New Theory about 0.999... and 1
- From: "said_yam" <said_yam@xxxxxxxxx>
- Date: 27 Apr 2006 12:24:17 -0700
There is a new theory that may end the debate about 1=0.999...
the whole theory can be found in this link
http://www.geocities.com/humood_theory/index.htm
here is a Quote about cracking the famouse proof:
Proof (A)
x=0.999...
10x= 9.999...
10x-x=9.999... - 0.999...=9
9x=9
x=1
Quote:
--------------------------------------------------------------
I will start by cracking this proof and show it is in fact a Paradox
not a proof.
Let x = 0.999...
10x = 9x + x = [ 9* (0.999...) ] + 0.999... [ I expressed 10x as (9x
+x)]
10x - x = ( 9x + x ) - x = [ 9* (0.999...) ] + 0.999... - 0.999...
[Positive 0.999... goes with Negative 0.999...]
9x = 9* (0.999...)
x = 9* ( 0.999...) / 9 = 1 * 0.999...
x =0.999
We did not end with x=1
In fact what made it looks like a proof was the mistake we did in this
step:
10 x - x = 9.999... - 0.999... = 9
This step assumes the value of infinite 9s after the decimal point in (
10x ) equals the value of infinite 9s in (x)
The infinite decimals in x = 0.999... have different behavior than the
infinite 9s in 10x =9.999... Because 9s in x= 0.999... Go to infinity
faster than 9s in 10x=9.999...
(10 x ) is always one decimal behind (x)
When x = 0.99 then 10x=9.9
When x = 0.9999 then 10x=9.999
When x = 0.999999999 then 10x=9.99999999
Value of Infinite decimals in (x) - Value of infinite decimals in
(10x) =
Lim x-> 8 9 / (10^x)
So the difference between the two values equals an infinitely small
fraction which is Lim x-> 8 9 / (10^x)
This means in Proof (A) when we say: 10 x - x = 9.999... - 0.999... =
9
We are already Assuming that a number minus an infinitely small
fraction is the number itself.
Which means we are already assuming that Lim x-> 8 1 - ( 1/x ) =
0.999... = 1
So, in Proof (A) we already assumed 1= 0.999... Then no wonder we
finally ended with our assumption, which make it not a Proof at all.
You cannot prove any statement by assuming it to be true from the
beginning.
But when we explain (10 x ) in a very fundamental method no body can
argue with, we will say:
10x =0.999... + 0.999... + 0.999... + 0.999... +0.999... +0.999...
+0.999... +0.999... +0.999... +0.999...
And when we subtract x = 0.999... from it :
10x - x =0.999... + 0.999... + 0.999... + 0.999... +0.999... +0.999...
+0.999... +0.999... +0.999... + 0.999... - 0.999...= 0.999... * 9
The infinite 9s in x after the decimal is gone with only one of those
0.999... Not all of them as we did in Proof (A).
------------------------------------------------------------------
All the theory and its results can be seen in the link:
http://www.geocities.com/humood_theory/index.htm
For comments please write it here in this gruop or send it to the
said_yam@xxxxxxxxx
.
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