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7/14/2013 8:10:57 AM
9
Sorry, but that is not how mathematics works. You cannot subtract a repeating decimal from whole number with a repeating decimal to prove that a repeating decimal is a whole number. No.
While I agree that for all practical purposes 0.999... = 1, it does not in the world of mathematics. Take a computer for example, a computer cannot hold the number 0.999... because it would require infinite amounts of memory to do so.
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I'm sorry to burst your bubble, but this is how mathematics works. If you are not satisfied by the algebraic proof, the proof using infinite series is sufficient to prove the same point. In fact, I prefer this proof as it seems more comprehensive and definitive than the algebraic one. 0.999... = 1 is a mathematical fact; there is no need to invoke practical purposes.
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What you're saying is akin to my saying that the function 1/X approaches a numerical value as X--> infinity. It does not. We can SAY that it's limit is 0, while is reality that is not the case.
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At infinity, it IS 0.
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Of course, now define infinity:) you can't, and x will NEVER equal 0.
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Have you taken a calculus class? I am not asking in condescension; I am genuinely curious. Dealing with infinities is the mathematics of calculus. If the decimal 0.99999999999999 stopped after ten trillion 9s, then that [b]would not[/b] equal 1 technically. But since 0.999... is never ending, it is mathematically equal to 1. Infinity is simply a word we ascribe to the notion of an entity without limit.
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Yes, I've taken multivariable calculus, vector calculus, and a class dedicated to the calculus of orbital mechanics. My professors must have just taught things differently from yours. I see where you're going with your proof and everything, and I accept that for ALL real world applications the repeating decimal 0.999... = 1, but i do not agree that it does in the world of mathematics. Things get complicated when you deal with situations where a variable, or for that matter the number of significant figures, go's to to infinity. It's all just a matter of how you were taught I suppose.
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Do you accept that the sum of an infinite geometric series is equal to the quotient of the first term in the sequence and the difference between the 1 and the common ratio (written below)? S = a1 / 1 - r
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Do you also accept that 0.999... can be expressed as infinite series defined by the general term 0.9(1/10)^n with the start of n = 0 and steps of 1?
