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Sekolah Alam Shah

Posted: Thu Dec 25, 2003 7:03 am

Guest

Okay, this is NOT going to be a message arguing for the above
ridiculous assertion.I am just going to suggest the reason for why
some people think that it is, because years ago I once had the same
trouble convincing myself of its falsity.

Thought of as a limit, 0.999... = 1 is readily obvious. But those who
insists on the opposite actually has a philosophical basis for this
assertion. I contend that these same people would, if you let them,
insist that 1 + 1 = 2 is false. They, (and I did) view "1 + 1" as an
object having two properties, the value assigned to it, and "1 +
1"-ness itself. "1 + 1" lives happily on one side of the Platonic
realm, "2" on the other, and these objects share in common only the
value property. These people have trouble restricting the meaning of
the equality sign to that narrowly defined in mathematics. (I am
somehow reminded of the concept of equivalence classes).

Well, maybe its muddled thoughts, but I hope it may help you
understand this confusion.

Guest

Posted: Thu Dec 25, 2003 7:06 am

Sorry, the ridiculous assertion is 0.999.... does NOT equal 1.

"Sekolah Alam Shah" <minnesotamichigan@yahoo.com> wrote in message
news:13b31031.0312250403.7eeb5e8d@posting.google.com...

Quote:

Rene

Posted: Thu Dec 25, 2003 1:15 pm

Guest

You may have hit a point here.

If one insists that 0.999... is a sequence, then it is not equal 1. However,
in mathematics, we use to indicate the limit of the above sequence with the
same notation. Likewise, for infinite sums.

However, I believe the true reason for troubles with 0.999... is denial of
the concept of infinity.

"Sekolah Alam Shah" <minnesotamichigan@yahoo.com> schrieb im Newsbeitrag
news:13b31031.0312250403.7eeb5e8d@posting.google.com...

Quote:

Paul Holbach

Posted: Thu Dec 25, 2003 1:42 pm

Guest

Quote:

hasnor_lot@hotmail.com> wrote in message news:
ppAGb.187931$Eq1.174027@twister.rdc-kc.rr.com>...

Quote:

Sorry, the ridiculous assertion is 0.999.... does NOT equal 1.

It certainly does!

Just try to subtract 0.999... from 1:

1 - 0.999... = 0

Reason:

There is no real number between 0.999... and 1, and, therefore, "they"
must be one and the same number!

Regards
PH

Paul Holbach

Posted: Thu Dec 25, 2003 1:49 pm

Guest

Quote:

minnesotamichigan@yahoo.com (Sekolah Alam Shah) wrote in message news:
13b31031.0312250403.7eeb5e8d@posting.google.com>...

Quote:

"1 + 1 = 2"

can be rendered as

"The number which results from adding the number 1 to itself is
identical with the number 2."

Or

"The expressions '1 + 1' and '2' refer to the same numerical object,
namely, the number 2."

"1 + 1 = 2" is certainly true!

PH

The Ghost In The Machine

Posted: Thu Dec 25, 2003 4:00 pm

Guest

In sci.logic, <hasnor_lot@hotmail.com>
<hasnor_lot@hotmail.com>
wrote
on Thu, 25 Dec 2003 12:06:45 GMT
<ppAGb.187931$Eq1.174027@twister.rdc-kc.rr.com>:

Quote:

Sorry, the ridiculous assertion is 0.999.... does NOT equal 1.

From a mathematical standpoint, you (hasnor_lot@hotmail.com)
are of course correct.

If one assumes a = 0.999... != 1, then a = 1 - d for
some d. Compute b = a/10 + 0.9. b has the same expansion
as a, since all the 9's are simply shifted right one place.
Therefore, 0.999... either equals 1 (since b = 1 - d/10 = a
= 1-d; therefore d = 0 and a = b = 1) or 0.999...describes
an entire slew of "infinitesimally close to 1" numbers,
and heaven help the poor user who wants to compute things
such as 0.999... - 0.999... or 0.999... / 0.999...

:-)

Or one gets into even marshier ground, as one is forced
to conclude that two numbers with the same digit expansion
(which will never cease, admittedly, but they're the same
for any desired distance) are somehow different.

Arguably, the most rigorous method for approaching this
issue is Cauchy sequences: if a = 0.999..., then
a_1 = 0.9, a_2 = 0.99, a_3 = 0.999, and so on; this
sequence is provably convergent, and the only sensible
value is 1. Note that a_{n+1} = 0.9 + a_n/10 = a_n + 9*10^(-(n+1)).

