7?ie=UTF8&psc=1

DEGREES OF SEPPERation
http://investmentwatchblog.com/dr-doom-scenario-it-looks-like-1914-again-world-war-coming/

http://www.blacklistednews.com/President_Roosevelt%E2%80%99s_Campaign_To_Incite_War_in_Europe%3A_The_Secret_Polish_Documents/32414/0/38/38/Y/M.html

http://www.cracked.com/article_20835_5-organizations-that-gamed-system-screwed-public.html

http://whatreallyhappened.com/

When making statements about mathematics one should use mathematical, not some popularized (or mysticized) terminology.

The expression 1+2+3+… = +infinity is a popularized way of saying that the limit of 1+2+3+…+n as n tends to infinity is +infinity. In this sense 1-1+1-1+1-… would stand for the limit of 1-1+1-…+(-1)^n as n tends to infinity, which does not exist, as every student of first year calculus would know. You can derive all sorts of nonsense from something that does not exist, like e.g. 1/0.

>>The physicists are happy bending mathematical rules to suit what they find in their real world experiments<<

This is true to some extent, although “replacing intuition for mathematical rigour” would be a better description of this: For instance, the Dirac function, that sounded like something mathematically self-contradictory, was used by physicists until Laurent Schwarz came with the mathematically precise concept of generalized functions (distributions) that fitted also the Dirac function. Also the Feynmann path integral, used in quantum field theory, still lacks a mathematically rigorous definition (at least did so in my time).

However this is not our case: any physicist would have learned about convergent and divergent series in a first year maths course .

Foxy,

The silly video is not about the concept of infinity (or many infinities as developed by Cantor) as such but about the concept of convergent (to infinity) and divergent series.

My thought is that infinity in this case includes all the negative and all the positive numbers, so in reality what is proved here is that when you take all the bad things and subtract it from all the good things your down the tube by 1/12, which proves mathematically Fraser was right "Life was not meant to be easy"

I am most likely wrong but the way I interpret the result is that the set in this case includes all the negative numbers as well as the positive numbers therefore the common sense answer should be zero and in fact what is proved is that 1-1 +2-2 +3-3...=-1/12. If we permit the use of negative numbers in our proof we must include them in the original equation.

Dear individual,

The numberphile crew do explore the number 666. If you are interested the link is below.
http://www.youtube.com/watch?v=UkZqFtYtqaI

Dear OUG,

Your reticence to countenance the notion is understandable. For 'God in his infinite wisdom' to become 'God in his -1/12 wisdom' kinda doesn't have the same ring to it does it. ;)

Dear belly,

Another thing the Numberphile crew explore is whether or not the Simpsons live in a base 8 world seeing as they all have 4 fingers. But there is one character in the Simpsons who has 5 fingers and it is the reason why they live in a base 10 world. Watch the video below to find out who that character is.
http://www.youtube.com/watch?v=UkZqFtYtqaI

Dear Foxy,

As I said earlier I am a Math cripple as well but understand enough to get a real kick out of this sort of stuff. If it has you going off to do more exploring as it did for me then that can only be a good thing. What has captured my imagination as well is the apparent contest of ideas between mathematicians and physicists. I have liked the two to accountants and artists respectfully. If fact I see physics as the language of the Gods.

Here is a video exploring 4 paradoxes of infinity you might find fun, Hilbert Hotel, Gabriel's Trumpet, The Puzzle of the Dart Board, and Double Your Money.
http://www.youtube.com/watch?v=dDl7g_2x74Q

Dear George,

Welcome.

I'm finding it fascinating that mathematical types really go after these two physicists right from the get go as you have just done, questioning their capabilities and their worth.

Here are a couple of quotes from another discussion I have been having;

“It really worries me that the video has over a million hits from people most probably having not studied university level maths, as it is tricking them (and yourself) into believing it can be that simple. I have done first year university maths and from the study over several weeks that we did on sequences and series it can be seen why his proof just doesn't work.”

“Both those physicists are full of it”

“Just because a certain number of people believe it doesn't make it right (although I am sure you would love to talk religion on that matter). 
To me rather than seeing that those physicists believe it, just to me discredits their reputation, and has serious implications for the institutions that employ them.”

Yet why is it that I can accept that the physicists and the editor were dumbing things down to make a Youtube friendly clip while others seemed to be incapable of it. Here is a link to their more complete treatment of the concept, even that is probably a far cry from what they are capable of.
http://www.youtube.com/watch?v=E-d9mgo8FGk&sns=em

Anyway I suppose it is incumbent on me to provide you something with greater weight so he is a Fullbright Fellow Carl Bender tackling the subject of the 1-1+1-1+1...=1/2 which seems to be the sticking point for so many.
https://www.youtube.com/watch?v=VvqeJkT3uyo#t=3834

And a question from me is how do you view the Ramanujan Summation? Did he have it wrong or is your issue purely with the video?

FOXey/quoTE..,,..<<the "point at infinity" is defined as the
intersection of all parallel lines.>>

THIS Reminds..me of a simulate
THE star of david..HAS THREE SETS..of paralleled lines
WITH 6 POINTS OF INTERSECTION..forming the hex-0-Gone..AND THE 6 ISOLATED SMALLER triangles..[fiefdoms]..

IN REALITY THE PARALLELED..lines never CAN CROSS
except by clever trick/points of commonality ..[hypothesized intersection]..or 'artistic license'..BUT VISUALLY ALONE ..[Known as the vanishing point..]

[OUT OF SIGHT=OUT OF our DULLED LOGIC Of mind]..
Feigned seeing=feigned believing...if its nonsense THE MATH wONT ADD UP...[and it wont compute].

thIS INSANITY REMINDS ME OF THE TRAIN..THAT NEVER ARRIVES..by the same insane logic...[YET TRAINS DO ARRIVE]..CALL that fuzzy logic..or intersecting vanishing points..or satanic illusIONS..but god alone=1]

Dear Steele,

This is becoming quite a learning experience
and fun as well. Thank You for opening up a whole
new world for me.

