Infinite 0.9 Not Equal To 1 - Page 2 - XP Math - Forums

 XP Math - Forums Infinite 0.9 Not Equal To 1

 05-02-2008 #11 jjbsports7 Guest   Posts: n/a Wll its close like its 99% out of a 100%. So I say its not equal to one!
 05-08-2008 #12 tajjet Guest   Posts: n/a Hmm... I plugged in on the computer's calculator: 0.9999999999+ 0.1111111111 and it came up with 1.1111111111 so therefore it considers 0.99999999 to equal 1
 05-08-2008 #13 hunter34 Guest   Posts: n/a Calculators can be made to be wrong. Not a nice way to prove ideas.
 06-11-2008 #14 0.9recurringequals1 Guest   Posts: n/a staying away from any mathematical proofs at the moment, would the doubters amongst you please tell me what the difference between 0.9 recurring and 1 is, if you believe they are not equal
 06-12-2008 #15 hunter34 Guest   Posts: n/a 0. Another Explanation: Perhaps the most common development of decimal expansions is to define them as sums of infinite series. In general: $b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\frac{1}{10}}) + b_2({\frac{1}{10}})^2 + b_3({\frac{1}{10}})^3 + b_4({\frac{1}{10}})^4 + \cdots$ . For 0.999… one can apply the powerful convergence theorem concerning geometric series: If |$r | < 1$ then $ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}$. Since 0.999… is such a sum with a common ratio $r=\textstyle\frac{1}{10}$, the theorem makes short work of the question: $0.999\ldots = 9(\frac{1}{10}) + 9({\frac{1}{10}})^2 + 9({\frac{1}{10}})^3 + \cdots = \frac{9({\frac{1}{10}})}{1-{\frac{1}{10}}} = 1$ This proof (actually, that 10 equals 9.999…) appears as early as 1770 in Leonhard Euler's Elements of Algebra. The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebra proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999…. A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis. A sequence $(x0, x1, x2,\cdots)$ has a limit x if the distance $|x - xn|$ becomes arbitrarily small as n increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit: $0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1$ The last step — that $\lim \frac{1}{10^n} = 0$ — is often justified by the axiom that the real numbers have the Archimedean property. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small". Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1.
 06-12-2008 #16 0.9recurringequals1 Guest   Posts: n/a ok, i am putting forward the challenge for all the people who do not believe that 0.9 recurring and 1 are equal to tell me the difference between them, as there must be a difference if you believe them not to be equal.
 06-13-2008 #17 Sillysidley   Join Date: Oct 2006 Posts: 822 Meh hunter, I doubt that most people on here calculus... at least I don't I don't even remember the formula for the sum of a general geometric series, I just derive it when I need to. But for the $a+ar+ar^2+ar^3+...$, the formula's $\frac{a}{1-r}$ when $|r<1|$, the proof works. __________________ .
 02-23-2011 #18 Anthony.R.Brown   Join Date: Feb 2011 Posts: 1 2011 New Year - New Way of Looking at Infinite-Recurring 0.9 2011 New Year - New Way of Looking at Infinite-Recurring 0.9 OK! The Way I would like to put forward is...to find The Negative,The Root,and The Positive.* Below is an Example Using 100* 100 Negative = 99.9* 100 Root = 100 ( THE ROOT IS THE START! THE ORIGIN! ) (The Root) is Not The Square Root)* 100 Positive = 100.1* A Better way of Looking at the above is 100N(99.9) 100R(100) 100P(100.1)* ---------------------------------------------------------------------------------------------------- -------- So Now all we have to find is...* Infinite-Recurring 0.9 Negative = ?* Infinite-Recurring 0.9 Root = ?* Infinite-Recurring 0.9 Positive = ?