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'[OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding'
2001\06\02@231056 by

"Thomas McGahee" <tom_mcgaheeSIGMAIS.COM> wrote, over and over and
over:

> Whether you like it or not, mathematically the mumber 1 exactly
> equals .9 repeating.

and David VanHorn <PICLISTMITVMA.MIT.EDU> replied:

> That's as nonsensical as saying that 2=3 for large values of 2.
> ....
>
> 0.99(followed by any finite or infinite number of nines) is by
> definition, not equal to 1.0
>
> The value of 0.99(inf..) is less than 1.0 by an infinitely small (but
> non-zero) amount.

[Sorry to single you out, Dave; this applies not only to you,
but to everyone else who's posted essentially the same thing.]

It's been a long time since I've written a "God DAMN it, you peopole
are pissing me off" email to the PICLIST, and since James Newton has
imposed a "no profanity" rule, I guess I won't write one now.

Still, though, you people ARE pissing me off.  0.9 repeating IS
exactly equal to 1.  Someone posted a simple algebraic explanation
earlier; was that TOO simple?  Here, for your edification, is a more
complex explanation, courtesy of the sci.math FAQ (where the
"0.999... = 1" question used to be #1 on the list):

11Q:  Why is 0.9999... = 1?

A:  In modern mathematics, the string of symbols "0.9999..." is
understood to be a shorthand for "the infinite sum  9/10 + 9/100
+ 9/1000 + ...." This in turn is shorthand for "the limit of the
sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 +
9/1000, ..."  Using the well-known epsilon-delta definition of
limit, one can easily show that this limit is 1.  The statement
that 0.9999...  = 1 is simply an abbreviation of this fact.

oo              m
---   9         ---   9
0.999... = >   ---- = lim  >   ----
--- 10^n  m->oo --- 10^n
n=1             n=1

Choose epsilon > 0. Suppose delta = 1/-log_10 epsilon, thus
epsilon = 10^(-1/delta). For every m>1/delta we have that

|  m           |
| ---   9      |     1          1
| >   ---- - 1 | = ---- < ------------ = epsilon
| --- 10^n     |   10^m   10^(1/delta)
| n=1          |

So by the (epsilon-delta) definition of the limit we have

m
---   9
lim  >   ---- = 1
m->oo --- 10^n
n=1

Does that make it clearer?

It's a basic mathematical FACT that 0.9 repeating is equal to 1.  If
you don't "get" this, ask questions.  Say, "Gee, that sure seems
counter-intuitive."  Find your old schoolbooks and see if it's
explained there.  Post a message to the sci.math newsgroup if you
want a thousand people to tell you to read the FAQ... But for God's
sake, if you just can't understand it no matter how hard you try,
don't ARGUE about it; that does nothing but advertise your
unwillingness or inability to learn.

-Andy

=== Andrew Warren - fastfwdix.netcom.com
=== Fast Forward Engineering - San Diego, CA
=== http://www.geocities.com/SiliconValley/2499

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>It's been a long time since I've written a "God DAMN it, you peopole
>are pissing me off" email to the PICLIST, and since James Newton has
>imposed a "no profanity" rule, I guess I won't write one now.

Hmmm...got stressed. It's bad.

>Still, though, you people ARE pissing me off.  0.9 repeating IS
>exactly equal to 1.  Someone posted a simple algebraic explanation
>earlier; was that TOO simple?  Here, for your edification, is a more
>complex explanation, courtesy of the sci.math FAQ (where the
>"0.999... = 1" question used to be #1 on the list):

So tell me why my teacher doesn't let me go to the next class/year if I mark 4.9 on her exam??? :o)))

---8<---Corte aqui---8<----

Alexandre Souza
taitoterra.com.br
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who cares?
-----Original Message-----
From: Andrew Warren <fastfwdIX.NETCOM.COM>
To: PICLISTMITVMA.MIT.EDU <PICLISTMITVMA.MIT.EDU>
Date: Saturday, June 02, 2001 10:14 PM
Subject: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

"Thomas McGahee" <tom_mcgaheeSIGMAIS.COM> wrote, over and over and
over:

> Whether you like it or not, mathematically the mumber 1 exactly
> equals .9 repeating.

and David VanHorn <PICLISTMITVMA.MIT.EDU> replied:

> That's as nonsensical as saying that 2=3 for large values of 2.
> ....
>
> 0.99(followed by any finite or infinite number of nines) is by
> definition, not equal to 1.0
>
> The value of 0.99(inf..) is less than 1.0 by an infinitely small (but
> non-zero) amount.

[Sorry to single you out, Dave; this applies not only to you,
but to everyone else who's posted essentially the same thing.]

It's been a long time since I've written a "God DAMN it, you peopole
are pissing me off" email to the PICLIST, and since James Newton has
imposed a "no profanity" rule, I guess I won't write one now.

Still, though, you people ARE pissing me off.  0.9 repeating IS
exactly equal to 1.  Someone posted a simple algebraic explanation
earlier; was that TOO simple?  Here, for your edification, is a more
complex explanation, courtesy of the sci.math FAQ (where the
"0.999... = 1" question used to be #1 on the list):

11Q:  Why is 0.9999... = 1?

A:  In modern mathematics, the string of symbols "0.9999..." is
understood to be a shorthand for "the infinite sum  9/10 + 9/100
+ 9/1000 + ...." This in turn is shorthand for "the limit of the
sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 +
9/1000, ..."  Using the well-known epsilon-delta definition of
limit, one can easily show that this limit is 1.  The statement
that 0.9999...  = 1 is simply an abbreviation of this fact.

oo              m
---   9         ---   9
0.999... = >   ---- = lim  >   ----
--- 10^n  m->oo --- 10^n
n=1             n=1

Choose epsilon > 0. Suppose delta = 1/-log_10 epsilon, thus
epsilon = 10^(-1/delta). For every m>1/delta we have that

|  m           |
| ---   9      |     1          1
| >   ---- - 1 | = ---- < ------------ = epsilon
| --- 10^n     |   10^m   10^(1/delta)
| n=1          |

So by the (epsilon-delta) definition of the limit we have

m
---   9
lim  >   ---- = 1
m->oo --- 10^n
n=1

Does that make it clearer?

It's a basic mathematical FACT that 0.9 repeating is equal to 1.  If
you don't "get" this, ask questions.  Say, "Gee, that sure seems
counter-intuitive."  Find your old schoolbooks and see if it's
explained there.  Post a message to the sci.math newsgroup if you
want a thousand people to tell you to read the FAQ... But for God's
sake, if you just can't understand it no matter how hard you try,
don't ARGUE about it; that does nothing but advertise your
unwillingness or inability to learn.

-Andy

=== Andrew Warren - fastfwdix.netcom.com
=== Fast Forward Engineering - San Diego, CA
=== http://www.geocities.com/SiliconValley/2499

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> 11Q:  Why is 0.9999... = 1?
>
> A:  In modern mathematics,

Modern mathematics. Not old common sense and logic

That's where I make a distinction and where my query about
0.999999r being infinitely close to 1 came from - a logical point
of view, not arguing mathematical proofs. If mathemeticians
in a mathematical world choose to represent 0.99999r as 1 then
I have no problem with that and respect your opinion Andrew

In the physical world "a miss is as good as a mile"

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>
>It's been a long time since I've written a "God DAMN it, you peopole
>are pissing me off" email to the PICLIST, and since James Newton has
>imposed a "no profanity" rule, I guess I won't write one now.

I'm sorry it's so upsetting to you.

0.9 is different from 1 by a tenth part.
0.99, by a hundredth.
No matter how far you go, the difference is still not zero.
It may become awkward to calculate, but tossing the remainder doesn't seem
fair.

You're telling me that I can take infinity as the number that makes
(1/N)*(1-0.9) = (1/N)*0 true.
It seems that it all hinges on what infinity means, and since people like
Feynman and Hawking are having trouble with calculations where infinity
pops in, I'm not bothered that it causes me trouble.

Still, science is that which, when you stop believing in it, is still works.
Can you show me something by which this is a consequence?
Something physical, that requires this to be true?
Or is this just a peculiarity of the current rule set.

