If you have a set of scales and you have a thing that shows on the scale that it weighs two, and another if that thing, that also shows on the scale that it weighs two, and you weigh both together you still see that it weighs five. That's the point.
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It is generally understood, although not necessarily accurately, that the number "2" means "2.000...", that is the number two with an infinite set of zeroes after the decimal.
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(As an aside this is also exactly the same as a number one with an infinite sequence of nines after the decimal.)
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Not exactly.
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Replying to @JustCanadian7 @phyphor and
Yes, exactly. 1.99999999999999... means the limit of the series 1+0.9+0.09+0.009+0.0009+...., which is 2.
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Replying to @JustCanadian7 @IMJackRudd and
Well this is ironic I mean, sure, no one can force you to accept this as true if you don't want to -- that's the whole point of this conversation! Everyone has their own idiosyncratic language about numbers they use on their head to understand the world and that's fine!
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Replying to @arthur_affect @JustCanadian7 and
But from the point of view of standard, accepted, classroom mathematics you're just wrong 0.9999... = 1 is, according to the normal rules, true You can get into an extremely formal proof of it via real analysis, but you can demonstrate it informally with simple algebra
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Replying to @arthur_affect @JustCanadian7 and
Ever since this debate became an Internet meme there's been mathematicians pointing out that *you don't have to* accept this if you don't want to You don't have to do anything, in math, math is a construct you choose to adopt for your own purposes
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Replying to @arthur_affect @JustCanadian7 and
Your whole "equivalent to but not equal" thing, for instance, is a matter of much discussion - the = sign does indeed mean different things to different people at different times It's valid, you can say that if you want
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There are *consequences* though Phrased differently, saying 0.999999... cannot equal 1 is saying you reject the concept of a limit (it's really saying that there is no such specifically constructible number as 0.9999..., you refuse to allow the 9s to go on into "infinity")
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Replying to @arthur_affect @JustCanadian7 and
And the concept of a limit is what calculus depends on If you reject it, then you just reject calculus Which, again, is fine! You don't need to accept calculus if you don't want to! I only took AP Calculus because of pressure from my parents and my crush on the teacher!
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Replying to @arthur_affect @JustCanadian7 and
In fact there's plenty of specific evidence from physics now that calculus does not reflect "underlying fundamental physical reality" (whatever that might mean) The real world isn't actually continuous, it's discrete, it has pixels
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