1.999 repeating = 2

You wrote the following two statements: "why can't a rational number not be equal to an irrational number?" and "your definition did not include that a rational number cannot be equal to an irrational number. You say that it can't but us debaters on here are probably not just going to take your word for it. Saying that I could check it in a math book is not enough to show your case. Give us some exact wording that can be cross checked so that we can indeed see that a rational number can never be equal to an irrational number." The definition of a rational number is a number that CAN be expressed as a ratio of two integers (where the denominator cannot equal zero). The definition of an irrational number is a number that CANNOT be expressed as a ratio of any two integers. Therefore a rational number CANNOT equal an irrational number (I honestly don't know how to make this point any clearer). By the way, here are some references: http://en.wikipedia.org... http://en.wikipedia.org... First of all I want to clarify that by saying repeating numbers I specifically mean the repeating representation of the number, such as 0.333 repeating. The number 1/3 is a fraction, it is composed by only two integers, while the number 0.333 repeating is composed of infinitely many digits. These two numbers can be proven to be equal by using simply dividing 1 by 3. The math property of dividing 1 by 3 is a well defined property in mathemetics because it involves two finite numbers. Once you convert 0.333 repeating to 1/3 then you can use all the math properties (associativity, commutativity, addition, etc.) that you want on this representation of the number. However, you cannot add, multiply, substract, or divided, on the repeating representation of the number directly using math properties because repeating numbers go on infinitely. The reason for not been able to use addition, substraction, multiplication or division on infinite numbers is because this properties break down when dealing with infinity. For example: infinity + 1 = infinity substracting infinity from both sides gives: 1 = 0 This was a very simple example, but it shows that basic operations break down when dealing with infinity because these operations are undefined for infinity. Therefore, to FORMALLY do an operation on a repeating number, this number will have to be re-written as a fraction and then these basic operations can be performed on the fraction representation of the number. The main problem with 1.999 repating is that it cannot be written as a ratio of any two numbers. I will briefly write the proof of this again (PLEASE CHECK THIS ARGUMENT CAREFULLY TO UNDERSTAND WHY 1.999 repeating CANNOT equal 2): If we divide 6 by 3 (for example) we can only obtain 2. However, it is impossible to obtain 1.999 repeating by dividing 6 by 3 since the number of times that the divisor (the number 3 for this case) is contained in the dividend (the number 6 for this case) is 2. THIS SHOWS THAT IT IS IMPOSSIBLE TO OBTAIN THE NUMBER 1 AS THE FIRST TERM OF THE NUMBER 1.999 repeating. Since there are no two integer numbers whose ratio gives 1.999 repeating then we must conclude this is an irrational number, and as stated above, that 1.999 repeating CANNOT equal the number 2. You wrote: "You say that 19.999 repeating minus 1.999 repeating is equal to 18.000 repeating plus one. Maybe I should instruct you on what infinity is, so that you don't botch it in our debate. Infinity goes on forever, so you would never have the chance to put that one in there" I am really glad you wrote this statement down. What makes you think that I would never have the chance to put that one in there, but you would have the chance of substracting all the terms in the repeating representations of 19.999 repeating and 1.999 repeating? You wrote: "your definitions are simply not as convincing as actually doing the math, kennard. The math works, and your definitions seem somewhat indirect and possibly misinterpreted" Have you seen how many post are out there indicating that 1=2? These statements are posted by people who think that "the math works". However, people who really know about math realize that mathematical definitions must be used to PROVE things. That is why anyone who knows math well looks at the statement 1=2 and immediately realize that it involves the old trick of dividing by zero, which is (as you know) UNDEFINED. You wrote: "Why did you write this? The difference is .12 feet. If you meant to say 1.888 repeating, then the difference is .111 repeating feet" I apologize for not been clear on that line. The point that I was trying to prove there was that the fact that I cannot distinguish (visibly) the difference between a 1.9998888777 feet board and a 2 feet board those not make the two numbers equal. I really hope that now that I rewrote the definition of rational and irrational numbers, and how they clearly can't be equal to each other, that you proceed to prove that 1.999 repeating is a rational number.