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| Authored by: Anonymous on Friday, December 14 2012 @ 10:26 AM EST |
1 = 1
1 > 0.999...
1 approximately equal to 0.999...
Pick
whichever answer you want. It all depends on how exact you want to be. [ Reply to This | Parent | # ]
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| Authored by: Anonymous on Friday, December 14 2012 @ 10:55 AM EST |
(excuse my notation)
if 1 = 0.9r then
0.9r = 0.9r8 by the same logic
repeat this enough times and you end up with 1 = 0, as "=" is a
commutative operation (a=b & b=c => a=c)
As 1 != 0, your initial proposition is false by induction.[ Reply to This | Parent | # ]
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| Authored by: ukjaybrat on Friday, December 14 2012 @ 11:25 AM EST |
begin with the axiom that one-ninth = zero point one one one
repeating (0.111...)
1/9 = 0.111...
Multiply both sides by 9
9 * (1/9) = 9 * (0.111...)
Simplify
1 = (0.999...)
There are several other froms of proofs if you search for
"0.999..." on wikipedia
---
IANAL[ Reply to This | Parent | # ]
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| Authored by: GaryD on Friday, December 14 2012 @ 11:52 AM EST |
There happens to be something special about an infinite sequence of 9s,
basically because 9 is the highest digit in base-10 arithmetic.
The theory of infinite sequences relies on the concept of a limit: in plain
English, an infinite sequence is equal to some value if, whatever positive
non-zero error you choose, the sequence not only gets within that error of the
limit value at some point, but it also remains at least that close to the limit
from then on.
So, if you choose any small but non-zero positive value (call it E), 0.999...
will get closer than that to 1 once you've added log10(1/E) 9s after the decimal
point, and it'll never get further away. So the infinite sequence 0.999... is
equal to 1.
---
Gary Duke[ Reply to This | Parent | # ]
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| Authored by: Anonymous on Friday, December 14 2012 @ 12:32 PM EST |
I had opportunity to take both linear algebra and strengths of materials classes
from the same professor.
When we were in the math room, he wouldn't accept a decimal conversion of a
fraction such as 1/3; said we were just being "calculator lazy" and
there wasn't enough space on the page to represent the value accurately.
Over in the physics room, he wouldn't let us use more significant figures than
available in the data we had about the physical characteristics of our
materials; 3 sigfigs was the usual.
So we really did have equations which were correct in one room and wrong in the
other.[ Reply to This | Parent | # ]
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- Fiddler's Green - Authored by: Anonymous on Friday, December 14 2012 @ 01:38 PM EST
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| Authored by: Imaginos1892 on Friday, December 14 2012 @ 01:28 PM EST |
0.99999999... is an asymptotic expression that approaches a
limit of 1.0, but is NOT "== 1.0". There is always a difference.
---------------------
Gentlemen! You can't fight in here -- this is the War Room!![ Reply to This | Parent | # ]
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| Authored by: Anonymous on Sunday, December 16 2012 @ 12:38 AM EST |
You know that scene in the original _Men in Black_ movie,
where Will Smith shows up for an intake exam with a crowd of
young G-men only to ends up discomfiting the bunch by
raspingly dragging a table across the room while the rest of
the party comically tries and fails to stabilize their
papers on their chair cushions and knees? My answer to your
question is that scene recast with mathematicians as the G-
men. My answer will allow you to stop thinking about this
conundrum, but math adepts will not be satisfied with it. No
worry: At the end of this discussion we'll send them down
the hall for the eye exam so badly need. We begin dragging
our figurative table exactly now.
You have been shown that 1 and 0.999... are equal because
(as the proof tells us in a dualistic way), (a) "there is no
number between them" and (b) that since summing three
instances of the fractional equivalent to 0.999... (that is,
three instance of 1/3) adds up to exactly 3, each of those
instances of the fractional equivalent to 1/3 must therefore
be equal to exactly 1. Who are we to argue with the cabal?
Thanks to their training, mathematicians have no trouble
ending their thoughts on the matter right here, and it is
appropriate that they do.
We non-math folk, however, remained vexed because, stripped
of any association with a possible fractional
ancestry/equivalence, 0.999... forever asymptotically
_nears_ 1 but never ever reaches it. To us practical folk,
this superficial sight-read alone is proof of their
difference; if 0.999... _was_ equal to 1, _we'd write it as
1_.
Fundamental to _our_ dealing satisfactorily with this
conundrum must be this understanding: Mathematicians do not
own the concept of infinity any more than the religious own
the concept of god. The concept of infinity is your tool as
much it is theirs, and we can use our concept of infinity to
_disprove_ to our (ignorant in mathdom's eyes) the statement
that "0.999... and 1 are not different because there is no
number between them." It's simple: Those numbers--1 and 1
minus 0.999..., in which ... means 9s stretching forever--
differ by exactly by 0....1--that is, by forever--by
infinity--followed by a 1.
As to the issue of adding three 1/3s (which, represented as
decimal fractions, are each equal to 0.999...) and having
them add (of course!) to exactly 3 in their fractional form,
and by transference ultimately proving that the fractional
equivalent to 1/3, 0.999..., is therefore equal to exactly
1, I call "foul." 1/3 is not a number in the way that 1 is a
number; it is an equation that has yet to be solved. When we
attempt to solve that equation, which in effect is an
exercise of moving from a number system based on 3s to one
based on 10s, we discover that the answer never fully
resolves--or rather, that it resolves at infinity, whereas
1/1 or 3/3 sits fully complete on the paper or screen before
us, forever bereft of Forever. By convention, and wanting to
stop thinking about this time-drain and use results
practically, mathematicians agree to resolve that
unresolvedness by representing its perpetuality with
ellipses. Now the unboundable is bounded; but in fact, the
actual solving of the equation 1/3 can never end; in
practice, akin to the importance of having invented 0, we
are saved by being able to represent perpetuality; we plop
it into the equation, all the numbers add up, and we drive
on. In contrast to this, a nonrepeating number like 1 or
5.322 or (the solution to) 2/4 ends right here, right now,
in our heads; a repeating number like 0.999... doesn't,
ever. Thoughts that cannot end can trouble and even sicken
us--look at the unfortunates obsessed with chasing the
mirage of nonrepetetion in the decimal equivalent to 22/7--
and so we agree to save ourselves from this looping and get
on with living our finite lives with the help of transitive
equality.
Of _course_ 0.999... is less than 1; it endlessly
approaches, but never quite reaches, 1. That "there's no
number between it and 1" is a facile red herring; actually
that number space is merely hard to represent cognitively
rather than being an indivisible disjunct or null, because 1
_ends now_ but 0.999... doesn't. Of _course_ 0.999...
doesn't equal 1, it equals 1 minus 0.infinity1. But that
representation disrupts the hyperspatial symbology of the
mathematician's world; far more important to them, and
essential to building the long chains of objective causality
along which we have as a culture walked to harness nuclear
fission and achieve the theory of quantum electrodynamics,
is the _internal coherence and trustworthiness_ of the
cognitive structures that allow, that demand, every solved
("true") equation to be a tautology.
And now thanks to all you math folk empathetic enough to
have stayed with this discussion for so long. You epitomize
the best of what we have come to expect from years of
academic training. And so if you'll step down the hall to
your right, there's just one more test to administer, and
that's an eye exam.[ Reply to This | Parent | # ]
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