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Sorry, but your reasoning isn't logical. You accept that point three recurring is one third, presumably because you accepted it as a young child - but you don't
by rm445 16y ago
Sorry, but your reasoning isn't logical. You accept that point three recurring is one third, presumably because you accepted it as a young child - but you don't make the leap to accepting that point nine recurring is 3/3, by the simple correspondence that every digit is multiplied by three.
- deleted 16y ago[deleted]
- srgseg 16y agoWhat I'm trying to get at is that while 0.3333... is a "valid" number because it can be represented by a fraction, 0.9999... is not a valid number because it cannot be represented by a fraction. Is there any merit in this reasoning, and the idea that 0.3333... is more valid a number than 0.9999...?
- isleyaardvark 16y ago0.999... 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.
- deleted 16y ago[deleted]