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0.999... = 1 (?)

Poll: 0.999... = 1?
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0.999... = 1 (?)
Noah
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Posts: 146
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Joined: 2009-03
#101
2010-03-18, 12:59 PM
2147483647 Wrote:Sorry for reviving this thread, but I'm baffled by this. My Calculus teacher, a Ph.D. in Mathematics, told me today that the closed set [0,1) has no upper bound. Specifically, I asked him the question, "what is the highest possible number that can fit in that set"? He replied,"there is no highest number". Then I proposed that the highest number might be 0.999...9! and he replied that it isn't, because it equals 1, and that trumps the definition of the closed set.

=[

It's not too hard to understand, really.
Assume that you have a real number x such that 0 <= x < 1
If you furthermore assume that this number is the highest number existing in this set, then the equation
x < 1 has to be satisfied, and x + ε < 1 implies that ε = 0. Now, choose ε = (1 - x) /2, which implies that x + 2ε = 1.
Then, x + ε < 1, but ε > 0 because x < 1 => 0 < 1 - x. This results into reductio ad absurdum, which means some of our previous assumptions were wrong. As we can find a number 0 <= x < 1, then that means this x is not the highest number in this set, and that it is true for all x.

Noah
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Messages In This Thread
0.999... = 1 (?) - by Nikkey - 2010-03-11, 06:09 PM
0.999... = 1 (?) - by Link - 2010-03-11, 06:14 PM
0.999... = 1 (?) - by Milelke - 2010-03-11, 06:16 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 06:17 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 06:25 PM
0.999... = 1 (?) - by Kevo - 2010-03-11, 06:31 PM
0.999... = 1 (?) - by Link - 2010-03-11, 06:31 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 06:35 PM
0.999... = 1 (?) - by Nikkey - 2010-03-11, 06:37 PM
0.999... = 1 (?) - by Cyadd - 2010-03-11, 06:37 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 06:45 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 06:46 PM
0.999... = 1 (?) - by shouri - 2010-03-11, 06:59 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 07:00 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 07:03 PM
0.999... = 1 (?) - by shouri - 2010-03-11, 07:05 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 07:06 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 07:06 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 07:08 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 07:11 PM
0.999... = 1 (?) - by Nikkey - 2010-03-11, 07:15 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 07:21 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 07:26 PM
0.999... = 1 (?) - by Nikkey - 2010-03-11, 07:26 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 07:27 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 07:29 PM
0.999... = 1 (?) - by Milelke - 2010-03-11, 07:56 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 08:02 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 08:03 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 08:03 PM
0.999... = 1 (?) - by Khoi - 2010-03-11, 08:04 PM
0.999... = 1 (?) - by randompeep - 2010-03-11, 08:16 PM
0.999... = 1 (?) - by Tikey - 2010-03-11, 08:23 PM
0.999... = 1 (?) - by Milelke - 2010-03-11, 08:25 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 08:26 PM
0.999... = 1 (?) - by Manu - 2010-03-11, 08:37 PM
0.999... = 1 (?) - by Dusk - 2010-03-11, 08:39 PM