Would this be leading to a dissertation on the difference between the Euclidian & Lafayette Systems of Mathematics, thence on to Lyndon La Rouche connection with the Citizens electoral Council. Would it steelie? I would have thought that Lyndon La Rouche would have been to extreme right for you.

Thanks SR just minutes ago in ABC NEWS 24 a maths man spoke of the hidden maths in that show.
Homer for one has been seen writing obscure [to us] maths problems and other too including Arboo,

I fundamentally believe the world makes sense even if it is not intuitive. After looking at this question more closely I am satisfied that the result is misleading.
The salient points are the natural numbers are defined as positive whole numbers therefore the result of the calculation can not be negative or to put another way if you take all the apples in the world and add them together you can not end up with -1/12 of an orange. The chosen staring point for the calculation is arbitrary I.E zero. The calculation could just as well have started at -3 in which case the series would look like this -3-2-1-0+1+2+3. My conclusion is the original formula refers to N meaning a whole number and not N meaning a positive whole number.

Using the same logic, I can get infinity = -1/6 or -1/24 etc.

The reality is that ignoring the end as n-> infinity makes the calculation meaningless.

I think the flaw in the system the guy uses is that he shifts the 2nd line to the right. Why?

Interesting It like the problem of 2=2=5, is it.

In 75 I did a Basic Electronics Course by Correspondence through Stotts College. (now defunct) I was doing some cleaning out over Christmas & ran across the Course. I though to my self, "I should run through that again as a refresher." So I started on New years Day. Very interesting & lots & lots of Maths, Graphs, Algebra & Cartesian Co-ordinates.

Dear Shadow Minister,

You wrote;

“Using the same logic, I can get infinity = -1/6 or -1/24 etc.”

I would enjoy seeing your working out. Are you able to reproduce it here?

Dear George,

Sorry about the link, I usually check to see if the link contains the rascally 's'. Slipped through this time.

www.youtube.com/watch?v=VvqeJkT3uyo#t=3834

The clip starts just before the relevant section.

Dear Warmair,

So you find quantum mechanics makes sense? You are a better man than I.

If you get a chance have a look at the Carl Bender video I posted.

You will note he talks about changing the representation of the series into another form to get a finite answer or in his words “a meaningful sum of the series”. Note this is a divergent series he is talking about.

He then says in some circumstance he can prove that “the procedure rigorously produces the physical answer” eg in quantum mechanics, but in “electro dynamics, while I can use this technique I cannot prove it works”.

Therefore the reality as described by physical experimentation is returning provable finite answers to the sum of divergent series. The abstract on the other hand is the purely mathematical approach which is found lacking.

This is why it is so fascinating.

SteeleRedux,

>>I’m finding it fascinating that mathematical types really go after these two physicists right from the get go as you have just done, questioning their capabilities and their worth.<<

You might also find it fascinating how biological “types really go after” Intelligent Designers and others misrepresenting what Darwin’s theory and biological evolution is all about. Or how geological “types really go after” those who claim the Earth is about 6000 years old and use geological arguments to support that claim, leading to “derision from parts of the geological community”.

Thanks for the link. I do not know who is Carl Bender, but he makes it clear that he derives his conclusion from his ASSUMPTION that the series 1-1+1-1+… converges to a sum S, which is against what is well known in mathematics (unless he REDEFINES the concept of convergence in which case he is not speaking ordinary mathematics and then also his 1+2+3+4+5+ ... must be seen as meaning something different from what it normally means).

Well, you can derive all sorts of things also in biology or geology if you ASSUME that something is true. It is even worse, because in biology and geology you would need evidence to support your assumption (called hypothesis), whereas in mathematics you can actually PROVE that something does not exist.

SR,

Not really the space, but for an interesting short one

If 1+2+3+4... = X (1)
=> 0+1+2+3... = X (2)

(1)-(2) gives

1+1+1+1+1... = 0
0+1+1+1+1... = 0
0+0+1+1+1... = 0
0+0+0+1+1... = 0
............
add infinite series together and get

1+2+3+4+5... = 0

GEORGE/QUOTE..<<..Well, you can derive all sorts of things also in biology or geology..*if you ASSUME that something is true.>>

ASSUMING A 'TREE..OF LIFE'..has DECEIVED MANY
now we compare//the dna..we are looking aT A FORREST..not a single..'tree'..of life

[IE MANY NEW Genus..Not many 'evolutions'.]

its gotten so..'presumptive'..THE MOST COMMON EDIT..IS Undo

http://en.wikipedia.org/w/index.php?title=Encyclopedia_of_Life&action=history

but how do we un-confirm..a FRAUD Thesis?.
...THE MATH DONT ADD UP..BUT tHE MEDIA SIMPLY DONT TELL YOU ITS REFUTED..and so..SO THE DELUSIONS GO ON,[then the cleanup mob comes in//deletes MY ..refutation..AS IT DELETES ITS gross claims

[surely ..you RECALL BIGGER PAGES..with pretty drawings..but the frauds..run by faulse imagined or invisible numbers [and revised/edited]..on the qt]..

and the/TRICKS DECEPTIONS lies go on.
MAKING IT UP AS WE SWALLOW THE LAtest lie

more INCONVENient study
http://wattsupwiththat.com/2014/01/28/inconvenient-study-arctic-was-warmer-than-the-present-during-the-medieval-warm-period/

leSS RE-lie-ABLE DATA
El Niño sea monitoring system may fail – half dead already

http://wattsupwiththat.com/2014/01/28/el-nino-sea-monitoring-system-may-fail-half-dead-already/
From the “send money or the instrumentation gets it” department comes news that the TAO array may already be toast due to budget constraints. One wonders if money sucked into climate programs might be a factor.