It wasn't that long ago that "zero" didn't exist in mathmatics. I'm sure
the scholars of that day thought their mathematics was quite complete and
correct as well.

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----- Original Message -----
From: "Andrew Warren" <fastfwdIX.NETCOM.COM>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Saturday, June 02, 2001 10:10 PM
Subject: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

{Quote hidden}

Now, without trying to upset anyone, I find it hard to believe that an FAQ
is the end all answer to this.

{Quote hidden}

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>> In the physical world "a miss is as good as a mile"

In the physical world, you don't need to get anywhere NEAR an infinite
number of 9s after your decimal point for .999... to be "equal to" 1.0
Say you're talking meters - at about 10 9s ("0.9999999999 m") you're about
one atomic diameter from 1.0

BillW

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It seems that it all hinges on what infinity means, and since people like
Feynman and Hawking are having trouble with calculations where infinity
pops in, I'm not bothered that it causes me trouble.

Infinity is fun.  Not very intuitive, but fun.  The sort of infinity we're
talking about here is pretty well understood, mathematically.  Someone said
recently here that "infinity < (2 * infinity)"  Sorry, that's wrong.
infinity = 2*infinity
Furthermore, you have bits like the number of integers is the same as the
number of real numbers (infinity, aleph-null (?), "countably infinite"), but
the number of irrational numbers is larger ("uncoutably infinite")

Can you show me something by which this is a consequence?
Something physical, that requires this to be true?

Physical things don't require infinity to work...  See previous message

Or is this just a peculiarity of the current rule set.
Sure.  That's "all."

BillW

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Andrew Warren wrote:

{Quote hidden}

Hmmm. I feel like the child in the "emperors new
clothes". These experts all backing each other up re
what they agree is "right". Sort of like politicians
but with better formulas. ;o)

Surely any child could point out that with the
0.9999 issue that ANY POSSIBLE STEP can only reduce
the error to a smaller amount. The *number* of steps
is completely confusing and superfluous, the error
must always exist as there is NO POSSIBLE STEP
that can eliminate the error, only reduce it a bit
more. And you really don't need impressive formulas
to understand that reality? :o)
-Roman

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>
> Surely any child could point out that with the
> 0.9999 issue that ANY POSSIBLE STEP can only reduce
> the error to a smaller amount. The *number* of steps
> is completely confusing and superfluous, the error
> must always exist as there is NO POSSIBLE STEP
> that can eliminate the error, only reduce it a bit
> more. And you really don't need impressive formulas
> to understand that reality? :o)
> -Roman

Roman, thank you. You gave me the words to deal with this.
Above, you said:

"The *number* of steps is completely confusing and superfluous"

However, 'infinitely many steps' is _not_ a number of steps. Rather it is
the process carried out for (if I may) an _infinite_ number of steps.

Remember: 'infinity' is not a number, its a concept.

That is the difference.

Bob Ammerman
RAm Systems
(contract development of high performance, high function, low-level
software)

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Earlier in this thread, the most simple (IMO)) proof that
0.999(Repeating) = 1 was shown using no magic, mirrors or divides by 0.

There are 3 thirds in a whole (ie 3*(1/3) = 1)
and, 1/3 = 0.33333333(repeating forever) (divide it out and see!)
Multiply that by 3 and get 0.99999999(repeating forever) (try it!)

Therefore 1 = 0.99999999(repeating forever)

It may be uncomfortable, but it is true!

Show me any flaw, trick, or error in this example.

David W. Gulley
Destiny Designs

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>Earlier in this thread, the most simple (IMO)) proof that
>0.999(Repeating) = 1 was shown using no magic, mirrors or divides by 0.
> There are 3 thirds in a whole (ie 3*(1/3) = 1)
>  and, 1/3 = 0.33333333(repeating forever) (divide it out and see!)
>    Multiply that by 3 and get 0.99999999(repeating forever) (try it!)
>    Therefore 1 = 0.99999999(repeating forever)
>It may be uncomfortable, but it is true!
>Show me any flaw, trick, or error in this example.

It seems to be something like "angel's gender" but let's revise it.

When you part one something MATERIAL (numbers aren't material, they just TRY to represent it), you get three thirds. One equals another. That's ok.

When you divide the number one, by three, you get ALMOST three equal parts. ALMOST because a calculator cannot represent the part who tend to infinity. If you divide something physical, you can get all three parts with the same volume or weight. But a calculator cannot represent this.

When you bring these three parts togheter (ops, where is my english lexican?) it becomes 1 again. But in a calculator, that ROUND NUMBERS PER SE, it will not bring you the same result. It's valid for any calcule that brings the "dizima periodica" (anyone who speaks portuguese, can tell me how we say this expression in english? It means the infinite .33333333 of the 1/3), because a calculator has a limit of HOW FAR can it store numbers.

So, I think it's nice to speak that 0.9999999999999 TENDS to 1. But it can never be 1.

And of course, In my HP48 I trust :o) And it gives me 0.99999999999999 when I multiply 0.33333333333 x 3 :o)

---8<---Corte aqui---8<----

Alexandre Souza
taitoterra.com.br
http://planeta.terra.com.br/lazer/pinball/

---8<---Corte aqui---8<----

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You trust your calculator? :)

----- Original Message -----
From: "David W. Gulley" <dgulleyDESTINYDESIGNS.COM>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Sunday, June 03, 2001 7:57 AM
Subject: Re: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

{Quote hidden}

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>You trust your calculator? :)

In HP I trust :o)

It remembers me an old phrase...

"In God we trust. All others pay cash"

---8<---Corte aqui---8<----

Alexandre Souza
taitoterra.com.br
http://planeta.terra.com.br/lazer/pinball/

---8<---Corte aqui---8<----

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----- Original Message -----
From: "David W. Gulley" <dgulleyDESTINYDESIGNS.COM>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Sunday, June 03, 2001 9:57 AM
Subject: Re: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

{Quote hidden}

IMnotsoHO the only flaw here is that many simple ratios cannot be precisely
represented by decimal notation.  But (being ignorant and hard headed) I
still fail to see how this proves anything except that decimal notation is
sometimes imprecise.  I guess it only serves to show that mathematics (much
like statistics) can be used (and abused) to show that anything is true.
Sort of like geometry, being based on axioms and postulates, where we
"assume" that certain basics are true.  Then we take these things and
extrapolate further information using "proofs".  How can we really say that
these "proofs" prove anything, when they are entirely based upon assumptions
(granted that these assumptions have never been shown to be false, yet).  I
realize that this is more philosophy than math but I cannot resist. ;-D
When taken to an extreme simple logic, reasoning and mathematics can be
shown to fall apart.  Mathematical infinity is such an extreme.  Taking
Kepler's laws of motion as an example, they worked fine most of the time
(until simple observation of Mercury indicated that something was critically
wrong).  Enter Albert Einstein who showed us why they didn't always work,
while introducing a new world of paradox's and problems such as parallel
lines that eventually meet.  Even the so-called "Big Bang Theory" relies on
an "assumption" that during the first miniscule amount of time after the
bang that all of the apparant "laws" of physics and quantum mechanics were
somehow not in effect.  Mathematics is flawed, plain and simple.
Demonstrating these flaws and then arguing about them probably serves no
practical purpose.  However without debate and questioning advancement will
not occur.  Standing up and shouting that something is true because an FAQ
says so is not proof, neither is a mathematical proof necessarily equal to
absolute truth.  Nothing has ever really been proved, only temporarily shown
to be apparantly true.  As time rolls on we find that their seems to always
be an exception to the rule.  Reality is not (and never has been) what
mathematics tells us that it is.  Mathematics (like a yard stick) is a
wonderful tool, but it does not dictate absolute truth.  It seems that the
only thing that is sure is that we can never be absolutely sure (only sure
until someone "proves" that we are not).  At any given time througout
history, there exists at least one known mathematical paradox.  This seems
to serve the purpose of contiunally reminding us that something is
fundamentaly wrong with our reasoning.  If people didn't question apparant
truth we would all still be living on a flat earth in a geocentric universe.

michael the fool
(not trying to upset anyone, just trying to stimulate some neurons and
healthy debate)

