This is a fun one, because of course the kneejerk rejection of the statement "1+2+3+4+5+... = -1/12" is correct Under everyday, ordinary grade-school arithmetic, the answer can't be "-1/12" -- but the answer can't be any other number either, the operation itself is not possiblehttps://twitter.com/tomgabion/status/1289857027381002241 …
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You cannot do anything an infinite number of times The answer to "1+2+3+4+5+... = x" is that x can't be anything because you will never actually finish adding up numbers and you will never get an answer "Infinite" is another way of saying "nonexistent"
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You can't say x="infinity" or ∞ because in ordinary arithmetic that's not a number, it's a meaningless word
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The *rules of arithmetic* say this -- numbers are *defined* by the fact that if you add 1 or subtract 1 from a number, you get a different number, if "∞" breaks this rule, it's not a number at all and you can't use it
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So people's intuitions about this are correct but they don't take it far enough Caught between the dueling intuitions between "Well the answer to this question can't be any ordinary number" and "It must HAVE an answer though"
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All new forms of math are built on that second part "Okay, I get that this is a stupid-ass question and Pythagoras or whoever would've just said 'shut the fuck up' if I asked him but WHAT IF you COULD add up numbers an infinite number of times"
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All summations of infinite series are based on making up new rules and saying "Okay let's pretend you can do this, what happens if you do, what new stuff do you discover if we just fuck around and act like this makes sense"
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So like, let's be clear This classic series: 1+1/2+1/4+1/8+1/16... Is, from a pure old-school POV, just as bad as the other one Even though this one looks like it has an answer (it adds up to 2 in the end)
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Replying to @arthur_affect
You got the wrong series, what you wrote is the harmonic series, which is strictly divergent. The series 1 - 1/2 + 1/3 - 1/4 + 1/5...., the alternating harmonic series, is the one that sums not to 2, but the natural logarithm of 2.
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I think you misread what I wrote, it's 1+1/2+1/4+1/8..., not 1+1/2+1/3+1/4...
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