
Why is 0.99999…=1? - ColinWright
http://www.xamuel.com/why-is-0point999-1/
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rskar
If you are inclined to believe that 0.3333... = (1/3), can you explain why
0.9999... = 1 is a big leap for you? Whether the case is 0.3333... or
0.9999..., each is the result of a convergent series. (
<http://en.wikipedia.org/wiki/Convergent_series> )

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."

\----

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

~~~
rskar
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.

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srgseg
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?

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rm445
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.

~~~
srgseg
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
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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coreyspitzer
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.