>
> David W. Gulley
> Destiny Designs

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On Sun, Jun 03, 2001 at 12:22:50PM -0300, Alexandre Domingos F. Souza wrote:
> >Earlier in this thread, the most simple (IMO)) proof that
> >0.999(Repeating) = 1 was shown using no magic, mirrors or divides by 0.
> > There are 3 thirds in a whole (ie 3*(1/3) = 1)
> >  and, 1/3 = 0.33333333(repeating forever) (divide it out and see!)
> >    Multiply that by 3 and get 0.99999999(repeating forever) (try it!)
> >    Therefore 1 = 0.99999999(repeating forever)
> >It may be uncomfortable, but it is true!
> >Show me any flaw, trick, or error in this example.
>
>         It seems to be something like "angel's gender" but let's revise it.
>
>         When you part one something MATERIAL (numbers aren't material, they just TRY to represent it), you get three thirds. One equals another. That's ok.
>
>         When you divide the number one, by three, you get ALMOST three equal parts. ALMOST because a calculator cannot represent the part who tend to infinity. If you divide something physical, you can get all three parts with the same volume or weight. But a calculator cannot represent this.
>

But this is exactly the point. When you divide the number 1 by 3 you get
exactly (note EXACTLY!!!!) 3 equal parts. However the representation of that
exactly is .3 (repeating).

Everyone is getting confused by the repeating. "There is a difference at
every step therefore there must be a difference at the 'infinite' step." is
close to the quote I've read in this thread.

There is no infinite step. The threes stop ending at the same place that
you find the biggest integer! Now would anyone care to discuss what the
biggest integer is?

The repeating series is the only way to represent the value exactly. Any
attempt to cut off the repeating and it's no longer an exact representation.
You have to take the baby with the bathwater. 1/3 is exactly .3 (repeating) and
three times each value is 1 and .9 (repeating) which are exactly (note
EXACTLY!!!) the same.

And no finite string of 3's or 9's, no matter how long the string is is
exactly equal to the infinitely repeating string. And that's why this works.
Any attempt to make the string finite gives a value less than the exact value.
However the infinite string is the exact value.

Let's consider a couple of cases. Consider two numbers x1 and x2. x1 is
exactly equal to 9 (repeating). x2 is equal to 2^1024 9's after the decimal
point (a lot of 9's to be sure). Now we should be able to agree on a couple
of things.

1) That x1 > x2. x1 has more 9's than x2, so it's bigger.
2) That 10*x2 - x2 is less than 9. Consider:

10 * 0.9999 (2^1024 of them) = 9.99999 (2^1024 - 1 9's after decimal point)
(-x2)       -0.99999 (2^1024 9's after decimal point)
---------------------
8.99999 (2^1024 -1  9's and a 1 on the end)

No problem right? Now do the same with x1 and you see the difference.

10 * .9 (repeating) = 9.9 (repeating)
-0.9 (repeating)
-----------------
9.0 (repeating)

Now the difference is that since in both cases the .9 (repeating) are exactly
the same. That's the difference with infinity. Arithmetic operations on
infinity result with exactly equal infinite values. That's the catch. With
any finite string, multiplying by 10 and subtracting has a difference however
small. But the same operation on a infinite string gives no, zero, 0 , nada
difference. Not an infintessimally small difference. NO DIFFERENCE!!! It's
like trying to define the smallest real number larger than 0. That number is
0. Sounds strange, but it's true because the limit of an infinite series
approching 0 is 0.

Everyone who's arguing the difference between the sum of an infinite series
and the limit of the series isn't getting that when you deal with infinity
there is no end, therefore there is no difference.

Just remember that to justify your argument you must, absolutely must, be
able to define the largest integer. Once you can do that, then everything
else falls into place.

Good luck. ;-)

>         When you bring these three parts togheter (ops, where is my english lexican?) it becomes 1 again. But in a calculator, that ROUND NUMBERS PER SE, it will not bring you the same result. It's valid for any calcule that brings the "dizima periodica" (anyone who speaks portuguese, can tell me how we say this expression in english? It means the infinite .33333333 of the 1/3), because a calculator has a limit of HOW FAR can it store numbers.
>
>         So, I think it's nice to speak that 0.9999999999999 TENDS to 1. But it can never be 1.
>
>         And of course, In my HP48 I trust :o) And it gives me 0.99999999999999 when I multiply 0.33333333333 x 3 :o)
>

Calculators don't help. They're finite. Infinity has different rules than
finite. I'm sure your calculator also has a largest integer or real that it
can store. So does that mean that the next larger integer or real (which
clearly does exist) does not actually exist because your calculator says so.

So to summarize. 1/3 = .3 (repeating). 3*(1/3) = 3 * .3 (repeating). Therefore
1 = .9 (repeating). End of story.

And by the way if you fell compelled to refute this argument, please bring
along the largest possible integer with you. You'll need it.

BAJ

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Ok, too crazy to me. Back to binaries :o)

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| Earlier in this thread, the most simple (IMO)) proof that
| 0.999(Repeating) = 1 was shown using no magic, mirrors or divides by 0.
|  There are 3 thirds in a whole (ie 3*(1/3) = 1)
|   and, 1/3 = 0.33333333(repeating forever) (divide it out and see!)
|     Multiply that by 3 and get 0.99999999(repeating forever) (try it!)
|     Therefore 1 = 0.99999999(repeating forever)
| It may be uncomfortable, but it is true!

Hi,

Leaving behind calculators and computers, mathematically "it can be proof"
(i'm not doing that in this post)that the following are two different
decimal series development for the same number:

1)    1

2)    0.99999999...  (infinite periodic digits)

A quick view: Starting with, for example, the number: 0.999 and "appending"
by steps a 9 to the rightmost digit, the number obtained step by step is
greater. You can find in every step infinite real numbers between  the
number obtained and 1. Remember that in the real set of numbers, there's
always an infinite quantity of numbers between two different numbers. This
is not what it happens in the limit, You can't find infinite (in fact, You
can't find *any* number...) real numbers between  0.999...  and 1.

So the mathematical proof of the equality is based on showing that between
0.9999... and 1 there's not an infinite quantity of real numbers.

Best regards,

S.-

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What about the math proof someone posted a while ago...

x = .999...
10*x = 9.999...
9*x = 10*x - x   (the infinite .9999 cleanly subtract from one another -->)
9*x = 9.999... - .999... = 9.0
x = 1

No divisions by zero or other math-destroying magic required...

BillW

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> What about the math proof someone posted a while ago...
>
> x = .999...
>10*x = 9.999...
> 9*x = 10*x - x   (the infinite .9999 cleanly subtract from one
> another -->)
> 9*x = 9.999... - .999... = 9.0
> x = 1
>
> No divisions by zero or other math-destroying magic required...
>
> BillW

Making something look like something else isn't how "I" define
proof. I'll be blunt and say that type of mathematical manipulation
is just card tricks. And as for "a miss is as good as a mile", look
at the number of particle smashes needed to detect neutrinos or
other sub-atomic bits, which are far far smaller than atoms. How
many head-on collisions were an infinitely small "that much" off ?
But understand that I'm quite comfortable with the fact that there
is a mathematical world and a physical world

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----- Original Message -----
From: "Jinx"

> But understand that I'm quite comfortable with the fact that there
> is a mathematical world and a physical world

This concept can be seen in 'Donald Duck in Mathematical World'! :)

-Mg

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Reminds me of a mathematician and an engineer in the same class, where
the teacher presents the problem:

Suppose  a beautiful woman (or man, take your pick ;-) is opposite a
room from you, and you can approach her/him with the restriction that
each movement can only halve the distance between you and her/him.  Will
you ever reach her/him?

The mathematician responds:
No.  You may get infinetely close, but you will never reach her/him.

The engineer responds:
You will get close enough for most intents or purposes...

Of course, conceptually, 0.999... does not equal one.  It equals
0.999....  But using calculus and other methods you see that 0.999...
approaches 1, and, for all intents and purposes, is equal to 1.  It
really depends on what level of precision one is asking for.  If one
want infinite precision then one cannot say that 0.999... = 1 unless one
can prove that 0.999... approaches one faster than the needed precision
approaches infinity.

