0.99 Repeating does Not equal 1.00

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/(1-x) The limit of x in this equation is 1. If x Ever reaches the value of 1 then the denominator will turn into 1/(1-1) 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/(1-x) 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 ten-thousandth off .99999, well you are a hundred-thousandth 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/(1-x) 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