I personally like using fractions of 9. 1/9 = 0.111... 2/9 = 0.222... 3/9 = 0.333... ... 8/9 = 0.888... 9/9 = 0.999... What's neat is that this tr

I personally like using fractions of 9. 1/9 = 0.111... 2/9 = 0.222... 3/9 = 0.333... ... 8/9 = 0.888... 9/9 = 0.999... What's neat is that this trick works for any repeating decimal, with any number of digits in the repeating part. For instance: 123/999 = 0.123123123... 999/999 = 0.999999999... Multiply or divide by powers of 10 as necessary to shift the decimal point, and add the non-repeating part. Once you accept this mapping, it's trivial to treat 0.999... as 9/9 (or 99/99, or 999/999, etc). Which can be simplified to 1.