1.999 repeating = 2

First of all, I want to indicate that this is not a philosophical debate. This is a mathematical debate, and anyone that has taken a course involving mathematical proof knows that definitions in mathematics are the basis or building blocks of any proof. You indicated that: "There is no reason why a rational number cannot be equal to an irrational number". By mathematical definition (not philosophical), an irrational number cannot be equal to a rational number. You can check this definition in any mathematics book. You indicated that: "Your definitions are great, but they don't necessarily preclude any possibility that the two could be equal to each other, only that the two cannot be IDENTICAL to each other". Well, first of all, there are not "my" definitions of rational and irrational numbers. This are the definitions that have been established for a long time, you can find them in any math book, and all mathematicians agree with them. And the definition clearly indicates that an irrational number cannot be equal to a rational number. Again, this is a mathematical debate, not a philosophical one were people can debate whether an apple can equal to an orange or whatever. You indicated that: "Tarzan's proof is rock solid and your semantical arguments are attempts at struggling to find any way you can not to believe something that is hard to accept". I don't see how your subjective opinion is PROOF that 1.999 repeating equals 2. You indicated that: "A board that is 1.999 repeating feet long is just as long as a board that is 2 feet long, because you can never identify the difference in length. How much less than 2 is 1.999 repeating? If they are in fact different values, than you should be able to extrapolate the difference between them" The fact that I cannot identify the difference between a 1.999 repeating feet board and a 2 feet board does not mean these two are equal. Using your logic, then 1.88 is equal to 2 as well, since I am pretty sure I wouldn't be able to identify the difference between a board that is 1.88 feet long and a board that is 2 feet long. You indicated that: "Also, a full third is 0.333 repeating. But if you simply look at the number, 0.333 repeating SEEMS like it is *just* a little less than a third, because the numbers don't do it justice, in the same way that 1.999 repeating doesn't quite do justice to represent 2" I indicated above that 0.333 repeating comes from the ratio of two integers, namely 1/3. If you calculate this ratio, and divide 1 by 3, you get 0.333 repeating. Therefore this shows clearly that 1/3 equals 0.333 repeating, and that 0.333 repeating is a rational number. You indicated that: "If you multiply .333 repeating by 3 you get .999 repeating, which is equal to 1 because three thirds are a whole" This is the origin of the problem, repeating numbers cannot be multiplied by integers since this operation is undefined in a similar way that 1/0 (one divided by zero, not the limit, just the number one divided by zero) is undefined. What I mean by "repeating numbers cannot be multiplied by integers" is that there is no mathematical definition or property that indicates that 3*0.333 repeating = 0.999 repeating, and you cannot get from one point to the other by using mathematical properties or definitions. However, you could show that 2*0.333 repeating equals 0.666 repeating by using the following mathematical properties: 2*0.333 repeating = 2*(1/3) This comes from the mathematical definition of dividing 1 by 3 = (2*1)/3 Associative property of multiplication = 2/3 Identity property of multiplication = 0.666 repeating This comes from the mathematical definition of dividing 2 by 3 The problem that arises with 0.999 repeating is that you cannot obtain it by dividing any two integers. Therefore you CANNOT show that 3*0.333 repeating = 0.999 repeating. You indicated that: "How can you not add a repeating decimal??? Does this mean I can't add three thirds to get a whole? Aren't all numbers repeating decimals, with zeros on each end?" As I indicated, you cannot add, multiply, substract, or divided, repeating numbers directly using math properties. There are properties that allow you to do this operations on integers or fractions, but not on repeating (or infinite numbers). And yes, all numbers can be written as repeating numbers, but the difference is that if you can write a repeating number as an integer or a fraction then you can perform addition, multiplication, substraction, or division on these forms of the number (NOTE: all numbers can be written as repeating numbers, however not all of them can be written with zeros at the end). In Tarzan's proof the flaw is in the fact that he multiplies 10 by A, then substracts A, and gets 18. What you should really get if you perform 10*A minus A, you will get 18.0000...(infinite zeros)...0001 As you can see, there will always be a one at the end of this infinite number (I know that for the human brain this is hard to grasp, but that is infinity).