Little math tricks I remember from school

• .999999... = 1
  • Let ... indicate repeating digits (I can't do a bar over the number)
  • 1/3 = .3333...
  • 2/3 = 1/3 + 1/3 = .3333... + .3333... = .6666...
  • 3/3 = 1/3 + 2/3 = .6666... + .3333... = .9999...
  • However, 3/3 = 1. So, .9999... = 1.
  • Interesting: there are two decimal representations for every number
• How do you express .345345345... as a fraction?
  • Let x = .345345345...
  • Then 1000x = 345.345345...
  • Thus, 1000x - x = 345.345345... - .345345... = 345
  • 1000x -x = 999x = 345
  • x = 345/999
  • To express repeating digits as a fraction
    • Write out the number you want to repeat (i.e. 123)
  • Divide by the same number of 9's: 123/999 = .123123123...