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
My math class are far away but, the way I interpret that is : as 0.3333... is exactly 1/3, 0.9999... is exactly 3/3, or 1.
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Replying to @Dominic11B4 @JustCanadian7 and
Yeah but that's not a rigorous proof because you haven't actually proven that 0.3333... is 1/3 either What this fight is about at heart is whether you're "allowed" to write numbers with that "..." after them and talk about an infinite series of digits
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Replying to @arthur_affect @JustCanadian7 and
That said, it's a good *demonstration*, because people typically aren't nearly as intuitively resistant to the assertion that 0.3333... = 1/3 as they are 0.9999... = 1, even though they are, as you say, equivalent
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If you want to get right to the meat of the issue you don't need the 1/3 Just point out that 0.99999... and 1.0000... are two ways to say the same thing If 1.0000... is "allowed" to = 1 then so is 0.99999...
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Replying to @arthur_affect @JustCanadian7 and
I think an intuitive obstacle to understanding this is that most people are familiar with something that resemble that, it's impossible for a particle with mass to reach the speed of light. So, in physics, you can get close to c but, never reach it. In maths, infinity exists
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Replying to @Dominic11B4 @arthur_affect and
It's impossible for a particle with *rest mass* to go the speed of light.
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End of conversation
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