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The 0.999…=1 Fallacy

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The 0.999…=1 Fallacy The 0.999…=1 fallacy. This fallacy has its origins in Euler’s wrong ideas. The only support comes from Euler’s definition of infinite sum, which is both illogical and spurious. Ignorant academics have unfortunately subscribed to Euler’s wrong ideas and a sewer load of theory has arisen therefrom. But what can one expect from a man who believed that = 0 ? After the advent of Georg Cantor’s poisonous ideas on set theory and non-existent real numbers, things grew progressively worse. A proof for simpletons: Consider that if ≠ 0.333 … , then 0.999 … ≠ 1. The support for the = 0.333 … equality comes from ignorance. However, for this short proof I will assume that it is correct. After I debunk the equality, I will proceed to show that non-terminating decimals are a fallacy that is based on incorrect logic (more like the lack of it). Proof: We know that: 3 1 1 = () = 1 − 10 3 10 Now followers of Euler (academic morons) like to think that 3 1 = 10 3 In which case, the direct implication is that (∞) = , but as we already know, (∞) does not exist as a well-defined concept. If indeed (∞) were possible, then the graph of () = 1 − would have no asymptote. In fact, the concept of limit in this respect, would also not exist. Therefore, the equality = 0.333 … is ill-formed and false. Radix representation: Representing numbers in radix systems is just a way of finding another equivalent fraction in that radix system. For example: to represent in base 10, we write 0.25 which means + = = . It is impossible to find an equivalent fraction for in base 10, because 3 is not one of the prime factors of 10. To argue ignorantly that + + + ⋯ = , is claiming that this statement regarding prime factors is FALSE. + + + ⋯ is not an equivalent fraction because the ellipsis is meaningless nonsense. Infinity is an ill-formed concept that has no place in rational thought and is not required in any mathematics. Debunking the popular notion that = . … because of long division. I begin this proof by stating that the belief that = 0.333 … has absolutely nothing to do with long division. I suppose that it would be helpful to see why long division works in the first place. Almost all mathematics professors have no idea why it works. So they should pay careful attention to this proof. The long division algorithm has its origins in Euclid’s Algorithm. Assume that and are two integers greater than 0. Furthermore, let denote the quotient and the remainder (if any). For long division to work, we must have: = ( × )10 + ( × )10 + ( × )10 + ⋯ + ( × )10 + ( × ) + So, = × … + = + where 0 ≤ < This last statement is exactly Euclid’s Algorithm. So how do we apply it to ? 1 = (3 × 0) + 1 Now, where in the previous statement do you see anything resembling that process mathematicians use to find 0.333 … , that is, the so-called long division algorithm? Nowhere! You might be tempted to argue as follows: = × + 1 = × 3 + = × 3 + = × 3 + where the quotient is given in red, the divisor in green, the remainder in light brown and the dividend in blue. Whence, 1 = + + + . Observe that = 3, and is always greater than . Furthermore, no stopping condition exists because is never equal to 0. The reason for this, is that long division has nothing to do with the process ignorant academics refer to in arriving at the fallacy = 0.333 … The process used to arrive at the ill-formed conclusion that = 0.333 … is as follows: 1 1 10 10 1 = × = × 3 3 10 3 10 1 1 1 = 3 + × 3 3 10 1 1 1 1 = 3 × + × 3 10 3 10 1 1 1 1 10 = 3 × + × × 3 10 3 10 10 1 1 10 1 1 = 3 × + × × 3 10 3 10 10 1 1 10 1 = 3 × + × 3 10 3 10 1 1 1 1 = 3 × + 3 + × 3 10 3 10 1 1 1 1 1 = 3 × + 3 × + × 3 10 10 3 10 So, we can continue in this fashion or notice that the 3s repeat. Or we can be smart and simply write: = 0.3( ) + × where is an integer greater than or equal to 0. The method just described has ABSOLUTELY NOTHING to do with long division. Yet idiot academics have been claiming this for the last few hundred years. Strange? You decide. Representing magnitudes and numbers: Since is a well-defined number, one ignorantly supposes that it can be represented in any radix system, in particular base 10 as 0.333 … This assertion is false because placing an ellipsis (…) behind the last 3 does not make it valid or meaningful. Infinity does not belong in well-defined concepts. In fact, infinity has no place in mathematics. 