I know this is really old, but I see this all the time still and it bugs me. Step 2 doesn't work, because you can't minus X on both sides. You can't say x = 0.99~ and then start a new formula like that, because 10x = 9.99 is actually equal to 0.999, not 1, since there is only one variable and you already defined it. You can't define X twice in the same formula for 2 different values. You would have done better with this: 3/3 = 1 1/3 = 0.333~ 0.333*3=0.999~ But since the fraction is infinite, you can never truly say 1=0.999~ because the number doesn't end. and If you replaced x to find the value, you would get: 10(0.999) = 9.999. 10x is still a multiplication, and so -0.999 would actually be this: 10(0.999)-0.999 = 9 which solves as : 9.999-0.999=9 9=9, or 1=1 or 10x-0.999=9, and if you solve this, it becomes: 10x=9+0.999 10x=9.999 x=9.99/10 x=0.999 You lose
We might say that .99~ = 1 but if we ever achieve the technology to travel at light speed (or faster) in the next thousands of years or so then that will not hold, as if you substituted 1 for .99~ in an equation used to calculate your destination, your destination would most likely be wrong.
take a paper, a whole paper. Cut off 9/10 of it. Place this in a pile. The pile contains .9 of a paper. Take the remaining tenth, cut off 9/10 of it and place it in the pile. The pile now contains .99 repeat the cuts an infinite number of times and place everything into the pile. This works out to .9999~ being equivalent to the entire paper. more rigorous definition http://en.wikipedia.org/wiki/Dedekind_cut .999999~ defines the same dedekind cut as 1. other definition. .99999~ is a cauchy sequence. Cauchy sequences converge. other definition .9... = 1 - .1^n where n is the number of 0s hence .99999~ = 1 - limit n->infinity .1^n limit n->infinity .1^n is defined to be 0 hence .99999~ = 1-0