As for "1 + 1 = 2", it depends on how one defines "1", "+", "1",
"=", and "2". One hopes that the two "1"'s are defined equally,
although they need not be; in the notation "12321.2321" the first
1 relate to 1 * 10^4, the second one to 1 * 10^0, the third
to 1 * 10^(-4).

Note that "+" in some systems need not be commutative;
one could define "+" as string concatenation: "1" + "1"
= "11", and in Java [*] that's exactly what one would get.
In such a system "1" + "2" = "12", but "2" + "1" = "21".
In C++ one can define something that gets silly real fast:

* * *

class A {
public:
A(int v) { }
};

A operator+(A a, A b) {
std::cout << "Not likely!" << std::endl; return A(0);
}

* * *

which "defines" A(1) + A(1) as A(0), and prints out a message
while adding.

As one book put it, there are four concepts:

what the result is
what the name of the result is
what the result is called
what the name of the result is called

I'm not sure that's quite right but one has to be careful; in
computers, it is possible to interpret M_PI (which in binary
double-precision floating format is 0x400921fb5442d18
= (1 + 0.570796326794896619231321691) * 2^(0x400 - 0x3ff)
= 3.141592653589793238462643383 [+]) as a very large long long integer,
and come up with a rather silly answer. Students (and even yours
truly!) occasionally write things such as

double p = atan(2);
printf("%d", p);

and then wonder why the result looks so strange, before they
notice the format specification error. (Many C/C++ compilers,
Gnu GCC among them, will warn of the mismatch.) We computer
software engineers run into this sort of thing routinely.

In memory, M_PI is actually a pattern of little charges; on
disk, it's either pits (CD-ROM) or magnetic domains. On
one's video screen it's a bunch of glowing bits.

In mathematics, I would say "1 + 1 = 2" is an equation and
a proposition. It's an equation because of the "=" sign,
and a proposition because it's a logical statement, which,
as it turns out, happens to be true if one defines the
terms in the "normal" fashion and uses standard logical
axioms. "1 + 1 = 3" is false if one defines the terms
in the "normal" fashion and uses standard logical axioms,
but one could just as easily define it in such a way as to
be true, meaningless, or illustrative of some other concept
that may or may not make sense without using metaphors.

(One interpretation method: "1 + 1" relates to man and
woman, and "3" relates to man, woman, and resulting baby;
"=" relates to the passage of time and other things.
Of course that's rather whimsical, and far from obvious;
the Martians in H.G. Wells (which reproduced by budding)
would probably not have understood this at all.)

In logic, statements such as x = 2 do not have an assigned
truth value; this is because there's a free variable.
One can of course quantify it: (Ax)(x = 2), which is
clearly false, or (Ex)(x = 2), which is either true or
false, depending on how one defines "existence" (it's true
if one includes concepts, and numbers are definitely concepts).

I can't say I know Plato's philosophy from that of Kant or Nietzsche,
but I do know what works. :-)

Quote:

"Sekolah Alam Shah" <minnesotamichigan@yahoo.com> wrote in message
news:13b31031.0312250403.7eeb5e8d@posting.google.com...
Okay, this is NOT going to be a message arguing for the above
ridiculous assertion.I am just going to suggest the reason for why
some people think that it is, because years ago I once had the same
trouble convincing myself of its falsity.

Thought of as a limit, 0.999... = 1 is readily obvious. But those who
insists on the opposite actually has a philosophical basis for this
assertion. I contend that these same people would, if you let them,
insist that 1 + 1 = 2 is false. They, (and I did) view "1 + 1" as an
object having two properties, the value assigned to it, and "1 +
1"-ness itself. "1 + 1" lives happily on one side of the Platonic
realm, "2" on the other, and these objects share in common only the
value property. These people have trouble restricting the meaning of
the equality sign to that narrowly defined in mathematics. (I am
somehow reminded of the concept of equivalence classes).

Well, maybe its muddled thoughts, but I hope it may help you
understand this confusion.

[*] http://java.sun.com, created, published, and distributed by
Sun Microsystems. It's a nice language but discussion of
its benefits or foibles is probably best relegated to newsgroups
such as comp.lang.java.advocacy. :-)

[+] 1 sign bit, 11 "excess 0x400 or 1024" bits for the exponent,
1 "hidden" 1 bit, and 52 bits for the mantissa. There is an
IEEE spec for this number format, and Intel microprocessors,
among many others, use this format.

--
#191, ewill3@earthlink.net
It's still legal to go .sigless.

|-|erc

Posted: Thu Dec 25, 2003 7:25 pm

Guest

----------------------------- <^> <(·¿·)> <^> -----------------------------

"The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in

Quote:

Or one gets into even marshier ground, as one is forced
to conclude that two numbers with the same digit expansion
(which will never cease, admittedly, but they're the same
for any desired distance) are somehow different.