But this is all academic, and is akin to proving that the chicken came
before the eggroll.  If anyone wants to prove me wrong, then sit down
and write a 0, a decimal point, an infinite series of nines, and equal
sign, and a one (no shorthand, please!) before hitting reply.

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David VanHorn <PICLISTMITVMA.MIT.EDU> wrote:

> I'm sorry it's so upsetting to you.

Thanks, Dave.  I don't know WHY this thread makes me so crazy --
it really shouldn't -- but I just can't help it.  I know you're
not deliberately trolling for an argument.

> 0.9 is different from 1 by a tenth part.
> 0.99, by a hundredth.

Ok, maybe it's easier if we turn the original assertion around:

What if I said that the DEFINITION of "infinite" is "the number
of 9s you'd have to append to 0.9 in order to make that number
equal to 1"?  Would that make sense?

Perhaps you could look at it in this casual, common-sense way:

If 0.999... APPROACHES 1 as the number of 9s APPROACHES infinity,
then 0.999... EQUALS 1 when the number of 9s EQUALS infinity.

Do either of those approaches help?

> It seems that it all hinges on what infinity means

Well, yeah.

> people like Feynman and Hawking are having trouble with
> calculations where infinity pops in

Feynman hasn't been having trouble with ANY calculations since
he died, and if Hawking could speak, I'm pretty sure he'd tell
you that 0.999... is equal to 1.

> I'm not bothered that it causes me trouble.

That's fine... But the fact that you have difficulty with the
concept isn't a good reason to assert that mathematics is wrong.

> It wasn't that long ago that "zero" didn't exist in mathmatics.

"Zero" has ALWAYS existed in mathematics: "If Julius has X
apples, and he gives III to Brutus and VII to Biggus Dickus, how
many apples remain?"

It's just the "0" SYMBOL which hasn't always existed.

> I'm sure the scholars of that day thought their mathematics was
> quite complete and correct as well.

That's an argument AGAINST your point of view, not for it.

-Andy

=== Andrew Warren - fastfwdix.netcom.com
=== Fast Forward Engineering - San Diego, CA
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>         So tell me why my teacher doesn't let me go to the next class/year
if I mark 4.9 on her exam??? :o)))

Because you did not score the repeating 9's after the 4.9?

Wouter

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> > The value of 0.99(inf..) is less than 1.0 y an infinitely small (but
> > non-zero) amount.

> 11Q:  Why is 0.9999... = 1?

Mathematically speaking the propositions "The value of 0.99... is less than
1.0 by an infinitely small amount" and "0.99... = 1" are the same! Maybe
that helps to understand a counter-intuitive fact... Infinitely small must
be interstand as "when I name an amount, *any amount at all*, the infinitely
small number is still smaller". Think of infinitely small as a very verbose
(but very usefull) way of specifying zero. In fact it is this way of
thinking about zero (or equality) that makes calculus possible.

Wouter

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>
>     What if I said that the DEFINITION of "infinite" is "the number
>     of 9s you'd have to append to 0.9 in order to make that number
>     equal to 1"?  Would that make sense?

No.
If, at any point we can just say "aw to hell with it, it's almost 1, so
it's 1", I have a problem with that.

However, as a tool in calculus, to allow calculations that wouldn't
otherwise be possible, and with the proviso that we are approximating, and
not exactly calculating an answer, I have no problem with it.

The root problem here appears to be that there some numbers which the
decimal system is ill equipped to represent, in a manner similar to roman
numerals having problems with large numbers, only in a deeper manner.

1/3 is easy to deal with, but can't be represented with complete accuracy
in decimal form.

>     Perhaps you could look at it in this casual, common-sense way:
>
>     If 0.999... APPROACHES 1 as the number of 9s APPROACHES infinity,

Ok

>     then 0.999... EQUALS 1 when the number of 9s EQUALS infinity.

No.
I see no requirement that it do so.

>     Feynman hasn't been having trouble with ANY calculations since
>     he died, and if Hawking could speak, I'm pretty sure he'd tell
>     you that 0.999... is equal to 1.

For any practical matter, yes.
However, in an absolute sense, this is just sweeping some ugliness under
the carpet.

> > It wasn't that long ago that "zero" didn't exist in mathmatics.
>
>     "Zero" has ALWAYS existed in mathematics: "If Julius has X
>     apples, and he gives III to Brutus and VII to Biggus Dickus, how
>     many apples remain?"

If you can't represent it, then you can't do calculations with it.

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----- Original Message -----
From: David VanHorn <dvanhornCEDAR.NET>

> >     Perhaps you could look at it in this casual, common-sense way:
> >
> >     If 0.999... APPROACHES 1 as the number of 9s APPROACHES infinity,
>
> Ok
>
> >     then 0.999... EQUALS 1 when the number of 9s EQUALS infinity.
>
> No.
> I see no requirement that it do so.
>
> >     Feynman hasn't been having trouble with ANY calculations since
> >     he died, and if Hawking could speak, I'm pretty sure he'd tell
> >     you that 0.999... is equal to 1.
>
> For any practical matter, yes.
> However, in an absolute sense, this is just sweeping some ugliness under
> the carpet.

I haven't seen this used in this thread, so I'll describe it this way, as my
old college calc prof used this as his example, try this:

Using decimal representations:
1/3 = .33333333333 (to infinity)

1/3 * 3 = .3333333333 * 3 = .99999999999 (to infinity)

However, using fracional simplification:
1/3 * 3 = 3/3 = 1/1 = 1

Therefore 1 = .9999999999 (to infinity)

Is that good enough for you?

> If you can't represent it, then you can't do calculations with it.

What is wrong with the overbar? Are you requiring all numeric
representations be in decimal?  So I can't use Pi in any of my calculations?
e?  i?

Jeff

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>
>1/3 * 3 = .3333333333 * 3 = .99999999999 (to infinity)
>
>However, using fracional simplification:
>1/3 * 3 = 3/3 = 1/1 = 1
>
>Therefore 1 = .9999999999 (to infinity)
>
>Is that good enough for you?

Works for me.
It's just awkward to represent in decimal
The fractions don't "simplify" it, they are a better representation.
Simplification implying discarding accuracy to achieve an end.
I'll agree with 1 = the sum of the infinite series 0.999.. strictly as a
simplification, because the result isn't likely to be significantly different.
But I don't agree that 1 = 0.999...
The key point is that by making the first statement, you are defining what
"sum of the infinite series" means, not stating the equality 1 = 0.999...

> > If you can't represent it, then you can't do calculations with it.
>
>What is wrong with the overbar? Are you requiring all numeric
>representations be in decimal?  So I can't use Pi in any of my calculations?
>e?  i?

'Snot what I said.
1/3 is a precise representation, 0.3333 isn't
PI works, 3.141... isn't precise.
When you force any of these trancendentals into decimal form, you can't
represent them accurately. It's a fundamental limitation of the decimal form.

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so in a nutshell
_
1/(1-0.9)=infinity

1/infinity=0

therefore
_
1-0.9 = 0 ?

it's certainly not intuitive but i think it is true.  also seems pretty academic for practical purposes.

it would be interesting to look at this thread (well the rounding thread) from a point of view of how many siginificant digits do i need so that i'm not significantly biasing my calculations ie: my robot won't be off course more than a 1" after traveling 20' using my optical encoder or accelerometer or whatever.

is anybody still reading this thread?  :)

-pete

On Mon, 4 Jun 2001, Jeff DeMaagd wrote:

> {Original Message removed}
----- Original Message -----
From: Peter William Green <pwgreenENGINEERING.UIOWA.EDU>

> it would be interesting to look at this thread (well the rounding thread)
from a point of view of how many siginificant digits do i need so that i'm
not significantly biasing my calculations ie: my robot won't be off course
more than a 1" after traveling 20' using my optical encoder or accelerometer
or whatever.

