0.999... is equal to 1

I`m having a blast discussing this topic. The crux of my counter-argument for Round 2 in refuting Con`s points will be the irrefutable fact that if you take 1 away from infinity, you still have infinity. Looking back at my Proof #1 under Part 3: Mathematical Proof, we notice under step 2 that when multiplying 0.999. .. by 10, we are obviously left with 9.999. .. . In other words, taking a single 9 away from an infinite amount of them still leaves you with an infinite amount of nines. I will reference this later. Now, on to the rebuttal! "If 0.333. .. = 1/3, then: 1/3 x 3 = 0.999. .. We got 3 whole thirds, so they should equal 1, not 0.999. .. " There is no reason to believe this disproves the notion that 0.999. .. and 1 are the same number. If they were (and are), then there would be nothing wrong with these simple arithmetic points. Saying "I believe they aren`t the same because they look different" isn`t sound reasoning in math. "Since there's no number between 0.999. .. and 1, they must be the same. I can't really refute that with a counter example. .. " I agree. "1 x 2 = 2" Yes. "0.999. .. x 2 = 1.999999999. .. .8" I can see why you would think this. I will try my best to explain why this is also false. An infinitely-repeating, neverending sequence of numbers is just that, neverending. It is non-terminating, meaning there is no "last" nine. The number 1.999. ..998 has a terminating number of nines if you can slap an 8 at the end of it and be done. If you have a ". .. " between two numbers, then that ellipsis cannot represent a non-terminating sequence because it is bound by two other numbers. If can only represent infinity when it is not followed by anything else. When you multiply 2 by 0.999. .. , you actually get 1.999. .. . The nines still never end. Another way to think of this is by using another non-terminating rational number. 2/3 = 0.666. .. (2/3) x 3 = 1.999. .. , or in other words, 2. HOWEVER, when you look at the terminating decimal 0.666 for example, and multiply it by 3, you do in fact get 1.998. That`s because in that case the sixes terminate, or they end. 2/3 in decimal form does not. The problem with a lot of people`s conception about infinity is they treat it like a number. If you do this, however, mathematics starts to break apart. For example: ∞ + 1 = ∞ Therefore, by subtracting ∞ from both sides, 1 = 0. Infinity is a concept, it is not a number. People also view a sequence like 0.999. .. by looking at each individual number being written out one at a time, and so it must have a finishing point. This is also obviously not a good way to look at the number, or to look at the concept of infinity. There is not enough space in the universe to contain all these flipping nines, there`s literally infinitely many of them. There is no last 9, ergo, there is no 8 to terminate the sequence when doubling the number. My final rebuttal is to look at one other point you made. "Since the number is 1.99999. .. .8, there's a number between it, and 2; 1.99999. .. Under the philosophical law, no number must exist between two numbers in order for them to be the same. " What happens when you take 1.999. .. and divide it by 2, then? Obviously, you get 0.999. .. , which is the same as 1. I hope I was able to convince my opponent that these two number are in fact equivalent. Please vote Pro. Thank you.