* A Better way of Looking at the above is IR0.9N( ? ) IR0.9R( ? ) IR0.9P( ? )* Infinite-Recurring 0.9 Negative = ? Put Here I/R 0.9 as a Negative Value* Infinite-Recurring 0.9 Root = ? Put Here what you think I/R 0.9 is ?* Infinite-Recurring 0.9 Positive = ? Put Here I/R 0.9 as a Positive Value* Meanwhile, can you find a number between 0.9999... and 1? = 0.9(.9)1* I have seen Many Example Calculations for Infinite-Recurring 0.9* But no One is Showing the Math from Root as Explained in my Example to it Becoming or is equal to 1* Below is an Example Using 100* 100 Negative = 99.9* 100 Root = 100* 100 Positive = 100.1* Can the above be Shown in the same way for Infinite-Recurring 0.9 Now we are Getting somewhere! Let me show what is wrong with the Examples Below...* 0.99999... - 0.1 = 0.8999999... = 0.8 + 0.09999... = 0.8 + 0.1 = 0.9* 0.99999... = 1* 0.99999... + 0.1 = 1.0999999... = 1 + 0.09999... = 1 + 0.1 = 1.1* First of all you Cannot Add or Subtract Etc. from Something that is Continuous! In this case Infinite-Recurring 0.9 As soon as you do one or more of the .9s is no longer Continuous!* In the Negative Example Below a .9 has become .8* 0.99999... - 0.1 = 0.8999999... = 0.8 + 0.09999... = 0.8 + 0.1 = 0.9* In the Root Example Below All the .9s Have Stopped being Continuous! By Assuming They All = 1* 0.99999... = 1* In the Positive Example Below...Which is the Worst case! All the .9s Have Stopped being Continuous! Plus There is Now a .1* 0.99999... + 0.1 = 1.0999999... = 1 + 0.09999... = 1 + 0.1 = 1.1* What the Above shows is that you Cannot Apply Calculations to Something that is Continuous!* This is the Mistake everyone is doing when trying to Prove Infinite-Recurring 0.9 = 1* The Next Main Mistake everyone is doing is Trying to Go Against the Above...by Calculating the Ten Times Calculation 10 x Etc.* There are Three types of Ten Times Calculation Which Concern the Infinite-Recurring 0.9 Problem! Only One is ever being used None of the other Two are mentioned as Possibilities!?* The First is the Normal 10 x 0.9 = 9 This is OK if the 0.9 is a Single 0.9 But I have seen many Example where it is Trying to be Applied to 0.999...Etc. Against the Above.* The Second is The 10 x 0.9 Which could Equal (.9999999999) That is Ten of the Continuous .9s But How can you Separate The Ten From the Rest? Again Against the Above.* And the Third is 10 x All the .9s Again Not Possible Because of the Above.* So to End this... Infinite-Recurring 0.9 Must always be Known and Shown as Having Continuous .9s* ----------------------------------------------------------------------------------------------* The Philosopher Quotes: “ I Think Therefore I am “* The Mathematician answers “ I Continue Therefore I’m Recurring “* ----------------------------------------------------------------------------------------------* GeniusIsBack.
 05-20-2015 #19 eliseo   Join Date: May 2015 Posts: 5 Correct way of viewing 0.9 repeating 1. Well you think that just because 0.999.... looks different from 1 you think it's not 1. It's like saying 1/2 does not equal 0.5 or ice is not water. 2. Subtract 0.999.... from 1. What do you get? 0.000....0001. This is EQUAL to zero because there are INFINITE zeros before the one. Therefore we will never reach the one. If the difference is zero, the numbers are the same. 3. Referring to reason 2. How are 1 and 1.000.....001 the same? Well, with decimals that go on forever, there is no last number. The last number of 1.000.....001 is not 1, because we never reach the end. Same with 0.999.... There is no end. 0.999.... goes on forever, because it's recurring. So 0.999....5 = 0.999....02 because we never reach the 5 or the 02. 