Well, that kind of depends on how complex your computations are and how
loose the mechanical tolerances are on your robots (mechanical error vs.
round off error).  --My robots don't need many significant digits at all...
:')

-Dal

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First, I'd like to thank everyone for their comments on this topic. They
have made me review and re-think the basic concepts of mathematics in
general. I've spent way too much time in the past few days re-reading text
books and searching for other texts on the various fundamental algebraic
operations, definitions and, axioms. However, I don't consider this wasted
time because I firmly believe that blind faith in anything is bad and, I
have not reviewed the fundamentals in the past decade or two.

I especially would like to thank Dave V. for the link to Swarthmore Colleges
excellent Dr. Math project. http://forum.swarthmore.edu/dr.math

There is only one thing I feel comfortable stating with 100% certainty after
my research and review.
If you are a student and the question of, does .999... = 1, comes up on a
I could not find one single reference by any professor/teacher/school
district/college or university that would allow you to answer no and be
given credit for a correct answer. The most complete web resource for info
on this topic I found is,
http://forum.swarthmore.edu/dr.math/faq/faq.0.9999.html. If you read the
page be sure to also read the five linked pages at the end of the FAQ

> The root problem here appears to be that there some numbers which the
> decimal system is ill equipped to represent, in a manner similar to roman
> numerals having problems with large numbers, only in a deeper manner.
>
> 1/3 is easy to deal with, but can't be represented with complete accuracy
> in decimal form.

This statement has brought much needed closure in my mind as to what the
objections are to the simple algebraic proof I presented in a post last
week. I had been wondering what step(s) in the proof were potentially
flawed. From this statement I see that the disagreement comes with the very
first statement of the proof.
"Given:
x = 0.9(repeating)"
The problem is that repeating decimals are considered by some to not be
accurate representations of numbers like 1/3. This statement appears to be
somewhat supported by the work of Georg Cantor from the late 19th century.
For 1/3 to be <> 0.333... all that needs to be true is that long division
does not work in some cases. Personally I don't believe this but to my mind
it is certainly possible that at some point in the future this may be proven
to be the case. After all similar events have happened regularly throughout
the course of human history. For those interested I recommend James Burkes,
"The Day the Universe Changed" either the book or television series from the
mid 1980's.

I have decided that in the future if a question regarding repeating decimal
representations of fractions comes up on the PICLIST, I will ask if the
questioner believes that 0.333... = 1/3. If they do I'll try to help but if
they don't I'll ask that the question be restated without the need for
repeating decimals before attempting to help.

Paul

PS - I was a bit shocked by some of the assertions that mathematics is all
magic tricks, I had thought that everyone accepted mathematics as the
official language of science and felt there was no magic to it. I do expect
people to look skeptically at things but to attribute magic to anything is
way more than I can accept.

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>
>The problem is that repeating decimals are considered by some to not be
>accurate representations of numbers like 1/3. This statement appears to be
>somewhat supported by the work of Georg Cantor from the late 19th century.
>For 1/3 to be <> 0.333... all that needs to be true is that long division
>does not work in some cases.

:) Work it out, and let me know what you get :)
I would assert that it does not work here, since you can't ever finish the
problem.
Feel free to prove me wrong, but show your work. <VBG>

>  Personally I don't believe this but to my mind
>it is certainly possible that at some point in the future this may be proven
>to be the case. After all similar events have happened regularly throughout
>the course of human history. For those interested I recommend James Burkes,
>"The Day the Universe Changed" either the book or television series from the
>mid 1980's.

I'm in agreement here.
Science guides us into theories and systems that are imperfect, but
workable representations of how the universe works. As time progresses, we
refine our theories and systems.

>PS - I was a bit shocked by some of the assertions that mathematics is all
>magic tricks, I had thought that everyone accepted mathematics as the
>official language of science and felt there was no magic to it. I do expect
>people to look skeptically at things but to attribute magic to anything is
>way more than I can accept.

I don't think it's "magic" or anything of the sort. Just a convenient way
to deal with inconvenient numbers.  It works, till something better comes
along.

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> I don't think it's "magic" or anything of the sort. Just a convenient way
> to deal with inconvenient numbers.  It works, till something better comes
> along.

I'm very sorry, I should have stated that you where completely excluded from
this final comment (since I had mentioned you by name earlier in the post).
I found your arguments to be excellent with absolutely no hints of anything
mystical.

It was some other members who made statements like this including one that
actually said "mathematical manipulation is just card tricks".

Again, sorry for unintentionally implying that you where in that group.

Paul

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On Sun, Jun 03, 2001 at 11:22:48PM -0400, M. Adam Davis wrote:
>
> Of course, conceptually, 0.999... does not equal one.

Nope. It equals 1. No intents or purposes necessary.

>  It equals
> 0.999....  But using calculus and other methods you see that 0.999...
> approaches 1, and, for all intents and purposes, is equal to 1.  It
> really depends on what level of precision one is asking for.
>  If one
> want infinite precision then one cannot say that 0.999... = 1 unless one
> can prove that 0.999... approaches one faster than the needed precision
> approaches infinity.

Precision implies finiteness. It implies that there will be a cutoff of the
9's at some point in the string.

The 9's end at the same place that the largest possible integer resides. Care
to define that number?

>
> But this is all academic, and is akin to proving that the chicken came
> before the eggroll.  If anyone wants to prove me wrong, then sit down
> and write a 0, a decimal point, an infinite series of nines, and equal
> sign, and a one (no shorthand, please!) before hitting reply.

There's no need for that. The term 0.9 (repeating) is a sufficient
representation for that infinite series. Folks here are failing to grasp that
mathematics for infinity doesn't follow normal finite mathematicl rules.
Equations as simple as 'infinity*2 = infinity' are completely nonsensical
for nonfinite values but true for infinity. Any argument for precision or
truncation of infinity is akin to stating that 'The largest integer is x.'
There's simply to truth to that statement.

In infinity arithmatic 0.9 (repeating) = 1. Not approximates. No approches.
No precision. No error. The proof has been given in this thread multiple times.

I really liked the argument I saw yesterday. I'll repeat it and ask if anyone
can propose a solution: "If 0.9 (repeating) < 1, then there are an infinite
number of real numbers that exists between 0.9 (repeating) and 1. Give one
such number."

And try as you might, you'll find that there is no such number. You'll find
in your search that no matter what number you pick, it will not be between
0.9 (repeating) and 1. Therefore the two values must be the same because not
only can you not define an infinite number of real numbers between the two,
you can't even pick one.

There's no pretense here. Infinity mathematics is exact. Not approximations.
It doesn't exist in the real world, only in the concepts of our minds.

So for the refuters you now need to bring two items to the table. The largest
possible integer, and a number between 0.9 (repeating) and 1.

Good luck.

BAJ

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Heh.  Next we should argue about

lim     sin(x)/x = 1
x->0

Lots of signal theory being based on sine waves, this is probably
of more practical application (?)  And it's even less intuitive.

:-)
BillW

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I think that one of the fundamental laws of arithmetics says that a number
is only equal to itself. This includes rational numbers (like 0.(9) and
1). This means that 1 != 0.(9) by definition. Proving this is rather
interesting (i.e. I missed that class). I believe that it can be reduced
to absurdum somehow.

Peter

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> one that actually said "mathematical manipulation is just card tricks"

It wasn't meant to imply any sort of chicanery or mumbo jumbo. David
VanHorn put it better. " I don't think it's "magic" or anything of the sort.
Just a convenient way to deal with inconvenient numbers.  It works, till
something better comes along". It's just how these convoluted means
to an end look to a mathematical outsider

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I think that one of the fundamental laws of arithmetics says that a number
is only equal to itself. This includes rational numbers (like 0.(9) and
1). This means that 1 != 0.(9) by definition.

But ".999 repeating" is not a "number", per se.  It's shorthand for an
algorithm that produces a value (that happens to be 1.0)

BillW

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> 'Snot what I said.
> 1/3 is a precise representation, 0.3333 isn't
> PI works, 3.141... isn't precise.

Full circle!!! :-)
This is where this topic started.

Pi to 4 significant figures / 2 decimal places should be 3.142 as hammered
out in brain burp rounding.
(But I know you knew that and it wasn't the point you were making)

Quite coincidentally, Pi for most any engineering purpose is 355/113
High by about 0.00000026676...
Easy to remember 113355 and split it in two and divide.
Courtesy HP long ago.

