Why is 0.99999…=1?(xamuel.com)
xamuel.com
Why is 0.99999…=1?
http://www.xamuel.com/why-is-0point999-1/
3 comments
I wonder if this Xamuel guy is a bit off his nut. He invents a word, "pseudo-real," to classify 0.9999... and other such representations. He eventually castigates the reasons behind declaring 0.9999...=1, and then asks a profound question: "The real question is not why 0.9999...=1, but rather, what on earth are real numbers in the first place?" It's OK to ask, of course, but he's asking it in a profoundly silly way.
For starters, 0.9999... is no less a real number than 0.9999 (i.e. 9999/10000). If we wish to continue on his line of thought, then we should all concede that any scrawling used to represent a quantity - any quantity, real number or no - is a pseudo-quantity. A scrawling is just not the real thing, whether "quantity" is in fact something real or simply the product of human imagination. Hence, "1" is also a pseudo-real, along with "I", "a", "0.9999...", "cos 0", etc. It's anyone's guess if there is such a thing as a unit of anything at all.
For starters, 0.9999... is no less a real number than 0.9999 (i.e. 9999/10000). If we wish to continue on his line of thought, then we should all concede that any scrawling used to represent a quantity - any quantity, real number or no - is a pseudo-quantity. A scrawling is just not the real thing, whether "quantity" is in fact something real or simply the product of human imagination. Hence, "1" is also a pseudo-real, along with "I", "a", "0.9999...", "cos 0", etc. It's anyone's guess if there is such a thing as a unit of anything at all.
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?
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?
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.
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...?
Is there any merit in this reasoning, and the idea that 0.3333... is more valid a number than 0.9999...?
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. That fails.
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. That fails.
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The way I like to explain it to get it to quickly click in anyone's head (math geek and non-math geek alike) is so:
What is true about any two distinct numbers? Answer: there are an infinite number of numbers between them (e.g. between 4.00005 and 4.00006 there are 4.000051, 4.0000501, 4.00005001, etc. You can always add more zeros). But one cannot name a single number between 0.9999... and 1 because they are the same number.
Not really an actual, rigorous proof, but it serves its purpose.
What is true about any two distinct numbers? Answer: there are an infinite number of numbers between them (e.g. between 4.00005 and 4.00006 there are 4.000051, 4.0000501, 4.00005001, etc. You can always add more zeros). But one cannot name a single number between 0.9999... and 1 because they are the same number.
Not really an actual, rigorous proof, but it serves its purpose.
On the word "fraction," do you take it to mean the result of dividing an integer by another integer, and that it must be less than 1? An "improper fraction" is still a fraction, nonetheless; hence 1 is also a fraction, namely (1/1), among other representations. ( http://en.wikipedia.org/wiki/Fraction_(mathematics) )
Fractions aside, 0.9999... is a legitimate, albeit strange, way to say "one."
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The grade-school approach (which can be used for 0.3333... too):
(1) y = 0.9999...
(2) 10y = 9.9999...
(3) 10y - y = 9.9999... - 0.9999... = 9y
(4) 9y = 9
(5) y = 1