From Nature News: Nearly half of the moored buoys in the Tropical Atmosphere Ocean (TAO) array have failed in the last two years, crippling an early-warning system for the warming and cooling events in the eastern equatorial Pacific, known respectively as El Niño and La Niña.

Scientists are now collecting data from just 40% of the array.
[the 40%..with the 'right-data/pro-philes?]

Noninfringing uses, in contrast, are less tangible
and less apparent at the onset of a technology.

http://www.techdirt.com/blog/innovation/articles/20131220/00174325644/innovation-asymmetry-why-copyright-industry-always-freaks-out-about-new-technologies.shtml

The costs of infringing uses can be quantified. They roll off copyright owners’ tongues: $250 billion in losses to the U.S. economy each year from IP infringement. 750,000 jobs lost annually from infringement. It doesn’t matter if the figures are correct. For even if they are completely disproved, the mere articulation of numbers promises a precision that is difficult to dislodge from the audience’s consciousness.

On the flip side, the power of potential innovation is much harder to quantify.

In contrast, noninfringing uses are less tangible, less obvious at the onset of a technology, and not advanced by an army of motivated advocates. First, they are less tangible. Noninfringing uses are difficult to quantify. How do we put a dollar figure on the benefits of enhanced communication and interaction? Estimates of future noninfringing uses will be less powerful than the actual, hard-dollar figures presented by copyright owners.

Second, they are more fully developed over time

For infinity to have any meaning - if that is not oxymoronic - it cannot be written as a number even if it can be written as an expression, surely?

(Perversely, I like to imagine that infinity could be enumerated before the universe inflated by a factor of 10 to the sixtieth power in less than 10 to the negative thirty seconds and, ever since, infinity has been getting even bigger because of space expanding at some 74.3 plus or minus 2.1 kilometers per second per megaparsec (Mpc =approx 3 million light-years.)

Transcendental or what?

INFINITY..isnt finite
[number's ARE FINITE..]..
YOU..either have one..or none..
[or owe AN..other-one..or owe..THE other]

WN/QUOTE..<<..and, ever since, infinity has been getting even bigger because of space expanding at some 74.3 plus or minus 2.1 kilometers per second per megaparsec (Mpc =approx 3 million light-years.)..>>

we live iN AN EXPANDING UNIVERSE..[thIS IS]..RELATIVELY SPEAKIng..constanT

BUT..THE RULER/measure..by which we could expand ON,..THE CONCEPT OF EXPANSION..is expanding proportionately...as that being measured

IE..IF THE Me-asured and the measure..both expand..at the same degree of error..[EX-change]..the change..of MEASURE WILL BE UNNOTICED..THUS UNMEASURED.

BUT I SUSPECT..BY..the APPLICATION..of the measure of light..it..ISNT A THIRD DIMENSIONAL..lateral measure..as much as a space time uni-LATERAL..RE-direction..[thesis]..

IE..KEEP BUSY 'WERK'

The Expanding State
and the Bureaucratic Way
http://rinf.com/alt-news/latest-news/expanding-state-bureaucratic-way/

Infinity = Been nowhere & going nowhere in no time. ;-)

This is a bit like the old story of the missing shilling.

: Three guests check into a hotel, and their bill is 30 shillings.Each guest pays 10 Shillings. Later, the hotel clerk realizes he's made a mistake, and their bill is actually only 25 Shillings. To fix his problem, he gives the bellhop the extra 5 Shillings to return to the guests. When the bellhop goes to the hotel room, he realizes that he cannot split the 5 Shillings evenly between the three guests. Since the guests are unaware of the total of the revised hotel bill, the bellhop decides to give each tenant 1 Shilling, and keep the rest of the money for himself. Each guest received their 1 Shilling, so now each guest has only paid 9 Shillings to stay at the hotel. This brings their total money spent to $27, while the bellhop kept 2 Shillings. By doing a little math, 27 Shillings plus 2 Shillings equals 9 Shillings. So, where is the missing Shillings ?

Dear Shadow Minister,

As I stated before I am in many ways a Math illiterate but Bender does discuss this. He talks about there being a infinity of diverging negative numbers and another of diverging positive numbers with zero being the great divide, probably for the very reason you have provided us. It will be somewhere in the 4 odd hours of his lectures I have been watching but I will see if I can find it again.

Dear WmTrevor,

Thanks for that figure. I have always wanted to know, if we removed the influence of gravity, just how much the moon would move away from the earth over the space of a year. I should be able to derive it form what you have furnished.

Dear George,

In referencing Creationists you make a very valid point and one that made me pause when I thought about it early on since I have been known to take New Earth types to task.

But then I realised the opposite could be easily applied. Mathematics like religion deals very much in the abstract and can become very divorced from the real world. The great Professor Feynman (I'm hoping you have heard about him) discusses the difference between the mathematicians and physicists far more lucidly than I could put here.
http://www.youtube.com/watch?v=obCjODeoLVw

So in a way it is you who is asking me to eschew reality for the abstract wouldn't you agree?

It is this aspect I have found to be so intriguing, and the tension between the two disciplines is something I would not have believed existed before now.

Dear SteeleRedux
Do I understand quantum theory no only at very superficial level, but I still think it makes sense as it is based on two fundamental principles. First that there is a limit to how far you can divide something and secondly statistics.

