Correct way of viewing 0.9 repeating
1. Well you think that just because 0.999.... looks different from 1 you think it's not 1. It's like saying 1/2 does not equal 0.5 or ice is not water.
2. Subtract 0.999.... from 1. What do you get? 0.000....0001. This is EQUAL to zero because there are INFINITE zeros before the one. Therefore we will never reach the one. If the difference is zero, the numbers are the same.
3. Referring to reason 2. How are 1 and 1.000.....001 the same? Well, with decimals that go on forever, there is no last number. The last number of 1.000.....001 is not 1, because we never reach the end. Same with 0.999.... There is no end. 0.999.... goes on forever, because it's recurring. So 0.999....5 = 0.999....02 because we never reach the 5 or the 02.
4. Infinity can get us anywhere. Because the number 0.999.... is recurring, it equals one. Infinitely large is impossible to count, and infinitely small is nothing. Do not underestimate the size and power of infinity. In
5. If you think that 0.999.... is not 1 because it is recurring, and recurring decimals cannot equal natural numbers, well, you're absolutely wrong because 1 is recurring: 1.000000000000............
6. If 0.999.... does not equal 1, this pattern breaks down:
1/9 = 0.1 + 0.01 + 0.001 + 0.0001...
2/9 = 0.2 + 0.02 + 0.002 + 0.0002...
3/9 = 0.3 + 0.03 + 0.003 + 0.0003...
4/9 = 0.4 + 0.04 + 0.004 + 0.0004...
5/9 = 0.5 + 0.05 + 0.005 + 0.0005...
6/9 = 0.6 + 0.06 + 0.006 + 0.0006...
7/9 = 0.7 + 0.07 + 0.007 + 0.0007...
8/9 = 0.8 + 0.08 + 0.008 + 0.0008...
9/9 = 0.9 + 0.09 + 0.009 + 0.0009...
10/9 = 1.0 + 0.10 + 0.010 + 0.0010...
11/9 = 1.1 + 0.11 + 0.011 + 0.0011...
12/9 = 1.2 + 0.12 + 0.012 + 0.0012...
The ninth line shows 9/9, which is equal to 1. But it also equals 0.9 + 0.09 + 0.009 + 0.0009... which is 0.999.... So 0.999.... does equal 1.
Trust me, I am 99.999....(100) percent sure this is correct. I'm not on the "yes" side just because mathematicians are on the same side, but I know 0.999.... = 1 because I did the math. If you are still not convinced, you are not correct. Have any doubts about this topic? I'll be happy to answer any questions you have to bring up. By the way it's okay to be wrong, as long as you're right in the end. Losing a debate is fine, because one side has to win.
Eliseo
