I wanted to share the video but also had an extra copy of DLC so I just linked the GA to here, and it's only an hour long
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The fun thing about math is that you can do whatever you want with it, in order to prove whatever you want ;)
Math allows you to say "1 + 1 = 3 in the decimal system", but if you omit that you redefined the addition operator as "the + sign means left side plus 1 plus right side", which is basically what this video is all about, then you perform invalid math.
If you also make wrong assumptions about converging sets at the start, then you fuck up badly.
Now, I have a brain teaser for you:
1) What is the sum of 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... ?
2) What is the sum of 1 + 1/512 + 1/65536 + 1/2 + 1/8 + 1/32 + ... ?
3) What is the sum of 1 + 1/4 + 1/16 + 1/64 + 1/256 + ... + 1/2 + 1/8 + 1/32 + 1/128 + ... ?
(Please note than in all three scenarios we're using the exact same infinite set of numbers! They're not in the same order, though. But the numbers check out, all of them are present every time.)
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Indeed. ^_^
So when you ask how this can be true:
1 + 2 + 3 + 4 + 5 + . . . = (-1/12)
The answer is that it's a new definition for the meaning of infinite sums. You want to know how this can contradict the sum that you know of:
1 + 2 + 3 + 4 + 5 + . . . = ∞.
It's because it's using a different, older definition for infinite sums. It's possible to use different definitions in different parts of mathematics.
(via)
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Numberphile is not right. He proves that if the series is convergent then it converges to -1/12 (without mentioning the "if" part), then promptly ignores the fact that the series is non-convergent and concludes that it converges to -1/12.
It's a similar trick to all those "proofs" showing that 1=2.
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Well no, they don't say the series is convergent but assign a number to it anyway (this whole Cesaro summation thingie). Still, don't fully explain that and mathematicians be mad about it. It's a different story than saying 1 = 2, though, and actually has uses. So to me, again, it's more like saying e^(sqrt(-1)*pi) = -1 without explaining that you actually can't do sqrt(-1) in real numbers, but the result does make sense and is widely used everywhere in engineering.
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Yes. The sum of a series refers to where the series converges. A Cesàro sum of a series is the sum of of that series if and only if that series converges. In that video the claim is made that the sum of all natural numbers is -1/12, which is untrue because, in this case, the Cesàro sum is not the sum. This is a misdirection akin to those used in "1=2" proofs, which tend to sneak a "cancel the x on both sides" non-sequitor where it is unlikely to be noticed.
By the way, the phrase "can't actually do sqrt(-1) in real numbers" doesn't mean anything.
There exists no real number r such that sqrt(-1)=r, but the equation "e^(sqrt(-1)*pi) = -1" follows from that fact rather than contradicting it, so it is a poor example.
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NNnnno. S1 isn't convergent, therefore the infinite sum is undefined.
What he's doing is Cesàro summation. A Cesàro sum is NOT the same as sum. It's a tool used in math, but you must be careful not to confuse mathematical tools for REALITY.
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I'm not familiar with this cesaro summation thingie, but I know that it often happened you got results that don't make sense, you then just assume they make sense, and have a useful tool to work with. Like square root of negative one doesn't make sense in real numbers so you "invent" "imaginary numbers" and actually they are quite useful. If they can assign a number to Cesaro sum and roll with it and it proves useful, what's wrong with that, I wonder
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Absolutely nothing, except that it confuses non-mathematicians into thinking that 1+2+3+4...+∞ actually adds up to -1/12.
-1/12 isn't a SUM. It's something else.
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No. sqrt(-1) does not equal anything in real numbers, it's an unspecified result. I guess you activated my trap card?
My point is, equation that doesn't make sense in real numbers can still be true in real numbers because you use behind the scenes something else. And somehow to me it's the same as with being angry that you can't just assign a number to an infinite sum which is not convergent and expect the result to make sense.
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1 + 2 + 3 + 4 + 5 + 6 + 7 + ...... (The sum of all natural numbers up to infinity)? That's Numberwang!
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SEEMS LEGIT
If you want something more amazing, fun and which is actually correct (unlike the -1/12 thing), try approximating Pi by throwing hot dogs on the floor!
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you can't say "the -1/12 thing" is incorrect because it's still true to some extent, it could just be the way Numberphile is missing out certain things or their proof in this case that's not really accurate mathematically
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In maths, if you leave out assumptions necessary to make it correct, it's incorrect. When he equates the first sum to being equal to 1/2, he should clarify that he's no longer sticking to the field of real numbers, since up to that point that would be what most reasonable people would assume he's using.
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IMHO:
Cheers!
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Use your brain.
1 + 2 = 3 ... Good & Logical.
1 + 2 + 3 = 6 ... Still Good & Logical
1 + 2 + + 3 + 4 + ... = ? ... Cannot be solved for the fact that there is no end. No one can solve it because it does not exist. Simple as that.
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uh no, as the guy in the video has already said when infinite sums are in question you don't necessarily expect sensible results. the result, -1/12, is useful in certain applications of physics like string theory. it's like saying square root of -1 is unsolvable because it doesn't make sense for a negative number to have roots, but 'i' is very useful in many applications
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Hi
So one day I was being my usual self, happy and browsing the internet.
And someone asked me this:
What's
1 + 2 + 3 + 4 + 5 + 6 + 7 + ...... (The sum of all natural numbers up to infinity)
He showed me this
Oh, before I forget. Here's your reward.
GA has ended, but I'll leave this thread open to discussion. I'm just a high school student, but I find this discussion on this very interesting
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