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#151
01-10-2006, 08:53 PM
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Re: .999~ = 1, Agree?
[ QUOTE ]
[ QUOTE ] [ QUOTE ] I am pretty amazed actually. I never ever came across the concept that .999... was the same as 1. We're taught in British schools (or at least I was) that infinite series are approximations. 0.333... isn't really 1/3 for example, but good enough. And the proof of this was that 0.9999... <> 1!!! Can any UKers remember this from school? [/ QUOTE ] Are you serious? So in the UK 1/3 is not .3 Repeating? [/ QUOTE ] Sheesh. It's taught as an approximation that's as near as you can get. Was that not clear in the statement [ QUOTE ] We're taught in British schools (or at least I was) that infinite series are approximations. 0.333... isn't really 1/3 for example, but good enough. [/ QUOTE ] Should I stick to monosyllables (sorry, "small words") next time? [img]/images/graemlins/grin.gif[/img] (just kidding ya - my "deliberately obtuse" detector went apeshit at your message, and you needed a bit of kidding for ya nit-picking) PS: I do remember specific lessons on this, but I really feel uncomfortable representing the whole UK on this matter. Maybe it was my school/teacher, I dunno. (Don't bother with the moran jokes, I've a PhD, so blow that out yer ass). [/ QUOTE ] Misread what you said, nothing against the UK heh. 
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#152
01-10-2006, 08:53 PM
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Re: .999~ = 1, Agree?
I find it intereasting that all rational numbers can be represented as repeating decimals. Terminating decimals are actually an odd case. Since there are two ways to represent them, thier correct decimal representation is ambiguous.
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#153
01-10-2006, 08:56 PM
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Re: .999~ = 1, Agree?
[ QUOTE ]
[ QUOTE ] Quoted for emphasis, the problem I've always had with this proof is that, while technically correct, it doesn't really help explain "why" .9~=1. If you are trying to prove/explain to someone why .9~=1 it isn't very helpful to assume 1/3=.3~ because if they understood that they would understand (in most cases) that .9~=1. For what are IMO better proofs refer to the wiki article on the subject (the advanced proofs section). [/ QUOTE ] Right, it begs the question, and as such IS NOT A PROOF AT ALL. [/ QUOTE ] It isn't a bad proof for its audience, which is people who can divide 3 into 1 and see that it returns .33333... and can't find an analogy for .999.... since 1 into 1 yields 1. But they take as given that 1/3 = .333 repeating, and then the teacher uses that to show that .999 repeating is actually 1.
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#154
01-10-2006, 08:59 PM
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Re: .999~ = 1, Agree?
[ QUOTE ]
[ QUOTE ] [ QUOTE ] The problem with this post is that there is hard university level stuff underlying the basic algebraic proof. Given that, your seventh grader who votes yes is really doing nothing more than appealing to authority. It is reasonable for an intelligent person to hear the seventh grade explanation, question it, and not understand the deeper explanation. I bet if you were to ask 100 seventh graders whose teacher had just put the proof up the day before, the kid who still questioned it would be the smartest one in the room. [/ QUOTE ] I don't think we need to bring university level math into this at all. 0.3~ is the decimal representation of 1/3. The 0.9~ = 1 thing doesn't even have any mathematical significance in our context. 0.9~ is an oddity of the decimal system that arises because the decimal system is not perfect. [/ QUOTE ] Not really. 9/10 + 9/100 + 9/1000 + ... = 1 whether we use a base 10 decimal system or not. [/ QUOTE ] I'm referring to the notation that uses a decimal point. It doesn't matter what the base is. 0.4~ = 1 in base 5. Just like how that dot is called the 'decimal point' no matter what base we use, I used 'decimal notation' to refer to this way of representing numbers. Let me know if there's a better word for it.
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#156
01-10-2006, 09:07 PM
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Re: .999~ = 1, Agree?
The problem is stated exactly the same if you put the numbers into fractional form, in any base. I think what you are saying is that the weird thing with .9999... in base 10 is that unlike .33333... or .44444..., .9999.... has another equivalent decimal representation (1.0000000), which makes it unnecessary, whereas .333... is necessary b/c it's the only representation we have within the notation for a certain number.
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#158
01-10-2006, 09:24 PM
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Re: .999~ = 1, Agree?
[ QUOTE ]
1 - 0.1 = 0.9 1 - 0.01 = 0.99 1 - 0.001 = 0.999 1 - 0.0001 = 0.9999 1 - 0.00001 = 0.99999 1 - (1/10)^n = 0.999999999 (n instances of 9) n going to infinity = BOOYAKASHA [/ QUOTE ] This is a cleaner way to prove it than the OP.
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#159
01-10-2006, 09:38 PM
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Re: .999~ = 1, Agree?
[ QUOTE ]
The problem is stated exactly the same if you put the numbers into fractional form, in any base. I think what you are saying is that the weird thing with .9999... in base 10 is that unlike .33333... or .44444..., .9999.... has another equivalent decimal representation (1.0000000), which makes it unnecessary, whereas .333... is necessary b/c it's the only representation we have within the notation for a certain number. [/ QUOTE ] Yes, I think you understand what I'm trying to say. However, I didn't say anything about the necessity of 0.9~. Just that it is equivalent to 1.
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#160
01-10-2006, 11:08 PM
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Re: .999~ = 1, Agree?
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Linear Mode