To clarify my argument the complete set of numbers in the series is minus infinity to plus infinity subtracting all the negative numbers from all the positive ones gives the result of -1/12. The mistake in the thinking was to take part of the whole set and assume it was equal to the value of the complete set. The principle that was obviously violated was when N is a positive value N+N+N... can not equal a negative number.

http://physicsbuzz.physicscentral.com/2014/01/does-1234-112.html

Quote
¨We're leaving the post up for completeness, but it's just dead wrong. So don't waste your time. Read the correction instead. ~Buzz¨

Correction Below
http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html

Dear SteeleRedux,

>> Mathematics like religion deals very much in the abstract and can become very divorced from the real world.<<

It is perhaps theology rather than religion that derives statements from a priori made assumptions about a non-physical world that might give meaning to the life of many of us, but is not as directly related to the “actual” physical world the way mathematics is, c.f. Wigner’s “unreasonable effectiveness” of mathematics. Hence an erroneous reasoning leading to nonsense in theology does not effect what we can know about physical reality, in distinction to a silly mathematical arguing, which does. As I said before, in religion you have to BELIEVE in God (and/or some basic tenets) in mathematics you have axioms that CLEARLY DEFINE the concepts what you want to derive things from. (ctd)

(ctd)
>>The great Professor Feynman … discusses the difference between the mathematicians and physicists far more lucidly than I could put here. http://www.youtube.com/watch?v=obCjODeoLVw <<

I am not sure when the talk originated. Feynmann died in 1988 but the technical quality of the video (B&W, poor sound) seems to indicate that it was recorded much earlier. In my times, which are not as old as Feymanns’, pure mathematics and mathematical physics developed so to say parallelly for a couple of decades: We used to say jokingly that pure mathematicians knew many questions but did not know the answers, wereas applied mathematicians knew many answers but did not know what (mathematical) questions they were answering.

That has chages considerably in the recent decades: contemporary cosmologists, mathematical and theoretical physicists (the difference is rather subtle) need to use - and they do - very abstract mathematics, that would have been seen as purely self-serving and not very applicable, in my times.

>> So in a way it is you who is asking me to eschew reality for the abstract wouldn't you agree?<<

I did not ask you to eshew anyting, only respect mathematics when speaking of mathematics, and respect physics when speaking of physics. You introduced religion, so read it also as a an analogue of "Render to Caesar (Physics) the things that are Caesar's, and to God (mathematics) the things that are God's.

Mathematics without physical applications can, and does, exist, but is pretty useless. Physics, without proper mathematics does not exists, except in its most naive, degenerate forms. Explorers in physics originally use mathematics ad hoc (which does not mean using nonsense like 1+2+3+4+5... = -1/12) but they gradually need to make it more or less rigorous to make their physical theories more adequately reflect some features of reality.

>>It is this aspect I have found to be so intriguing, and the tension between the two disciplines is something I would not have believed existed before now.<<

I do not think this is a place to further elaborate on this "tension" from the perspective of mathematics or physics (or both).

Dear warmair,

Thank you for the sterling link. The correction is quite robust, persuasive and convincing to my unlearned eye. I see it was only posted yesterday and I will look forward, and read with interest, the responses it generates.

Dear George,

Thank you for your expansive post. We probably differ on the extent of the distinction between axioms and belief (as would Feynman I think) but so be it, the rest I am happy to take on face value.

Could I trouble you for a clarification though? By describing 1+2+3+4+5... = -1/12 as nonsense am I correct in concluding you see Ramanujan as having been completely wrong and if the equation appears to be a central tenet (even axiom) of String Theory (damn that is a lot of 'c' words) then the implication is that you perhaps see that entire field as nonsense also?

Ultimately I think the video and the debate it has generated has been great stuff, it has driven an interest in the topic from us plebs, seen some very high horses mounted along with potentially some spectacular dismounts, and for me at least given some inkling of just how romantic exploitative physics must be. These guys just seem like they are having so much fun.

Very instructive, Wm Trevor.

>>...ever since, infinity has been getting even bigger because of space expanding at some 74.3 plus or minus 2.1 kilometers per second per megaparsec (Mpc =approx 3 million light-years<<

But can you make that into a YouTube video? Remember Marshall McLuhan?

Talking of which, you might be interested in this. I was looking for the "Marshall McLuhan. What are you doing?" sequence on the Rowan and Martin Laugh-in. As you do. Instead, I found this...

"Berlin in twenty years from now, 1989. There was dancing in the streets today as East Germany finally tore down the Berlin Wall"

http://www.youtube.com/watch?v=oMPR9DalSPg

The short segment is at 4.13

For a comedy show, not a bad prediction, eh?

Lol. That should read exploratory physics not 'exploitative'. Darn spell checker.

BAFFLE Em with pictures..comes next

LOOK AT THE PICTures in-between..the quotes

<<..there will be pictures, and those are the most important parts.

Here we go . . .

I find it easier to understand things visually, so it occurred to me to plot out the series 1+2+3+4+ . . . at various points as you add the numbers. These points are called partial sums. This is what you get if you stop the sum after each step

for n=1 the partial sum is 1
for n=2 the partial sum is 1+2 = 3
for n=3 the partial sum is 1+2+3=6
for n=4 the partial sum is 1+2+3+4=10
for n=5 the partial sum is 1+2+3+4+5=15
etc.

If you draw the first five terms on a piece of paper, it looks like this

But you don't have to add all those numbers to calculate value at each point. The numbers, it so happens, are a sequence (1,3,6,10,15, . . .) that you can calculate with a simple formula called a generating function. In this case the generating function is

G(n)=n(n+1)/2

To get a number at any point in the sequence, like say the 5th spot, just plug in 5 for n, and you get the answer

for n=5, G(5)=5(5+1)/2=15

As it turns out, you can also use the generating function to figure out what the values would be between whole numbers. Essentially, you replace the number n with an x, which can have any numerical value you like. If you plot the result, you get this, for positive x.

