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Can anyone comment on my non-mathematician reasoning here: 0.9999... cannot be represented as a fraction. 0.3333... can be represented as a fraction (1/3). T

by srgseg 15y ago

Can anyone comment on my non-mathematician reasoning here: 0.9999... cannot be represented as a fraction. 0.3333... can be represented as a fraction (1/3). Therefore we can apply meaningful operations to the fraction (1/3), but when we try to do this on the "intellectual concept" of an everlasting list of 9s, we're trying to operate on a number that cannot exist. To me, 0.9999... is not a valid number, in the same way that I wouldn't consider the intellectual concept of 5555.... to be a valid number (five recurring, i.e. not 0.5555... but 5555....) Anyone care to comment on my intuitive reasoning here?

rm445 15y 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 15y ago
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- srgseg 15y ago
  What 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 15y ago
    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.
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by srgseg 15y ago

rm445 15y 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 15y ago
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srgseg 15y ago
What 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 15y ago
  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.
- deleted 15y ago
  [deleted]