1 We can well define 3 as follows in base 10: 1 1 1 0.3(n times) n 3 3 10 [Well defined] Now if one puts any of n 1,2,3,4,5,... into [Well defined] and uses the same common false proof, that is, x0.999... 10 x 9.999... 9 x 9 x 1, then choosing n 7 gives: 1 1 1. x 0.3333333 7 3 10 1 2. 3x 0.9999999 10,000,000 9999999 1 10,000,000 3. 3x 10,000,000 10,000,000 10,000,000 4. 3x 1 1 5. x 3 1 1 0.9999999 3x 1 Observe from step 2 that 10,000,000 because , 1 x so 3 . Now, let's choose n ( is an ill-defined concept) and repeat the process: 1 1 1. x 0.333... 3 10 1 2. 3x 0.999... 10 999... 1 999... 1 10 3. 3x 10 10 10 10 4. 3x 1 1 5. x 3 And once again, even by using the ill-defined concept of infinity, we have the same 1 3x 1 x result, that is, 3 which is what we started out with. Euler incorrectly defined an infinite sum to be: a arn a a S lim lim 0 n1r n 1 r 1 r 1 r Where does Euler go wrong in his definition of infinite sum? a arn a1 a S lim lim lim 0101 n1r n 1 r 1 r n 10 1 r a 1 1 But we know that the difference 1 0.999..., 1 r 10 10 1 1 that is, 1 0.999... 1 0.999... , therefore 1 0.999... 10 10 1 Step 2 reveals Euler's error. In his definition, he simply discards , that is, Euler 10 treats this as zero, but it is not zero! This misconception arises from the fact that 1 lim 0 n . n 10 However, mathematicians don't really understand limits that well. In fact all of them don't know what is a number. What the last expression means is: 1 As n increases without bound, the least value (greatest lower bound) that 10n cannot attain is 0 . 1 This is the correct interpretation of limn 0 . n 10 When one uses well-defined objects, the results are always consistent. Note that step 2 proves 0.999... 1. Yet another proof: 1 1 1 0.3(n times) n Take the limit of both sides of 3 3 10 : 1 1 1 lim lim 0.3(n times) lim n n3 n n 3 10 The previous statement is equivalent to: 1 1 1 lim 0.333... lim n n3 n 3 10 [E] According to Euler's formula for an infinite sum: 3 1 1 n 3 3 3 10 10 0.333... lim lim 2 ... n n9 n 10 10 10 10 1 1 1 0.333... lim lim n n3 n 3 10 Putting this in [E]: 1 1 1 1 1 1 lim lim lim n lim n n3 n 3 n 3 10 n 3 10 [CR] 1 1 3 3 1 It can be seen that discarding the portion of 3 , that cannot be measured exactly in 1 1 lim n base 10, that is, n 3 10 , would have prevented us from reaching the correct conclusion of the RHS of [CR]. 1 1 lim n 0 Simply replacing n 3 10 with is therefore incorrect, that is, Euler's definition of infinite sum is flawed. A lesson on fractions. After several discussions with academics on Space time and the Universe forum: What does it mean to represent rational numbers in radix systems? I realized that all of them did not understand what it means for a rational number to be represented in a radix system. So I decided to include this section to clarify some of these details. The long division algorithm was used by mathematicians to facilitate the conversion of a rational numbers of the form into some radix form b n n1 b1 b 2 b3 bn an10 a n1 10 ...10 a 1 a 0 ... . 10 102 10 3 10n It was soon discovered that rational numbers whose denominator was not a factor of the radix could not be represented finitely in that radix system, that is, it was not a possible to represent the remainder of , because it cannot be measured by b the finite sum ONLY of the products of integers and reciprocal powers of that radix. 1 A common misconception is that 0.333... 3 1 It is easy to convert to base 10 if we use the following definition: 3 1 1 1 0.3(n times)... n 3 3 10 where n and n 1. The problem occurs when one tries to have n : 1 1 1 0.333..
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