The curve you see here goes up to infinity as x goes to infinity. So far, so good. But if we're going to plot the graph between the various values of n, we might as well look at negative values of x too.

When you do that, you get a graph like this>>..

THE BIG lie seems to be..on the vertiCLE AXIS
WHERE minus ONE TIMES Itself..EQUALS POSITIVE one..in both series
[THERE CANT BE A BELL CURVE]

it seems the phobia..is linked to nuthinness[ZERO/..oR RATHER FEAR..OF ZERO]

plot out the series -1+-2+-3+-4+ . . .
at various points..as you added the positive numbers. ..[These points are called partial sums.] This is what you get if you stop the sum after each step

for n=-1 the partial sum is -1
for n=-2 the partial sum is -1+-2 = -3
for n=-3 the partial sum is -1+-2+-3=-6
for n=-4 the partial sum is -1+-2+-3+-4=-10
for n=-5 the partial sum is -1+-2+-3+-4+-5=-15
etc.

If you draw the first five negative..terms on a piece of paper,
it looks like this..[..%..]..[OR THIS..[/]..not like this..[u]..

Rowan and Martin's News of the Future... were they prescient or Illuminati insiders?

"Remember Marshall McLuhan?"

Remember him, Pericles? Yep, studied Understanding Media at uni in the 70s though I didn't agree with classmates who thought the book was cool.

Speaking of divergent series, television that is, not mathematical...

Found this short [1:34] Stephen Fry introduced Open University blooper which could be substituted for some of this thread's discussion:

http://www.youtube.com/watch?v=2un9rO2ZF4g

Dear UOG,

I think the mistake you are making is not continuing past zero and sticking with whole numbers.

This is what I get when I plug in n1=n(n+1)/2

-0.9 -0.045
-0.8 -0.08
-0.7 -0.105
-0.6 -0.12
-0.5 -0.125
-0.4 -0.12
-0.3 -0.105
-0.2 -0.08
-0.1 -0.045
0 0
0.1 0.055
0.2 0.12
0.3 0.195
0.4 0.28

Graph those little babies up and I think you will be sweet.

Dear SteeleRedux,

If you are intersted in Ramanujan’s approach to “summation”, see e.g. http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF, where you will find sentences like

“The series 1 + 2 + 3 + 4 + · · · can be 'summed' by zeta function regularization.”

or

“Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ...“.

Note the quotation marks in both sentence. This is more or less what I had in mind when I wrote

>> unless he REDEFINES the concept of convergence in which case he is not speaking ordinary mathematics and then also his 1+2+3+4+5+ ... must be seen as meaning something different from what it normally means <<.

That redefinition is called zeta function regularization but I do not think this is the place to discuss zeta functions, which, anyhow, do not fall within my (past) specialisation. In particular, I know nothing about the application of zeta functions (and Ramanujan summation) to string theory but I am sure they do not use the relation infinity = -1/2 in this form.

Very roughly speaking, Ramanujan did to convergence of series what Lobachevsky did to Euclidean geometry: he did not disprove Pythagoras Theorem, only showed that in a some other (non-Euclidean) geometry it does not hold.

Dear George and warmair,

I had been happily conceding in the various discussions I've been engaged in that 1+2+3+4+5... does not = -1/12th based on warmair's link to physicscentral.

http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html?showComment=1391212975656#c9134991847062538477

Not that George hasn't provided some convincing rebuttals its just that I am a visual guy and needed the graph to kick me over the line.

But there is one issue that was troubling me and I'm hoping he might be good enough to sort it out, but I thought I might throw it to you guys as well. Looking at the below graph;

http://2.bp.blogspot.com/-F8iY78u3QWU/UuqOSIkcQyI/AAAAAAAAD9s/7Ici_NkPcmQ/s1600/graph+ABC.png

I might be further exposing my illiteracy here but why can't you subtract area A from area B and get left with area C which equals -1/12th?

Both A and B are the same size and surely it doesn't matter where they start on the axis as long as one is entirely positive and the other entirely negative, shouldn't they both cancel each other out? I concede the sum would then need to be written something like ...1+2+3+4+5...=-1/12th or whatever the correct mathematical equivalent would be.

Where are I going wrong?

steelredux/QUOTE..<<..why can't you subtract area A from area B and get left with area C..>>

TO BEGIN WITH..AREA..a and b..
are outside..THE AREA DEFINED..AS BEING OF C..

the vertical..liNE PASSES THROUGH..c
[IN THE FIRST DIAGRAM..it passes through..the Empty/unnamed..area..named a]

C IF IT RELATED TO ANYTHING..should relate to the unnamed AREAS
between b and a..BUT SUCH IS THE WORK OF DECEIT....IF YOU DID IT PROPER WAY[LIKE THE OTHER GRAPHS..it would b e more clear[see the first GRAPH]..THE VErticle line bisects c..yet not in your examp-le

think why.

LOOK IN A PROPER GRAPH..zERO..WOULD BE AT
THE central..FOCUS POINT

a proper graph WOULD HAVE ALL 'REAL NUMBERS...on one wing
then the negative numbers on the other...THEN at the intersewcting poiunt..thE VALUES..of each addition..[plus eACH NEGATIVE 'ADDITION']

THUS..ON ONE AXIS..would BE THE RESULT OF 1..plus 2..[IE 3...6..10..etc]

ON ITS opposite would lie
minus 3..minus 6 minus 10 etc

its all smoke and mirrors..[poiSENING THE WELL]
ITS DESIGNED TOI MAKE ALL OF YOU Feel dumb

look if george cant explain..it think WHY

ITS DESIGNED TO MAKE us feel dumb
but it in reality reveals even the clever are faking it

get bit?

if infinite = everYTHING
IT CANT BE JUST A FRACTION..OF ANYTHING

think of it like if god = the universe..or the sun
then jesus cant..be god..yetHE CAN REMAIN AS A SON..of the sun.