Russell McMahon

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> Easy to remember 113355 and split it in two and divide.
> Courtesy HP long ago.
>
OH NO another thing to remember. ;-D  I'll pass on this and continue
recalling what I already learned about pi.  3.1415926585 (PS I didn't cheat,
this is off the top of my head. So I could be wrong!)  If I ever need more
precision than this I will just look it up. ;-D  I doubt that I ever will,
since this is more than accurate enough to navigate to the moon should I
ever build a Pic controlled spacecraft.  Thanks everyone! It's been fun.  I
love a good argument.  I'm still not convinced though.  TTYL

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Russell McMahon wrote:

<snip>

>Quite coincidentally, Pi for most any engineering purpose is 355/113
>High by about 0.00000026676...
>Easy to remember 113355 and split it in two and divide.
>Courtesy HP long ago.
>
>
Depending on the accuracy required, 22/7 can work also. While not as
accurate as 355/113, it's only high by about 0.001264489... (about 0.04%). I
find the 22/7 easier to remember. :=)
Courtesy of a Forth Interest Group member, also long ago.

_________________________________________________________________

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I'll pass on this and continue recalling what I already learned
about pi.  3.1415926585 (PS I didn't cheat, this is off the top
of my head. So I could be wrong!)

My memory says 3.14159265*3*5...  (hmm.  Found at least one
online source that agrees with me (and another that says "42"))

BillW

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>
>My memory says 3.14159265*3*5...  (hmm.  Found at least one
>online source that agrees with me (and another that says "42"))

The indiana legislature nearly passed a law defining it as 3.2

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> >Quite coincidentally, Pi for most any engineering purpose is 355/113
> >High by about 0.00000026676...
> >Easy to remember 113355 and split it in two and divide.
> >Courtesy HP long ago.
> >
> >
> Depending on the accuracy required, 22/7 can work also. While not as
> accurate as 355/113, it's only high by about 0.001264489... (about 0.04%).
I
> find the 22/7 easier to remember. :=)
> Courtesy of a Forth Interest Group member, also long ago.

Seriously, I think 113355 is easier for those who have no real grasp of what
is going on but want to use Pi. I have met a few.
Some cannot remember 22/7 and have no idea what the 'real" number is.

You take them down the     "11   33  55" path and drum the pattern  in.
All odd numbers, Isn't that odd etc,
Then just divide in two and divide larger by smaller.
113355
355/113
bingo.

I personally just remember it to 14 places (which is where my brain decided
to stop for reasons known only to itself) which has been good enough fro
everything so far. 3.14159265358979

More than any Engineer will ever need.

RM

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Russell McMahon wrote:

> Seriously, I think 113355 is easier for those who have no real grasp of what
> is going on but want to use Pi. I have met a few.
> Some cannot remember 22/7 and have no idea what the 'real" number is.

> I personally just remember it to 14 places (which is where my brain decided
> to stop for reasons known only to itself) which has been good enough fro
> everything so far. 3.14159265358979
>
> More than any Engineer will ever need.

Ha ha! This is like the nerd olympics! See who
knows Pi to the most decimal places! ;o)
(no offense meant Russell!)

Gee I don't even own a calculator that I could
type that number into. <grin>
-Roman

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> > I personally just remember it to 14 places (which is where my brain
decided
> > to stop for reasons known only to itself) which has been good enough fro
> > everything so far. 3.14159265358979

> Gee I don't even own a calculator that I could
> type that number into. <grin>
> -Roman

Yes you do, if you have Windoze.

Bob Ammerman
RAm Systems
(contract development of high performance, high function, low-level
software)

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At 07:02 AM 6/5/01 -0400, you wrote:

>Yes you do, if you have Windoze.

But then he could just push the 'P' for 'pi' key and get:

3.1415926535897932384626433832795

Best regards,

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Spehro Pefhany --"it's the network..."            "The Journey is the reward"
speffinterlog.com             Info for manufacturers: http://www.trexon.com
Embedded software/hardware/analog  Info for designers:  http://www.speff.com
Contributions invited->The AVR-gcc FAQ is at: http://www.bluecollarlinux.com
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

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Anyone ever heard this one?

http://www.snopes2.com/religion/pi.htm

NOTE:This legend is FALSE!!!!!

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At 07:26 AM 6/5/01 -0400, you wrote:
>Anyone ever heard this one?
>
>http://www.snopes2.com/religion/pi.htm
>NOTE:This legend is FALSE!!!!!

But this one is not: (pi = 3.2)

http://www.urbanlegends.com/legal/indiana_pi_bill.html

Best regards,

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Spehro Pefhany --"it's the network..."            "The Journey is the reward"
speffinterlog.com             Info for manufacturers: http://www.trexon.com
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Well, we all pick on the Indiana legislature for this, but I'm pretty sure
the whole thing was a spoof even when it happened. Here is an excerpt from
the bill:

<QUOTE>

Section -3- In further proof of the value of the author's proposed
contribution to education and offered as a gift to the State of Indiana, is
the fact of his solutions of the trisection of the angle, duplication of the
cube and quadrature of the circle having been already accepted as
contributions to science by the American Mathematical Monthly, the leading
exponent of mathematical thought in this country. And be it remembered that
these noted problems had been long since given up by scientific bodies as
insolvable mysteries and above man's ability to comprehend.

</QUOTE>

Given that, IIRC, it has been _proven_ that trisection of an angle is
impossible (and I think the other items listed also); it seems to me that
this whole thing was done tongue-in-cheek.

Bob Ammerman
RAm Systems
(contract development of high performance, high function, low-level
software)

{Original Message removed}
Being from Indiana, let me offer this bit of trivia.  When I was in grade
school we went on a field trip to the capitol building.  The tour guide told
us that, according to Indiana state law, whenever a vote is conducted using
the electronic button voting thingy, that whatever is voted upon could
become a law, regardless of how silly it may be.  Also, only legislators can
conduct a vote (meaning only legislators can push the buttons).  So when
they test for an affirmation, whatever was "voted" on becomes law.
Consequently, there are a lot of strange laws in Indiana.

Michael Brown
Instant Net Solutions
http://www.KillerPCs.net

{Original Message removed}
Paul Hutchinson wrote:

> PS - I was a bit shocked by some of the assertions that mathematics is > all magic tricks...

More like a glass bead game - Math is a logical structure
erected on a foundation of assumptions (axioms), once held
to be self-evident.

Untold effort has gone into establishing that these axioms
have an existence independent of our assertion, to no avail.
Quite the contrary, in fact - The work of Godel apparently
demonstrates that this is not possible.

That math is often useful to describe the behaviour of the
universe is a happy circumstance, but so far has not been
shown to be more than that.
This has made a lot of very smart people unhappy for 60+
years now, but remains the state of the art, AFAIK.

regards, Jack

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John Gardner wrote:
>Paul Hutchinson wrote:
>
>> PS - I was a bit shocked by some of the assertions that mathematics is >
all magic tricks...
>
.........
>That math is often useful to describe the behaviour of the
>universe is a happy circumstance, but so far has not been
>shown to be more than that.
>This has made a lot of very smart people unhappy for 60+
>years now, but remains the state of the art, AFAIK.
>

By "... more than that ...", do you mean being able to "explain"
the meaning of life and the genesis of the universe, etc, as
opposed to simply being a way to "describe" it?

