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What is .99 repeating as a fraction?
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| February 2nd, 2012 at 2:57:38 PM permalink | |
| Doc Member since: Feb 27, 2010 Threads: 21 Posts: 2825 | I'm curious -- if you don't think the infinite repeating decimal .999... is equal to 1, how much do you think they differ by? Is that difference something that does not equal zero? On the chance that you think they differ by something equivalent to zero, how is that different from being equal? |
| February 2nd, 2012 at 3:34:01 PM permalink | |
| MathExtremist Member since: Aug 31, 2010 Threads: 46 Posts: 2521 |
An exponentially decreasing number? What does that mean? Let X = 1 and Y = 0.999... then X-Y is not an "exponentially decreasing number". It is a constant, because both X and Y are constants. Pop quiz: what is X-Y? "In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563 |
| February 2nd, 2012 at 3:34:33 PM permalink | |
| YoDiceRoll11 Member since: Jan 9, 2012 Threads: 7 Posts: 529 |
Please re-read my last post. I can't quantify how much .999..., is different to 1. Because there is always an infinitely smaller number between it and 1. Plain and simple. .999 does not equal 1. .99999999999999999999999999999999999 does not equal 1 .999999999999999999999999999999999999999999999999999999999999999999999999999999 does not equal 1. Extrapolate to your heart's content. |
| February 2nd, 2012 at 3:39:12 PM permalink | |
| MathExtremist Member since: Aug 31, 2010 Threads: 46 Posts: 2521 |
None of the numbers you just wrote are equal to 0.999... They are equal to some finite number of 9s following the decimal. Your idea is correct for any finite number of 9s following the decimal, but not for 0.999... which has an infinite number of 9s following the decimal. "In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563 |
| February 2nd, 2012 at 3:40:04 PM permalink | |
| weaselman Member since: Jul 11, 2010 Threads: 17 Posts: 1924 |
You are right as long as there is only a finite number of nines. If there are infinitely many of them, then obviously there is nothing bigger than it yet smaller than 1, because you can't add more digits to it, and you can't increase any of them. In fact, any (non-zero) terminating rational number has two alternative decimal representations, not just 1 for example: 1.25 = 1.24999999... etc. An important thing to understand is that this is not two numbers that happen to be equal, but rather two different ways to write down the same number. Much like 1.25 = 5/4 = 1+1/4 etc. Note, that this is not some kind of mind bugling fundamental property of mathematics, just a curios artefact of the positional numerical notation, that we use to express decimals. Roman numerals do not have such an oddity and neither do simple fractions. "When two people always agree one of them is unnecessary" |
| February 2nd, 2012 at 3:46:48 PM permalink | |
| MathExtremist Member since: Aug 31, 2010 Threads: 46 Posts: 2521 | Moreover, this property of repeating decimals is found in many other instances besides .999... Consider 5/11. In decimal notation, that equals 0.4545... (45 repeating). Since that is the definition of how to denote 5/11 as a decimal number, it would be silly to turn around and suggest that 0.4545... is somehow not equal to 5/11 based on the logic that 0.45 != 5/11, 0.4545 != 5/11, 0.45454545454545 != 5/11, etc. "In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563 |
| February 2nd, 2012 at 3:50:07 PM permalink | |
| Doc Member since: Feb 27, 2010 Threads: 21 Posts: 2825 |
I did not ask about the difference between 1 and any of those decimals with a finite number of digits. I asked how much 1 differs from the infinite repeating decimal .999... and whether you think that difference is something other than zero. I think you deliberately evaded answering the questions that were asked. Oh, by the way, I did re-read your last post where you made the erroneous statement (repeated in your next post):
At least that statement is false to anyone who understands what an infinite repeating decimal is. |
| February 2nd, 2012 at 4:33:06 PM permalink | |
| thecesspit Member since: Apr 19, 2010 Threads: 38 Posts: 3108 | Let x = 0.999... (recurring). x * 10 = 9.9999999..... = 9 + x x*10 - x = 9x => 9x = 9 + x - x => 9x = 9 => x = 1 QED. "Then you can admire the real gambler, who has neither eaten, slept through nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire, for a coup at trente-et-quarante" - Honore de Balzac, 1829 |
| February 2nd, 2012 at 4:34:21 PM permalink | |
| thecesspit Member since: Apr 19, 2010 Threads: 38 Posts: 3108 | DOUBLE POST "Then you can admire the real gambler, who has neither eaten, slept through nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire, for a coup at trente-et-quarante" - Honore de Balzac, 1829 |
| February 2nd, 2012 at 4:43:17 PM permalink | |
| s2dbaker Member since: Jun 10, 2010 Threads: 34 Posts: 1215 | .333... = 1/3 .666... = 2/3 .999... = 1 |
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