[you give me lemons
i make lemon aide.]

Dear SteeleRedux,

Let me first reiterate (also as a comment on warmair’s post) that you can “do” mathematcs (at any level) without knowing anything about quantum physics or string theory, but not the other way around. Physicists can talk about the APPLICABILITY to their problem of this or that part of mathematics, but PHYSICS as such cannot decide what is correct and what not in mathematics. As quoted in my paper http://www.onlineopinion.com.au/view.asp?article=15928 , “while our perception of the physical world can always be distorted, our perception of the mathematical truths can’t be”. Apparently, string theorists found the zeta function (which is not easy to define at this level) useful, and I think the Wikipedia article I linked to is a good explanation of why zeta(-1) = -1/12 can be associated with 1+2+3+4+ ...

I looked at the links you gave and I admit I did not understand how they relate the picture and the calculations to the zeta function. True, the two regions A and B both have an infinite area, actually they are of the same shape.

>>why can't you subtract area A from area B and get left with area C which equals -1/12th?<<

Because both these areas are infinite numbers, and you cannot subtract one infinite number from another, more precisely, you could reasonably assign anything to the result:

For instance, there are infinitely many natural numbers (1,2,3,4 ...) and there are infinitely many even numbers; if you “subtract” (i.e. remove) the latter from the former you get the set of all odd numbers, and there are again infinitely many of them. On the other hand, if you “subtract” from all natural numbers all those greater than 2014, again infinitely many of them, you get the set of all the natural numbers less or equal 2014, which is finite, its size being exactly 2014.

So in the first case we “derived” infinity - infinity = infinity, in the second case infinity - infinity = 2014.

Dear SteeleRedux
Frankly I am out of my depth on this topic, anyway thank you for bringing it up, I found it fascinating. I mentioned it to my nephew who is a mathematical whiz kid, I expect in about 6 weeks I will get a very detailed response which will be way over my head.

I did in the end however come to the same conclusion as George that infinity - infinity can equal whatever you choose.

Thanks George for providing a good deal of clarity on the issue.

Good afternoon to you STEELEREDUX...

I've carefully examined the conundrum you've kindly posed for us all. My only answer can be best described by my candid admission that having passed, six subjects in the (1957) NSW Leaving Certificate, general mathematics was not amongst them. Though I managed to scrape through in chemistry, much to the incredulity of Mr Potts, our Chem. teacher ?

In my former occupation, it was required that I perform a couple of years penance in the Academy. In those days, government thought it better they attract more undergraduates, ostensibly lifting the basic educational standards of police recruits. I recall on one occasion, listening in to a bunch of these recruit's arguing over the efficacy of adopting a modified type of anemometer (wind speed measuring device) as a traffic calming strategy ? They proposed all manner of mathematical formulae, some supporting the device, and others strenuously opposing such a strategy ! All based on these complicated math. type diagrams ? They all appeared, on the face of it at least, to be brilliant mathematicians ?

Interestingly though, later on while marking some of their law theory examination papers, many of these same individuals had made some truly awful (simple) spelling errors, together with an almost non-existent ability to structure their narrative style answers, using correct syntax and basic punctuation ?

Upon enquiring later on why this was so, I was duly informed, at university little attention was/is paid to either correct spelling or correct grammar, as long as their responses were clear and comprehensible, and naturally correct ?

I suppose on reflection, their mathematical adroitness is a much more important and desirable expertise, rather than spelling an answer incorrectly, with badly structured grammar ?

STEELEREDUX, how things have changed since many of us went to school ? I agree with FOXY, there's a whole new world out there. Consisting of formulae, theorems, symbols, reasoning, abstract concepts, etc etc ? A really good topic I reckon ! However, it would be a pity, if you allowed it to further inflate that hauteur of yours now ?

Dear George,

Firstly I want to thank you for taking the time to deliver expanded answers to a pleb like myself. I am grateful.

You wrote;

“I looked at the links you gave and I admit I did not understand how they relate the picture and the calculations to the zeta function.”

That's good since neither do I. But plugging the formula G(n)=n(n+1)/2 into a spreadsheet (using .1 increments) then graphing it directly delivers the result shown in my earlier link.

This is where I am struggling. If area C was moved anywhere to the left it would retain the value of -1/12th. But apparently if we move area A to the right by 1 then it is not allowed because it is infinity?

I had imagined that if we keep our calculations strictly to the areas and not the figures contained within then a sum of those areas; (-)A + B + (-)C then why wouldn't it leave us with (-)C?

The answer again it seems is because it is infinity. But for a pleb the question becomes if we can change the rules one way for infinity what prevents us changing them in other ways?

I just keep getting the feeling I have all these roads leading to Rome, the video, Ramanujan, Bender, Physics Central but I kept being told I'm heading in the wrong direction.

Cont...

Cont...

You also wrote;

“”For instance, there are infinitely many natural numbers (1,2,3,4 ...) and there are infinitely many even numbers; if you “subtract” (i.e. remove) the latter from the former you get the set of all odd numbers, and there are again infinitely many of them. On the other hand, if you “subtract” from all natural numbers all those greater than 2014, again infinitely many of them, you get the set of all the natural numbers less or equal 2014, which is finite, its size being exactly 2014.”

I get what you are saying yet I'm not sure it applies here. I would have assumed if you take away all the natural numbers (1,2,3,4...) from all the natural numbers (1,2,3,4...) that you got zero. From that doesn't it follow if you add all the positive natural numbers plus all their corresponding negative numbers you would again have zero?

