I'm sure most of you already know this. The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1. Here are the proofs that I've used so far: 1) 1/3 is precisely equal to .333... (to which he agreed). 3*.333... equals .999... and 3*1/3 equals 1. Therefore, since I multiplied equal numbers by the same number, they should still be equal...ergo 1 = .999... 2) If 1 is greater than .999..., then by how much, exactly, is it greater? After all, 1 is greater than .9 by .1. It is greater than .99 by .01. It is greater than .99999999999999999999 by .00000000000000000001. And so on, and so on. However, if the number of nines is infinite, then the number of zeros preceding the 1 will be infinite, and 0.000... is obviously equal to 0, which means that 1 is greater than .999... by exactly 0. 3) After this point, my friend revealed that he doesn't really believe in actual infinites. He thinks you will eventually reach the 1, at the end of the infinite series of 0s. Ridiculous as that may sound, I respected that as his opinion and posed this next point: If there are no infinites then the term equal to 1/3, which is usually considered .333..., would actually (eventually) terminate in a 4. I asked him if that was right. After some thought, he said "yes". So, I asked, if we multiplied this number by 2, would you logically expect to get the same number 6s as you had 3s, and then terminate in an 8 (since 4+4 is 8)? He said, "yes", not realizing that he had just implied that the infinitely repeating number .666..., which is equal to 2/3 must now terminate in an 8, instead of a 7 (as his reasoning would naturaly have assumed). Are there any other points I can use? I wasn't going to ask for help on this, until he talked to his math teacher, and the teacher agreed with him!! Now he's got this quaint little ipse dixe argument, "well, the math teacher said I'm right". Any new points or comments on my previous ones are appreciated.