.9999...Repeating exactly equals 1

Thank you for debating. I know the resolution is a concept that on intuition seems false to many people. However, it is a perfectly valid statement to make mathematically. Mathematics is like science and logic. There is an agreed upon process of performing operations. Without this agreed upon process 2 different people would get 2 different unequal answers. For example something as simple as 2+4x3. Thanks to the Order of Operations we know this is 4x3=12 then 2+12=14. Multiplication is done before addition. With this in mind I move to my first proof.1. 3 thirds1/3 = .3333... repeating That is the 3's go on forever. This is not an aproximation. The sign is not ≈ it is =.2/3 = .6666... This is becuase the fraction is a type of equation itself. It says 2 divided by 3.P1: 2/3 + 1/3 If we see this as an equation with no fractions it is (2 divided by 3) add (1 divided by 3)..6666... + .3333... = .999... Order of Operations PEMDAS [1], we do the division first. Then add together equaling .9¯P2:2/3 + 1/3 = 3/3 = 1 If we see this as adding fractions we have 3 over 3 or a whole. Which equals 1.Again all the rules of Math were followed. So we can not have 2 different people come up with 2 different "Quantities". Sure one person may right an answer as 8 and another write it as 2^3 (2 with exponent 3 or power of 3). 2 different ways of writing the same answer. But the answers are equal, they are the same quantity. Same equation, different but equal answers.C1: So 2/3 + 1/3 = .999... OR 2/3 + 1/3 = 1 Therefore .999... = 1My Opponent will have to show where I broke a rule of Math. Notice that I did not mix fractions with decimals. I added fractions to fractions, and I added decimal to decimal. It's important to see that they are not approximations of each other. But that the decimals are the result of a mathematical operation, division.2. Transitive Propertyx = .999... We start with an equation. As long as we perform the same operation to both sides, the equation stays equal.10x = 10 times .999...10x = 9.999...10x -x = 9.999... - x (which we know from earlier x = .999...)Becuase x and .9¯ are equal we can exchange them based on the Axioms of Equality of math. This allows us to calculate "like" terms. This more accurately is called the Symmetric Property.9x = 99x/9 = 9/9x = 1C2: So again we have x = .999... and we have x=1 therefore .999... equals 1. Or 1 = x = .999... This is called a transitive property of equality. [2]3. Number line explanantionWhen we have 2 numbers that are "not equal" and we place them on a number line, like 3 and 1. We know there is a quantity between them. We figure out this quantity by subtraction ie: 3 -1 =2. 2 is the quantity between 3 and 1.0-----1-----2-----3-----4 There are 2 "-----" between 3 and 1.P1: IF 1 does not equal .999... Then there should be a quantity between them.What is 1 - .999... = ?Well it is not .0001 Becuase the 9's go on forever. It is not .00000000001 for the same reason. This implies that the answer is a decimal point followed by an infinite amount of 0's and at the end 1. There is no end to infinity. This 1 never comes. So the answer is an infinite amount of 0's. Which is 0P2: The quantity between 1 and .999... is 0C3: There is no quantity between 1 and .999... therefore they are equal.Closing:This can be a hard truth to swallow. Maybe even harder than accepting quantum entanglement or negative energy."All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident" -Arthur Schopenhauer, German philosopherThis is not a mathematical riddle to trick someone, like the bellhop and missing dollar. This resolution is a fact. I obseved all mathematical conventions and await my opponent's rebuttals.[1]http://www.purplemath.com...[2]http://www.wausau.k12.wi.us...