Thanks again for the effort you are putting in to slap me into shape. Also I have just realised I have been using 'illiteracy' instead of 'innumeracy'. Sorry folks.

Dear warmair,

I'm glad you got a buzz out of it and believe me there would be a hell of a lot more depth under my feet than yours. My nephew is also a mathematical whizz kid. 17, already a double degree under his belt and about to embark on his masters. My usual retort when they get a bit dismissive on this one is to ask them to explain why I should be taking their word over that of Ramanujan, widely considered the most naturally gifted of mathematicians over the last 200 years.

Dear o sung wu,

I will endeavour not to let your appreciation of the topic to over-inflate your impression of my impression of myself.

Dear SteeleRedux,

I am still curious how the author of http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html got the value of zeta(n) for odd integers n other than -1, i.e. where he got his generating function for those n. (If you are just after the numerical results, the online calculator http://keisan.casio.com/exec/system/1180573438 will give them, though I have no idea what algorithm or programme they used.)

As to your question, first a clarification in terminology: You know the formula which gives the AREA (a number) of a triangle (a REGION). So the picture denotes the REGIONS A and B, which both do not have a finite AREA but it is reasonable to denote their area as infinity in the following sense:

In the picture, the part of B going from - n to -1 is the same (only flipped over) as the part of the region A going from 0 to n-1, hence both have the same area which you can calculate directly (or using Wolfram which I do not have access to) as (1/12)(2n+1)(n-1)^2 which clearly tends to infinity as n tends to infinity, i.e. as the partial region is becoming the whole of A. This is what one understands when saying that the area of A is infinity; the same for the area of B.
(ctd)

(ctd)
>> But apparently if we move area A to the right by 1 then it is not allowed because it is infinity?<<

You can move any region, finite or infinite, and its area will not change, so I do not know what you mean.

>>(-)A + B + (-)C then why wouldn't it leave us with (-)C?<<

Again, I don't understand what REGION you mean whose area is supposed to be (-)A + B + (-)C.

>> if we can change the rules one way for infinity what prevents us changing them in other ways?<<

You do not change any rules. Infinity here, as explained above, is a symbol to describe the behaviour of a sequence (of finite areas), NOT A NUMBER. If two sequences “tend to infinity” and you subtract them then the result will depend on which sequence “more strongly” tends to infinity. For instance, the sequence 1, 2^2, 3^2, 4^2, ... tends stronger (faster) to infinity than the sequence 1,2,3,4 ...

>> I would have assumed if you take away all the natural numbers (1,2,3,4...) from all the natural numbers (1,2,3,4...) that you got zero.<<

Another way of seeing that you cannot simply subtract one infinity from another was given in my last interpretation of infinity - infinity: I obtained TWO possibilities (infinity and 2014), and your example provided a THIRD possibility, namely 0, so it clearly does not make sense to ascribe any meaning to infinity - infinity.

george/quote..<<..so it clearly does not make sense to ascribe any meaning to infinity - infinity.>>..'numerically'..?

i never go to the extreme..right of my board..so searched..for the minus..[-]...but find it much the same as [-]..above the button...

Anyhow..infinity..not being a number..cant..be subtracted..from any other non number..SO [-]..CANT REASONABLY APPLY..thus i must presume..[-]..is intended.

infinite plus infinite=..infinite
INFINITE MINUS INFINITE=ZERO..[removing everything from everything/../must leave nothing]

MUCH IS MADE FROM THE SHAPE OF A ..AND B..BUT A AND B..ARNT RANGES..[arnt 'shapes'..they are inclined lineal progressions..ALL ONG an ever flattening arc..as the extra number forces divergence AWAY..FROM A DIRECT straight]-LINE THAT PLUS ONE WOULD DEFINE.

ANYHOW Clearly infinite equaling any FINITE=INSANE
LET ALONE INFINITY =..a fraction..

Forget zeta function..recall THE RULE OF ALPHA..it it sounds like non-sense..IT IS...[for the atheists among you..imagine IF SOME 'GOD BOTHERER'..SAID GOD =1/12..BUT HECK YOUR JUST TOO DUMB TO DO THE MATH..[OR ALTERNATIVELY..imagine an atheist saying god=1/12..THUS WE NOW PROVE god cant exist..as 1..

it remains fun..but till we agree..its simple make busy work..to distract from the real issues...there are demons out there..that want yOU TO LOOK AT 'OTHER'..[IE THE REAL..ISSUES http://www.infowars.com/#]..FUNNY THEIR TALKING NUMBERS TOO http://www.infowars.com/nfl-microcosm-of-the-corporate-class/ http://www.infowars.com/nj-gives-nfl-super-tax-break-for-super-bowl-and-pays-for-security/

BUT HECK..lets get back to the real 'game'..finite e;otes trying to steal the infinite from THE TOO Distracted..by roman game playing..to notice.

minus-NOTHING [ie everything]..FROM NUTHIN..will loose us all everything..[just do the math].

http://www.mathsisfun.com/numbers/infinity.html
Dear OUG
Lets start by understanding something about infiniy.

dear teacher..[warmair..from your link/
<<..Example: Isn't [eternity]../..[eternity]..; equal to 1?