4. Infinity can get us anywhere. Because the number 0.999.... is recurring, it equals one. Infinitely large is impossible to count, and infinitely small is nothing. Do not underestimate the size and power of infinity. In 5. If you think that 0.999.... is not 1 because it is recurring, and recurring decimals cannot equal natural numbers, well, you're absolutely wrong because 1 is recurring: 1.000000000000............ 6. If 0.999.... does not equal 1, this pattern breaks down: 1/9 = 0.1 + 0.01 + 0.001 + 0.0001... 2/9 = 0.2 + 0.02 + 0.002 + 0.0002... 3/9 = 0.3 + 0.03 + 0.003 + 0.0003... 4/9 = 0.4 + 0.04 + 0.004 + 0.0004... 5/9 = 0.5 + 0.05 + 0.005 + 0.0005... 6/9 = 0.6 + 0.06 + 0.006 + 0.0006... 7/9 = 0.7 + 0.07 + 0.007 + 0.0007... 8/9 = 0.8 + 0.08 + 0.008 + 0.0008... 9/9 = 0.9 + 0.09 + 0.009 + 0.0009... 10/9 = 1.0 + 0.10 + 0.010 + 0.0010... 11/9 = 1.1 + 0.11 + 0.011 + 0.0011... 12/9 = 1.2 + 0.12 + 0.012 + 0.0012... The ninth line shows 9/9, which is equal to 1. But it also equals 0.9 + 0.09 + 0.009 + 0.0009... which is 0.999.... So 0.999.... does equal 1. Trust me, I am 99.999....(100) percent sure this is correct. I'm not on the "yes" side just because mathematicians are on the same side, but I know 0.999.... = 1 because I did the math. If you are still not convinced, you are not correct. Have any doubts about this topic? I'll be happy to answer any questions you have to bring up. By the way it's okay to be wrong, as long as you're right in the end. Losing a debate is fine, because one side has to win. Eliseo
06-19-2015   #20
naruto beast

Join Date: May 2015
Posts: 136

really long post wow
Quote:
 Originally Posted by eliseo 1. Well you think that just because 0.999.... looks different from 1 you think it's not 1. It's like saying 1/2 does not equal 0.5 or ice is not water. 2. Subtract 0.999.... from 1. What do you get? 0.000....0001. This is EQUAL to zero because there are INFINITE zeros before the one. Therefore we will never reach the one. If the difference is zero, the numbers are the same. 3. Referring to reason 2. How are 1 and 1.000.....001 the same? Well, with decimals that go on forever, there is no last number. The last number of 1.000.....001 is not 1, because we never reach the end. Same with 0.999.... There is no end. 0.999.... goes on forever, because it's recurring. So 0.999....5 = 0.999....02 because we never reach the 5 or the 02. 4. Infinity can get us anywhere. Because the number 0.999.... is recurring, it equals one. Infinitely large is impossible to count, and infinitely small is nothing. Do not underestimate the size and power of infinity. In 5. If you think that 0.999.... is not 1 because it is recurring, and recurring decimals cannot equal natural numbers, well, you're absolutely wrong because 1 is recurring: 1.000000000000............ 6. If 0.999.... does not equal 1, this pattern breaks down: 1/9 = 0.1 + 0.01 + 0.001 + 0.0001... 2/9 = 0.2 + 0.02 + 0.002 + 0.0002... 3/9 = 0.3 + 0.03 + 0.003 + 0.0003... 4/9 = 0.4 + 0.04 + 0.004 + 0.0004... 5/9 = 0.5 + 0.05 + 0.005 + 0.0005... 6/9 = 0.6 + 0.06 + 0.006 + 0.0006... 7/9 = 0.7 + 0.07 + 0.007 + 0.0007... 8/9 = 0.8 + 0.08 + 0.008 + 0.0008... 9/9 = 0.9 + 0.09 + 0.009 + 0.0009... 10/9 = 1.0 + 0.10 + 0.010 + 0.0010... 11/9 = 1.1 + 0.11 + 0.011 + 0.0011... 12/9 = 1.2 + 0.12 + 0.012 + 0.0012... The ninth line shows 9/9, which is equal to 1. But it also equals 0.9 + 0.09 + 0.009 + 0.0009... which is 0.999.... So 0.999.... does equal 1. Trust me, I am 99.999....(100) percent sure this is correct. I'm not on the "yes" side just because mathematicians are on the same side, but I know 0.999.... = 1 because I did the math. If you are still not convinced, you are not correct. Have any doubts about this topic? I'll be happy to answer any questions you have to bring up. By the way it's okay to be wrong, as long as you're right in the end. Losing a debate is fine, because one side has to win. Eliseo
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