0.999... = 1 (?) - by PwNeR88 - 2010-03-11, 08:41 PM
0.999... = 1 (?) - by DarkPwnage - 2010-03-11, 08:43 PM
0.999... = 1 (?) - by Corn - 2010-03-11, 08:43 PM
0.999... = 1 (?) - by Hazzy - 2010-03-11, 08:57 PM
0.999... = 1 (?) - by y0y0y0y0shi0 - 2010-03-11, 08:58 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 09:07 PM
0.999... = 1 (?) - by shouri - 2010-03-11, 09:10 PM
0.999... = 1 (?) - by Tikey - 2010-03-11, 09:11 PM
0.999... = 1 (?) - by Hazzy - 2010-03-11, 09:12 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 09:13 PM
0.999... = 1 (?) - by Khoi - 2010-03-11, 09:15 PM
0.999... = 1 (?) - by shouri - 2010-03-11, 09:21 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 09:23 PM
0.999... = 1 (?) - by shouri - 2010-03-11, 09:25 PM
0.999... = 1 (?) - by Dusk - 2010-03-11, 09:35 PM
0.999... = 1 (?) - by Noah - 2010-03-11, 09:42 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 09:47 PM
0.999... = 1 (?) - by Beaner - 2010-03-11, 10:05 PM
0.999... = 1 (?) - by Hazzy - 2010-03-11, 10:07 PM
0.999... = 1 (?) - by Kalovale - 2010-03-11, 10:09 PM
0.999... = 1 (?) - by Noah - 2010-03-11, 10:09 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-11, 10:10 PM
0.999... = 1 (?) - by 2147483647 - 2010-03-11, 10:23 PM
0.999... = 1 (?) - by Sn1perJohnE - 2010-03-11, 10:40 PM
0.999... = 1 (?) - by Noah - 2010-03-11, 10:55 PM
0.999... = 1 (?) - by Shidoshi - 2010-03-11, 11:02 PM
0.999... = 1 (?) - by Russt - 2010-03-12, 12:28 AM
0.999... = 1 (?) - by 2147483647 - 2010-03-12, 01:29 AM
0.999... = 1 (?) - by Russt - 2010-03-12, 01:39 AM
0.999... = 1 (?) - by 2147483647 - 2010-03-12, 02:00 AM
0.999... = 1 (?) - by Nikkey - 2010-03-12, 02:45 AM
0.999... = 1 (?) - by 2147483647 - 2010-03-12, 02:47 AM
0.999... = 1 (?) - by Fiel - 2010-03-12, 04:55 AM
0.999... = 1 (?) - by Kalovale - 2010-03-12, 05:10 AM
0.999... = 1 (?) - by Russt - 2010-03-12, 12:15 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-12, 12:23 PM
0.999... = 1 (?) - by Russt - 2010-03-12, 12:27 PM
0.999... = 1 (?) - by Khoi - 2010-03-12, 12:32 PM
0.999... = 1 (?) - by Horusmaster - 2010-03-12, 12:39 PM
0.999... = 1 (?) - by Russt - 2010-03-12, 12:57 PM
0.999... = 1 (?) - by Hazzy - 2010-03-12, 02:05 PM
0.999... = 1 (?) - by Russt - 2010-03-12, 02:36 PM
0.999... = 1 (?) - by Nikkey - 2010-03-12, 03:39 PM
0.999... = 1 (?) - by shouri - 2010-03-12, 04:37 PM
0.999... = 1 (?) - by Milelke - 2010-03-12, 09:05 PM
0.999... = 1 (?) - by Hazzy - 2010-03-12, 09:15 PM
0.999... = 1 (?) - by Cancambo - 2010-03-12, 09:26 PM
0.999... = 1 (?) - by Milelke - 2010-03-12, 09:33 PM
0.999... = 1 (?) - by Jellyflower - 2010-03-12, 10:31 PM
0.999... = 1 (?) - by Russt - 2010-03-12, 11:50 PM
0.999... = 1 (?) - by Dusk - 2010-03-13, 12:10 AM
0.999... = 1 (?) - by Kabanaw - 2010-03-13, 12:21 AM
0.999... = 1 (?) - by shouri - 2010-03-13, 02:54 AM
0.999... = 1 (?) - by 2147483647 - 2010-03-13, 04:06 AM
0.999... = 1 (?) - by Locked - 2010-03-13, 06:12 AM
0.999... = 1 (?) - by Noah - 2010-03-13, 08:47 AM
0.999... = 1 (?) - by Fiel - 2010-03-14, 05:12 AM
0.999... = 1 (?) - by 2147483647 - 2010-03-14, 05:46 AM
0.999... = 1 (?) - by Shidoshi - 2010-03-14, 08:36 PM
0.999... = 1 (?) - by GummyBear - 2010-03-15, 05:09 AM
0.999... = 1 (?) - by 2147483647 - 2010-03-18, 02:44 AM
0.999... = 1 (?) - by shouri - 2010-03-18, 03:10 AM
0.999... = 1 (?) - by Dusk - 2010-03-18, 09:25 AM
0.999... = 1 (?) - by Noah - 2010-03-18, 12:59 PM
0.999... = 1 (?) - by Jellyflower - 2010-03-24, 11:30 AM
0.999... = 1 (?) - by shouri - 2010-03-24, 04:01 PM

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