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As I stated in an earlier (long winded) post.  I completely agree with this.
To paraphrase a Great book, you cannot build a house on sand [and expect it
to hold up].  You need to build a house on a solid foundation, unfortunately
mathematics has failed us in this respect.  If I were a lawyer that built a
case on unprovable (but "self evident") evidence then it would in all
probability be declared a mistrial.  Don't misunderstand me, mathematics is
a wonderful tool, it's just not quite a perfect tool.  The same applies to
electrical theory (I guess that's why it is called theory and not fact).
One should wonder why the formulae for physics, electrical theory, and
mathematics etc... are so simple).  One need not wonder why (when taken to
an extreme) that they fail us.  If we are imperfect creatures how can we
create perfection?  Not trying to be theological here, so please don't be
offended anyone.  I'm just stating my case.  Unfortunately the "state of the
art" techniques used in science tend not to be based on actual observation
and repeatability.  It's no longer science (but in reality a religion) when
we begin to discount actual observation and begin to look for reasons
(actually excuses) when observation doesn't match prediction.  It's hardly
accurate science when we need to make adjustments in calculations that
involve an entire order of magnitude (such as the mass of the universe or
the distance to the stars).  Or, for example, we accept things such as 90%
of the universe is missing (locked up in so called unobservable dark
matter).  I fully believe in scientific method, unfortunately modern
physicists don't seem to be much concerned about it.  We are rapidly
approaching (actually we seem to already be there) a point where modern
science requires "faith".  Finally, math is great, science is great but is
only 99.9 (repeating but not to infinity)% accurate.

michael the humble, uneducated, misguided, ignorant fool

{Original Message removed}
Michael Brown wrote:
....... Unfortunately the "state of the
>art" techniques used in science tend not to be based on actual observation
>and repeatability.  It's no longer science (but in reality a religion) when
>we begin to discount actual observation and begin to look for reasons
>(actually excuses) when observation doesn't match prediction.
.........

Michael, I think you are making some incorrect assumptions here.
Classically, science uses the discrepancies between current theory
and current observations as a bootstrap to create better theories.

Only a "bad" scientist would pretend that his/her pet theory is
the "one", "final", and "correct" answer. Having been in school a
couple of times, I have met some of those - but don't generalize
this to all science and scientists.

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> I think that one of the fundamental laws of arithmetics says that a number
> is only equal to itself.
Right

> This includes rational numbers (like 0.(9) and
> 1). This means that 1 != 0.(9) by definition.
So 0d10 != 0xA != 0o12 ?

Wouter

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> Finally, math is great, science is great but is
> only 99.9 (repeating but not to infinity)% accurate.

You should make a big distinction between math and science.

Math in itself has nothing to do with the real word, it is just deriving
consequences from a axioms. In math you can prove such a consequence. Sorth
of an error in the proof in math "once proven is forever true". But note
that there are a large number of mathematical systems, and what is true in
one system (there is no x such that x*x=-1) can be false in another system.

Science is (simplified) applying math to the real world. One guy just poses
some theory, and when others can not disprove it in reasonable time it is
said to hold. But that does not mean it is the truth, just that it is the
best we have up to now.

Wouter

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> Michael Brown wrote:
> ....... Unfortunately the "state of the
> >art" techniques used in science tend not to be based on actual
observation
> >and repeatability.  It's no longer science (but in reality a religion)
when
{Quote hidden}

(sorry to be long winded as usual but hey it's OT)
I firmly stand behind scientific method and "real" science.  As you pointed
out, it is the scientists with which I have a problem.  For years I have
listened to assertions and mathematical claims (statistical) that life
"must" only exist on the earth since the entire universe would have been
filled with life by now.  Then as soon as one little rock from mars is found
on the Earth everyone "scientists" seems to immediately jump ship and climb
on the band wagon purporting that the possibility of life in several
locations within our own solar system is plausible.  I don't see the
scientific method in this.  I still must ask, "Where is the (missing?) dark
matter?"  How far away are the stars really?  Why is gravity (an incredibly
weak force) solely responsible for shaping the universe.  Why is it not
possible that the electric force (an incredibly strong force) has had some
responsibility for shaping the structure of the universe.  Experiments
involving plasma and electric force seem to be better at creating tiny
models that more closely resemble the known universe, yet no one (gravity
based scientists) seems to pay any attention to this.  I feel that the norm
is becoming to accept anything that remotely supports the "mass accepted"
belief system, yet ignore or discount anything else that may "upset the tea
cart" until it becomes too painfully obvious to do so.  It just seems that
"faith based" science is the wave of the future.  Too many observations are
being ignored.  For example, whenever a new crystalline compound is
synthesized the compound seems to have trouble determining which structure
to assume, trying one structure and then another until it finally "decides"
which structure to use.  And then when these experiments are repeated the
compound somehow "knows" which crystal structure to assume and never again
tries to assume any other structure.  This may sound crazy, but check it out
if you don't believe me.  Also, how can it be explained that monkeys of some
species suddenly modify their (instinctive?) behavior by beginning to wash
food before eating, and just as suddenly the same species of monkeys
(thousands of miles away) modify their behavior in the same way.  Something
is going on here for which we have no fundamental explanation.  I guess what
I am trying to say is that this universe in which we live has things going
on that probably no amount of mathematical, inductive, or deductive
reasoning is going to explain.  BTW my favorite "wacky" science experiment
is the photon thru the slit(s) trick. ;-D

michael the 'head scratcher'

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Michael Brown wrote:

>(sorry to be long winded as usual but hey it's OT)
>I firmly stand behind scientific method and "real" science.  As you pointed
>out, it is the scientists with which I have a problem.  For years I have
>listened to assertions and mathematical claims (statistical) that life
>"must" only exist on the earth since the entire universe would have been
>filled with life by now.
...........

What you raised are all good questions, mostly for which there are
currently no good answers. Science works incrementally, a little
step at a time. Theories change daily. Egos get in the way. That's
human nature.

It takes a long time and a lot of work to tackle these issues. It
is only since the Hubble Telescope - just a decade now - that
we are really beginning to see the "big" picture concerning the
universe. This one instrument alone has widened the possibilities
immensely.

Maybe all of these open questions are better viewed as cause
for further conjecture, rather than cause for asking why all of
the answers aren't already known. Most of these questions, and/or
their progeny, will still be here when you and I are both dead.

best regards,
- dan
==================

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Michael Brown wrote:

For years I have
>listened to assertions and mathematical claims (statistical) that life
>"must" only exist on the earth since the entire universe would have been
>filled with life by now.  Then as soon as one little rock from mars is found
>on the Earth everyone "scientists" seems to immediately jump ship and climb
>on the band wagon purporting that the possibility of life in several
>locations within our own solar system is plausible.  I don't see the
>scientific method in this.
........

Ummm - actually this "is" the basis of scientific method. When new
evidence comes in, then new theories are formed - at least by some.

And interesting you should use this example too, because not all
scientists think what was found in that particular rock is actually
evidence of life. In every one of these areas, there are several
alternative theories. That's what makes all of this science. Just
for fun, take a look at the following book - it will give you a
few more questions to add to your list:

"Cradle of Life", by Wm Schopf, 1999

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2001\06\06@030606 by
> Only a "bad" scientist would pretend that his/her pet theory is
> the "one", "final", and "correct" answer. Having been in school a
> couple of times, I have met some of those - but don't generalize
> this to all science and scientists.

Flame shields up.

Try this exercise.

Search for comments from SUPPORTERS of the theory of evolution.
For this exercise, do NOT read ANYTHING by anyone who is NOT an ardent
supporter.
See how many statements you can find about this NOT being a theory but now
being proven self evident fact that is now elevated above the state of
theory.
While there, look for "proof" of this theory that would satisfy the
scientific method. (An unfair test as the scientific method seeks only to
disprove anything - but the point should be clear.)

Draw conclusions.

Report back.

Flame shields still up.

regards

Russell McMahon

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Russell McMahon wrote:

>
>Flame shields up.
>
>Try this exercise.
>
>Search for comments from SUPPORTERS of the theory of evolution.

Bad dog, Russell -->  ** YOU ** know full well you are not supposed
to use the 'e' word on piclist.

As past history has shown - again and again - pretty soon James
will have to come in and wield Maxwell's Silver Hammer on

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ROTFL!

Jack

Dan Michaels wrote:
{Quote hidden}

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> Try this exercise.
>
> While there, look for "proof" of this theory that would satisfy the
> scientific method. (An unfair test as the scientific method seeks only to
> disprove anything - but the point should be clear.)

This is a bit unfair to the *theory* of evolution, because it (sadly) works
for almost any scientific *therory* (note that there are no scientific
truths, only theories that have not yet been proven false). IMHO this is the
biggest misconception about science (found among the general public,
politicians, but also among scientists!): that science provides things that
can be called thruths. It does not. It does not, in about the same way that
democracy is *not* a very good way to organise a country. It is just the
least bad way we currently know.

