0.99 Repeating does Not equal 1.00
Voting Style:  Open  Point System:  7 Point  
Started:  9/14/2012  Category:  Science  
Updated:  5 years ago  Status:  Post Voting Period  
Viewed:  10,821 times  Debate No:  25599 
Only rules are keeping it on topic and proving to the voters that .9 repeating does or does not equal 1.
This is a common topic that is brought up all the time, at least in my experience, it is too often left at very beginning stages of argument and accepted as fact. I am here to convince the voters that .99 repeating = .99 repeating, not 1. I will commence opening statements after the challenge is accepted. I accept the challenge. Pro (mobikenobi) needs to prove that 0.99 repeating is not equal to one. Con (Baggins) needs to point our mistakes in Pro's argument and hence establish that 0.99 repeating can be treated as 1 for all practical purpose. Best of Luck to my opponent for the opening round. 

For the sake of time and word count from this point on .99r will be used to indicate .9 repeating.
Before we begin I will bring up the most commonly used proofs that .99r = 1. First you have 1/3 = .33 repeating .33 repeating * 3 = .99r 1/3 repeating * 3 = 1.00 1/3 = .33, .99r = 1.00 .99r = 1x 9.99r = 10x subtract x from each side 9 = 9x 1 = x These are the most commonly accepted examples of why .99r = 1, things that your teacher uses to blow your mind in 8th grade, but have no purpose past that, they are mere parlor tricks. The basis for .99r relies entirely on infinity, the basis for all arguments for .99r = 1 are based using the concept of infinity. That's right the Concept. The reason it is a concept is because it doesn't actually exist, your brain cannot comprehend it because everything in our universe is in fact finite. Everything you experience, see, do, its all of the finite nature, how many blades of grass are on earth, how many gallons water are in the ocean, how many hairs are on your head, its a finite number, it can be counted. This is relative because .99r is nine repeating, infinitely, it never reaches 1. PROOF 1: Simple Analysis The number 1 can be written in a variety of ways, but for our case here lets go ahead and right down 1 = 1.00 repeating, the 0s are repeating to infinity, it is another way to write 1. .99r can be rewritten as 0.99r, the 9s are repeating to infinity, another way to write .99r. From simple looking at the numbers, 1.0 > 0.99r simply because 0.99r has a 0, it has a decimal, it has to be less than one. Proof 2: Subtraction Though infinity cannot ever be "reached" depending on how you look at it, I would rather not start a debate on infinity, you can predict the outcome using things such as limits. Because infinity is infinite, I will simply show a pattern that repeats to infinity. 1  0.9 = 0.1 1  0.99 = 0.01 1  0.999 = 0.001 1  0.9999 = 0.0001 1  0.99999 = 0.00001 Continue however many times you want, literally continue this a trillion times, and you will never reach 0. Repeat this a trillion times to the trillionth power, repeat this for every atom there is in the universe. You will never ever reach 0, you will gain another 0, but the 1 will keep moving and never go away. Proof 3 Limit Lim x = 1/(1x) The limit of x in this equation is 1. If x Ever reaches the value of 1 then the denominator will turn into 1/(11) or 1/0. Because 1/0 is not possible, the value of x can never reach 1, the domain of x is {x:x=/=1} So a limit is defined basically as a value that you can approach but never reach, that's all that you need to do know for this situation. So as I said, the domain of x is ANY number that is not 1. 1/(1x) you can use the value 1 trillion if you want, you will get 1/999999999999, a very tiny number, but still in the domain of x. x literally can never equal 1 in this equation. It can reach any other number, and as the definition of the limit is that you are approaching 1 then lets see what the closest number to 1 is. .9, well you are still a tenth off .99, well you are still a hundredth off .999, well you are still a thousandth off .9999, well you are still a tenthousandth off .99999, well you are a hundredthousandth off The limit is 1 so you can approach 1 but never cross it So you can approach 1 for how long? Well infinitely, you can continue to go to the next decimal, .99999999999, but you can never cross the threshold between .9999r and 