No,..''because we really don't know how big infinity is,''>>

*IMPORTANT..[Recall previous]
..<<.."minus infinity..is less than any real number,
and infinity is greater than any real number">>

plus
<<..Infinity is Simple..Yes! It is actually simpler than things which do have an end. Because if something has an end, you have to define where that end is...>>

1/12=AN END..end=END[IE NOT INFINITE V2014..not 1/12
thats all the spin

first quote continues

<<.. so we can't say..that two infinities are the same.>>
ans moST CETAINLy not ant fraction..nor real/nor unreal number

they..are AN ending..
IE..[IN ETERNITY..is..in PROCESS..not definitive..THUs not..1/12

that doesn't make sense!
1/12..IS AN END

further..unending..times unending..=unending
ending..minus eterNITY =eternity
yet ETERNITY MINUS ENDING = ETERNITY

one eternity minus ONE..etERnity=END OF ETERNITY
[as all eternities alone must be equal

NEGATIVE un-dending times positive unending...cant be 1/12..[an ending]

there are three base errors at the link
think..instead of positive/negative numbers..think of opposites
like real/true numbers..versus untrue..unreal numbers

if eternal=unending
minus non ending = THE END...[AN OPPOSING TO NON ENDING]

BUT IM BOred of talking semantics..
ETERNAL=A EVOLVING..PROCESS..NOT A NUMBER
IT CANT BELONG TO MATH...AS IT HAS NO USEFUL..MEANS TO RESOLVE IT BY MATH.

IT MOST CERTAINLY CANNOT..ever 'be'=1/12
FOR THE SAME REASON..SEE EG..ETERNITY/ETERNITY

Dear George,

Sorry mate, I had my A and B around the wrong way.

Bernd on the Physics Central site answered my query this way;

“The crucial point in the comparison between the areas A and B is the following:

If the two integrals yielding the areas A and B were both convergent, i.e. if the areas A and B had finite values, then it would not matter at all whether the curve is centered around -1/2 (as it is) or around 0. Then one could actually shift the curve a little to the right, centering it around 0, without changing the areas A and B, as infinity is not changed by adding a finite number to it. And then it would be clear that A and B would have to be exactly equal to each other: A - B = 0.

But the two integrals yielding the areas A and B are divergent, so we cannot simply shift the curve and claim that A = B. Instead, neither A, nor B, nor even A-B are finite, they all correspond to divergent integrals. The parametrization of such divergent integrals is crucial when comparing them. It is not possible to simply compare A and B, instead you must specify that the areas you want to compare extend to the same finite number (up to the sign) on the left- and right-hand side, with this number going to infinity in your limit.

Such difficulties with divergent series or integrals make it complicated to relate our naive, non-expert's notion of areas etc. to what the true mathematical outcome is.”

I'm quite happy acknowledging I am a naïve non-expert' but I'm assuming when Buzz made this statement in his correction he wasn't kosher either?

“The area of the little region below the horizontal axis equals -1/12, and subtracting the infinite area under the curve on the left from the one on the right gives you 1+2+3+4+. . . , which goes to infinity as you add terms, not to -1/12.”

Thanks again.

quotes..from your link
http://www.mathsisfun.com/numbers/infinity.html
Infinity has no end
Infinity is the idea of something that has no end.

In our world we don't have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.

Just think "endless", or "boundless".
An idea..of something without an end.

..infinity does not behave like a real number.

[..mathematical shorthand for.."minus infinity
is less than any real number,..and infinity is greater than any real number"]

LET INFINITY=1
INFINITY/INFINITY=INFINITY..[ie 1/1=1]

Dear SteeleRedux,

Sorry, but that quote does not make much sense to me.

I can only repeat: You do not need any integration or Wolfram to see that the regions A and B are MIRROR IMAGES of each other (with respect to the parabola’s vertical axis), and BECAUSE OF THAT (nothing to do with “subtracting” infinities) both have the same (infinite) area.

I do not see where 1+2+3+4+... comes into it, except that if f(x)=x(x+1)/2 is the function whose graph is that parabola, then f(n) = 1+2+3+ ... +n. i.e. it is a partial sum of the infinite series 1+2+3+4+ ... . Thus if A(x) is the area of the region under the parabola above the segment [0,x] (for x>0), then its RATE of increase - i.e. the derivative dA(x)/dx which is f(x) - tends to infinity as fast as 1+2+3+4 ... . That is all that I can see about relating A (or B) to 1+2+3+ 4 + ... .

Dear George,

Many thanks for the effort you have put in on this thread and to the others who have contributed. As I no have only one nostril above the surface I think I had better call it quits. I'm off to study a little more math, and to practice a couple of neat card tricks the Numberphile crew have shown on their Youtube channel. Doing base 3 calculations in a few seconds is still not quite in my comfortable grasp but I think it is worth it. Impressed me.
http://www.youtube.com/watch?v=l7lP9y7Bb5g

Cheers mate.

Dear Warmair,

I thought you might be interested in a little update to the link you provided. Buzz has corrected his correction, at least to a degree, and Bernd has left in disgust.

“Anyway, I have the impression that further debates here are useless. You probably cannot understand what I have tried to explain through my many comments. So I will stop wasting my time any longer.”

And I'm left like a shag on a rock.

Still it looks like the debate will continue for some time yet.

Is there no end to this.

JAYB..infinity cannot bE 1/12th

why?

<<..[ITS..mathematical shorthand

for.."minus infinity
*is less than any real number,..

and *infinity is greater than any real number"

1/12 TH..IS LESS THAN ANY REAL finite..NUMBER..
Further...AS INFINITE..ISNT even..A NUMBER..[let alone a MINUS NUMBER.

BUT IF IT WAS IT WOULD BE A WHoLE NUMBER/
IT MUST BE GREATER..[or if negative lesser]..than any whole number

BUT..ITS not..a NUMBER
..1/12..IS A FRACTION..OF A WHOLE NUMBER

ie..[ITS ALL JUST NON*SENSE]
WALK AWAY..I have..some people with nuthin..want it all

BUT IF IT.s ALL..=..1/12..TH..is insane.
LIKE CALLING...EVOLUTION/A SCIENCE..when its a theory...WITH NO Falsifiable..[NO abILITY TO REplicate..TO CONFIRM..IT..

CAN 1/12 TH-INFINITY
OF COURSE NOT..case closed

Didn't get it, did he.