Is it true that if the digit expansion is the same for any desired distance
that the numbers are equal?

...............

RTP all numbers appear on the list of computable numbers.

1 A number is a sequence of digits.
2 The 1st digit appears on the list. P(n_1 at digit 1) = 1
3 If the digits up to q appear on the list, probabilistically/computationally digit q+1
together appears on the list.
4 Therefore by induction all digits appear on the list

5 Therefore all numbers appear on the list.

Quote:

As for "1 + 1 = 2", it depends on how one defines "1", "+", "1",
"=", and "2". One hopes that the two "1"'s are defined equally,
although they need not be; in the notation "12321.2321" the first
1 relate to 1 * 10^4, the second one to 1 * 10^0, the third
to 1 * 10^(-4).

Note that "+" in some systems need not be commutative;
one could define "+" as string concatenation: "1" + "1"
= "11", and in Java [*] that's exactly what one would get.
In such a system "1" + "2" = "12", but "2" + "1" = "21".

And "=" could be evaluated 1st.

1 + 1 = 2
1 + (1 = 2)
1 + f

If + is concatination then that equals the sequence <1,f>

Then the question becomes :
Does <1,f>?

The answer is no, number 1 doesn't f because noone believes me that the government
and media ganged up on number 1 and all the internet combined couldn't
help the truman Adam see Eve again.

Herc

The Ghost In The Machine

Posted: Fri Dec 26, 2003 11:25 am

Guest

In sci.logic, |-|erc
<trymyform@wwwadamskingdom.com>
wrote
on Fri, 26 Dec 2003 10:25:59 +1000
<z9LGb.34$ma.903@nnrp1.ozemail.com.au>:

Quote:

----------------------------- <^> <(·¿·)> <^> -----------------------------

"The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in

Or one gets into even marshier ground, as one is forced
to conclude that two numbers with the same digit expansion
(which will never cease, admittedly, but they're the same
for any desired distance) are somehow different.

Is it true that if the digit expansion is the same for any desired distance
that the numbers are equal?

I would think so, yes. Certainly both digit expansions each
define a Cauchy sequence, and they can be intermingled without
difficulty, resulting in a single limit.

Quote:

..............

RTP all numbers appear on the list of computable numbers.

1 A number is a sequence of digits.

A number is either a Dedekind cut or a Cauchy sequence. One
representation is as a sequence of digits -- an infinite,
non-repeating sequence for most numbers. I'll admit I'm
not entirely sure about this.

Quote:

2 The 1st digit appears on the list. P(n_1 at digit 1) = 1
3 If the digits up to q appear on the list,
probabilistically/computationally digit q+1
together appears on the list.
4 Therefore by induction all digits appear on the list

5 Therefore all numbers appear on the list.

An interesting deduction chain but I do have to ask the obvious
question as to whether induction works on uncountable ordinals.

Quote:

A possibility, although contrary to common usage. However, C uses
this facility rather often, usually in contexts such as

functioncall(..., a = expression, ...)

where the expression is an interesting intermediate value.
There's also the (usually) bug

if (a = b)

which is a miscoding of

if (a == b)

and can lead to much confusion.

Quote:

If + is concatination then that equals the sequence <1,f

Then the question becomes :
Does <1,f>?

The answer is no, number 1 doesn't f because noone believes me
that the government and media ganged up on number 1 and all the
internet combined couldn't help the truman Adam see Eve again.

Well, the simpler answer is that "<1,f>" is not a Boolean and
cannot be converted thereto. Smile

No paranoia required.

An interesting interpretation.

Quote:

Herc

--
#191, ewill3@earthlink.net
It's still legal to go .sigless.

Charlie-Boo

Posted: Fri Jan 02, 2004 7:49 pm

Guest

paulholbachSPAMBAN@freenet.de (Paul Holbach) wrote

Quote:

Neat "proof", but you are playing foot loose with the definition of =.
0.999... is an infinite series, a shorthand notation for .9, .99,
..999, ... 1 is an integer. The relationship is that 1 is a (the)
value for which, for every D>0 there is an N such that for all M>N the
value of the Mth number in the series is between 1-D and 1+D.

The problem occurs when people start saying that .999... "equals" 1.

Charlie Volkstorf

|-|erc

Posted: Fri Jan 02, 2004 8:43 pm

Guest

Quote:

Sorry, the ridiculous assertion is 0.999.... does NOT equal 1.

It certainly does!

Just try to subtract 0.999... from 1:

1 - 0.999... = 0

Reason:

There is no real number between 0.999... and 1, and, therefore, "they"
must be one and the same number!

