0.99 Repeating does Not equal 1.00

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/(1-r). 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 non-zero. 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.<k-1 zeros>1 = 1/(10^k) When we take the limit k as tending to infinity, we get: 1-0.9r = 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/(1-x): It has been correctly pointed out that domain of the function y=1/(1-x) 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.