0.999... can be represented by a fraction. 3/3 is a fraction. Or 4/4 or 17/17, they all equal one and equal 0.999... One of the reasons it looks confusing is b

0.999... can be represented by a fraction. 3/3 is a fraction. Or 4/4 or 17/17, they all equal one and equal 0.999... One of the reasons it looks confusing is because fractions are commonly used for numbers < 1. You see 0.33..., you learn it is less than one, you see 0.99..., and the notation is similar to commonly used numbers less than one. As long as you realize 0.333... is exactly 1/3, that it is simply the decimal notation for exactly that value, you can understand that 0.999... is exactly 1. The equation: x * 3 / 3 * 3 / 3 = x That's x = x, pretty simple and should hold for any value of x. Now say we plug in 0.333... and look at it this way: (x * 3 / 3) * (3 / 3) = x Now you've got 0.999.../3 which leads to ("some number less than one divided by three") times one = "one divided by three". That fails.