1.000r. so the limit approaches 1, or .99r so if the limit was defined as say a function of y y = .99r and we know from the equation 1/(1x) that you can never reach 1 y =/= 1 What's that? .99r = y but y =/= 1? .99r =/= 1 I believe this is enough to prove that .99r =/= 1 Q. How many mathematicians does it take to screw in a bulb? A. 0.9999… repeating == General Discussion == My opponent has made a comment, which highlights the problem with his approach. The basis for .99r relies entirely on infinity, the basis for all arguments for .99r = 1 are based using the concept of infinity. That's right the Concept. The reason it is a concept is because it doesn't actually exist, your brain cannot comprehend it because everything in our universe is in fact finite. Infinity is a concept which exists in mathematics. Since this is a mathematical debate, we will use this concept. In case, my opponent wants to construct a new kind of mathematics which does not contain infinity, the number 0.9r would be meaningless. After all 0.9r means infinite number of 9s after decimal. == My arguments == Apparently my opponent’s school teacher presented him a simple proof why 0.9r = 1 in standard 8th. He has presented the proof and asserted that: These are the most commonly accepted examples of why .99r = 1, things that your teacher uses to blow your mind in 8th grade, but have no purpose past that, they are mere parlor tricks. However he has not explained what is wrong with the proof. Is it wrong because it is too simple? Until he can present the problem with proof, I would like to submit that as the first proof in favor of my case. I will present an additional proof which is much more direct. This proof was first presented by Euler. Consider a decimal number (x = 0.abcd) where abcd are the digits. Then we can consider ‘x’ as sum of the series. x = (a/10) + (b/100) + (c/1000) + (d/10000) When defined this way we can see that, 0.9r = (9/10) + (9/100) + (9/100) +++ till infinite number of terms. This is an infinite geometric progression with a = 9/10 and r = 1/10. Sum of an infinite G.P with r < 1 is given by a/(1r). This means: 0.9r = (9/10)/{1(1/10)} = 1 This proof is more direct since it utilizes the way decimal numbers are defined in terms of series. == Rebuttals == 1. Simple Analysis: My opponent proposes that 1 can be written as ‘1.0000 repeating’. This according to my opponent is greater than 0.999999 by ‘simply looking at the numbers’. The flaw lies in the fact that both these numbers continue up to infinite terms. We cannot judge which number is greater by ‘simply looking’. A more clear proof is needed. For example my opponent would need to prove that difference between them is nonzero. 2. Subtraction: Mobikenobi actually tries to do subtraction and establish a pattern. I will use the same pattern to prove that 0.9r = 1. The pattern established by my esteemed opponent is: 1 – (0.99 repeating till k terms) = 0.<k1 zeros>1 = 1/(10^k) When we take the limit k as tending to infinity, we get: 10.9r = 0 My opponent challenges us to imagine the pattern extending to infinity. This is clearly not simple. Imagining things which are infinite is clearly difficult for our finite intelligence. However we can still imagine them with the help of mathematics, as I have shown. 3. Domain of y = 1/(1x): It has been correctly pointed out that domain of the function y=1/(1x) is: x={x:x is element R and =/= 1} Turns out that 0.9r is not in the domain either. If we put x=0.9r, the denominator becomes zero and the function becomes ill defined. However we just discussed that domain of this function includes all real numbers except 1. This must mean 0.9r=1 ==Conclusion== All my opponent’s arguments rely on same pattern. It is difficult to imagine a number which extends to infinite number of terms. Mathematics provides us the tools to extend our imaginations. Once we start our thinking through mathematics, it becomes immediately obvious that the concept 0.9r=1 poses no challenge to intellect. 

mobikenobi forfeited this round.


mobikenobi forfeited this round.
It is possible that my opponent got stuck in some important work. Best of luck to him. In the meantime, this looks like a simple win for me. 

mobikenobi forfeited this round.

mobikenobi  baggins  Tied  

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Total points awarded:  0  7 