Re: Question


José Carlos Santos wrote:
zuhair wrote:

The only representation of 1 is 1.0000......... and to me
0.9999........... < 1.
Then you are wrong. If that expression represented a number strictly
smaller than 1, call it x, then (1+x)/2 would be a number strictly
smaller than 1, and strictly larger than x.

What, pray tell, is the decimal representation of this number?
How come I didn't read that post?

Anyhow I should answer it.

I will change your terminology.

Let x = 1- 0.9999..............
That's a rather strange way to answer to Arturo, since he told you to
call _x_ to 0.9999... and you choose to call _x_ to 1 - 0.9999....

Now 1+ 0.99999... = 2 -x

Now (2 - x)/2 = 2/2 - x/2

now x/2= x
How did you get that? But if you think (as I do) that this is a true
statement, then the conclusion is, of course, that x = 0. Since you
defined _x_ as 1 - 0.9999..., it follows that 1 = 0.999....

Now, what about answering Arturo's question. If x = 0.9999..., what is
the decimal representation of (1 + x)/2?

I answered Arturo's question.

Let me repeat it again but I will use Arturo's terminology.

x= 0.9999.......

Let y = 1-x

it follows that x= 1-y

Now 1 + x = 1+ 1- y = 2 - y

Now y/2 = y

Two questions here:

1) How do you know that y/2 = y?

because y is an infinite number since, 0.9999...... is infinite, and if
you divide this
infinite number by any finite number you will get the same number.

1- 0.999........ = 1/(10^Aleph-0)

[1/(10^Aleph-0)]/n = 1/(10^Aleph-0)

This comes from the equation

[x/n] + [x/(n^2) ] + [x/ (n^3)] +.....+ [x/(n^k) ]= {x/(n-1)} - {
x/[(n^k)(n-1)]}

for n=2,3,4,5,........
K=1,2,3,4,.........

if we regard k as the number of terms in the sequence above.

Then when that sequence is infinite then = Aleph-0

so for x=9, n=10 and k=Aleph-0 we will have 0.9999...........= 1-
[1/(10^Aleph-0)]

Now whatever is the division operator in 1/(10^Aleph-0) is, the result
cannot equal zero.

and 1/(10^Aleph-0) > 0

So 0.9999........ <1

Zuhair

2) How come you don't deduce from this that y = 0 and therefore that
1 = 0.9999...?

Then ( 1+x)/ 2 = (2-y)/2 = 1 -( y/2) = 1- y = x = 0.99999............

You see the same number.

Best regards,

Jose Carlos Santos

.

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