Regards
PH

Neat "proof", but you are playing foot loose with the definition of =.
0.999... is an infinite series, a shorthand notation for .9, .99,
.999, ... 1 is an integer. The relationship is that 1 is a (the)

2 - 1 is a function, 1 is a number

they're still equal.
What is the number between 0.99... and 1?

Herc

Quote:

value for which, for every D>0 there is an N such that for all M>N the
value of the Mth number in the series is between 1-D and 1+D.

The problem occurs when people start saying that .999... "equals" 1.

Charlie Volkstorf

Daniel W. Johnson

Posted: Fri Jan 02, 2004 10:31 pm

Guest

Charlie-Boo <chvol@aol.com> wrote:

Quote:

Neat "proof", but you are playing foot loose with the definition of =.
0.999... is an infinite series, a shorthand notation for .9, .99,
.999, ... 1 is an integer. The relationship is that 1 is a (the)
value for which, for every D>0 there is an N such that for all M>N the
value of the Mth number in the series is between 1-D and 1+D.

The problem occurs when people start saying that .999... "equals" 1.

There are ways to associate a value with 0.9999... such that the value
does not equal 1. There are two immediate consequences, though:

1) 0.3333... does not equal 1/3 .

2) 0.9999... and 0.3333... (and other such values) are not real numbers.
--
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W

Thomas Bushnell, BSG

Posted: Sat Jan 03, 2004 12:46 am

Guest

chvol@aol.com (Charlie-Boo) writes:

Quote:

But this is exactly the definition of an infinite decimal. What do
you thing .999... means, if not the limit of the partial sums?

The Ghost In The Machine

Posted: Sat Jan 03, 2004 4:00 am

Guest

In sci.logic, Daniel W. Johnson
<panoptes@iquest.net>
wrote
on Fri, 2 Jan 2004 22:31:37 -0500
<1g6yh02.h3uybwz75oe4N%panoptes@iquest.net>:

Quote:

Charlie-Boo <chvol@aol.com> wrote:

Neat "proof", but you are playing foot loose with the definition of =.
0.999... is an infinite series, a shorthand notation for .9, .99,
.999, ... 1 is an integer. The relationship is that 1 is a (the)
value for which, for every D>0 there is an N such that for all M>N the
value of the Mth number in the series is between 1-D and 1+D.

The problem occurs when people start saying that .999... "equals" 1.

There are ways to associate a value with 0.9999... such that the value
does not equal 1. There are two immediate consequences, though:

1) 0.3333... does not equal 1/3 .

2) 0.9999... and 0.3333... (and other such values) are not real numbers.

Well, if x = 0.9999.... = 1-d, then (x/10 + 0.9)
= 0.9999.... = 1-d/10. So either we have some very
weird set of numbers all with the same decimal expansion
0.9999... (and another weird set of numbers all of which
are non-zero infinitesimals), or d = 0 and they're all
equal to 1. The second hypothesis is probably more
straightforward.

Also, 0.999... can be taken as a sequence, which turns
out to be a Cauchy sequence, with limit 1.

--
#191, ewill3@earthlink.net
It's still legal to go .sigless.

Russell Easterly

Posted: Sat Jan 03, 2004 4:17 am

Guest

"Daniel W. Johnson" <panoptes@iquest.net> wrote in message
news:1g6yh02.h3uybwz75oe4N%panoptes@iquest.net...

Quote:

To expand on this.
If you assume 0.999... and 1.000... represent real numbers,
then 0.999... = 1.000... by definition.

Real numbers can be defined
as the limit of a Cauchy sequence.
The limit of the sequence represented by .999... is 1.

You can claim 0.999... != 1.000... but you shouldn't
call them real numbers. "Real number" has a standard
definition.

Russell
- 2 many 2 count

Dave Seaman

Posted: Sat Jan 03, 2004 4:36 am

Guest

On 3 Jan 2004 03:55:08 -0800, Charlie-Boo wrote:

Quote:

2 - 1 is a function, 1 is a number

they're still equal.

A number is a special case of a function (zero arguments - actually,
everything is a special case of a function.) A number is not a
special case of an infinite series (which is not what we're talking
about anyway - we're talking about the limit of .999...)

One of the common ways of defining the real numbers is to consider the
equivalence classes of Cauchy sequences of rationals. According to this
definition, the sequences

9/10, 99/100, 999/1000, ...
and
1, 1, 1, 1, ...

are members of the same equivalence class, and therefore are
representatives of the same real number.

Quote:

What is the number between 0.99... and 1?

There is none, because .999... isn't a number, remember?

Wrong.

--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

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