Wouter

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michael brown <n5qmgAMSAT.ORG> writes:
> I firmly stand behind scientific method and "real" science.  As you pointed
> out, it is the scientists with which I have a problem.  For years I have
> listened to assertions and mathematical claims (statistical) that life
> "must" only exist on the earth since the entire universe would have been
> filled with life by now.

That's a well-known hypothesis, but it was not and is not the case that
ALL scientists believe it.  I doubt that even MOST scientists belive it.
And it wasn't even a scientific theory, because it is not falsifiable.
You can't prove a universal negative.

There are many other potential explanations for why life may exist
elsewhere in the universe but yet not have evidence seen from Earth.
Those hypothesis will also be difficult to test, by their very nature.

> Then as soon as one little rock from mars is found
> on the Earth everyone "scientists" seems to immediately jump ship and climb
> on the band wagon purporting that the possibility of life in several
> locations within our own solar system is plausible.

Scientists have been claiming that it is possible for life to exist in
other parts of our solar system for over a century.  It can be proven by
finding at least one example.  It would be incredibly difficult to
disprove.

> I don't see the scientific method in this.

You may have some misconceptions about science.  A few helpful hints:

1.  Not everything the media claims to be science is.

2.  Not everyone the media calls a scientist is one.

3.  Not everything that an actual scientist does or says is science.

4.  Most people have little concept of what science is, or what it means
for something to be scientific.  Most people lump anything that sounds
"technical" into a huge category they call "science".

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What I'm seeing here is a misunderstanding of the scientific
method and the difference between hypothesis and theory.  I
_AM_ a scientist.  What science is about is objectivity.  It
is a systematic and objective way of looking at stuff.  It
involves observation and testing to sort stuff-that-works
from stuff-that-doesnt-work.

Over and over on the piclist I see recommendations to use
good tools.  Rework stations, soldering irons, C compilers.
Good tools mean that you dont waste time fighting your tools
that you could be using to get your real work done.  If
someone doesnt like the brand YOU use, well, that's up to
them.  Just don't disparage tools you've never used.

I can't do parts of my work without being able to figure out
stratigraphic sequences.  I deal with the realities of
continental drift all the time.  One interesting example was
locating old buried channels of the Santa Ana River that had
been offset along the San Andreas Fault and subsequently
buried.  Well, I found them pretty much where I predicted
they would be.  And oil geologists use paleontology to locate
the correct reservoir rocks to develop oil fields.  And I
notoriously and openly locate water wells without dowsing,
and Im not a bit sorry.  I am interested, Russell, in what
advice you can give me on how to solve these problems without
methods that rely heavily on evolution, radioisotpic age
dating, and plate tectonics.

Just for the record, the scientific method isnt about proving
anything beyond a shadow of a doubt.  Its just a tool to get
at knowledge.  Tools arent perfect, or we wouldnt have so
many threads about C compilers.  But if it get you there, its
hard not to like.  And we can get emotionally attached to our
tools, I have a lucky screwdriver that I use on pic projects.
But so far I haven't mistaken it for actually having a clue
what I'm doing.

alice

{Quote hidden}

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----- Original Message -----
From: "Alice Campbell" <acampbellSCSENGINEERS.COM>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Wednesday, June 06, 2001 5:15 PM
Subject: Re: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

{Quote hidden}

I KNOW that dowsing works, but I am still waiting on a good "scientific"
explanation for it.  The problem is that "allot" of scientists would simply
refute it and not even investigate.  I worked, for a while, with a real live
astrophysicist from Denmark (who was also a programmer)  Really smart and
nice guy.  Extremely set in his ways and belief system.  We used to have
really, really fun debates about things like this.  BTW he was the guy that
explained the "statistical evidence" about the nonexistance of life in the
rest of the universe to me.  Valid math, flawed hypothesys.  The most
intriguing part of the statistical evidence is that it involved only a
portion of the universe, because the rest of it was too far away to matter.
Also the assumption seemed to be that any life in this part of the universe
would be sufficiently advanced that they (the aliens) would have already
been here (earth).  I was fascinated by this given the FACT that we (an
advanced form of life) are not capable of reaching even the closest star
(besides the sun of course) much less exploring the portion of the universe
in question.  He could also identify the (precise?) instant after the big
bang where the laws of physics kicked in and began to apply.  The funny
thing to me is that he didn't seem to be bothered with the question of why
they (physics,quantum mechanics) weren't in effect at the start of the big
bang.

{Quote hidden}

ROTFL - excelent point on the screwdriver

{Quote hidden}

now
> > being proven self evident fact that is now elevated above the state of
> > theory.
> > While there, look for "proof" of this theory that would satisfy the
> > scientific method. (An unfair test as the scientific method seeks only
to
{Quote hidden}

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>> This includes rational numbers (like 0.(9) and
>> 1). This means that 1 != 0.(9) by definition.
>So 0d10 != 0xA != 0o12 ?

0.(9) and 1 are expressed in the same base.

Peter

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> You need to build a house on a solid foundation, unfortunately
> mathematics has failed us in this respect.

Math is mostly the abstract science (or game) of manipulating certain
abstract symbols in certain abstract ways according to sets of
well-defined abstact rules. I don't think that it has 'failed' anybody. I
think that it does what it is advertised to do most of the time and the
'journey is the reward'. And it just so happens that some operations on
some corpii have limited practical uses.

Unlike physics where most operations have physical equivalents (or so they
say).

Peter

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By the way, is 0.(9) rational ? I am unable to find a simple fraction that
gives 0.(9). ;-(

Peter

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> From: Peter L. Peres [plpactcom.co.il]
>
> By the way, is 0.(9) rational ? I am unable to find a simple
> fraction that
> gives 0.(9). ;-(
>

0.9 repeating is rational and may be represented by N/N where N is any non-zero
integer.  Remember 0.9 repeating is exactly equal to 1 which is exactly equal to
N/N.

-Mike

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Hi, I'm doing homework for my engineering class,
can anyone tell me if 0.99999 (repeating) is
equal to 1.0 ?? Anyone...??

Hee hee !!! Sorry about that, really couldn't
resist. Sorry sorry sorry. :o)
-Roman

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----- Original Message -----
From: "Peter L. Peres" <plpACTCOM.CO.IL>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Wednesday, June 06, 2001 1:37 PM
Subject: Re: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

> By the way, is 0.(9) rational ? I am unable to find a simple fraction that
> gives 0.(9). ;-(
>
> Peter
Just divide it by 1.  The "real" 1 that is.  This way we can make any number
rational.  Maybe we need to have a new axiom. ;o)

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----- Original Message -----
From: "Roman Black" <fastvidEZY.NET.AU>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Thursday, June 07, 2001 1:31 PM
Subject: Re: [OT]: ENOUGH! (was: "Re: [OT]: Brain Burp Rounding??")

> Hi, I'm doing homework for my engineering class,
> can anyone tell me if 0.99999 (repeating) is
> equal to 1.0 ?? Anyone...??
>
> Hee hee !!! Sorry about that, really couldn't
> resist. Sorry sorry sorry. :o)
> -Roman
Just search the archives after James puts in a new hard drive just for that

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James Newton, PICList Admin #3
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{Original Message removed}
All repeating numbers are rational. There is a relatively simple proof of
this.

Given a number N which ends in a repeating group of 'p' digits:

Then the number (10^p) * N ends with the same repeating block.

Now if we subtract:

(10^p)*N - N

the 'repeating blocks' cancel out and we have a terminating decimal (the
part of N before the repeating block). Note, of course that this terminating
decimal is rational.

Thus, N(10^p-1) == N(10^p-1) is rational. If we now divide this by (10^p-1)
we have:

N(10^p-1)
N == -------------------
10^p-1

Note that both the numerator and denominator are rational, therefore the
entire value is rational, therefore N is rational.

By the way: this proof is from my 11th grade trigonometry book.

Bob Ammerman
RAm Systems
(contract development of high performance, high function, low-level
software)

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Lets try (1/11) + (10/11)

First fraction is 0.090909(09)r,
Second fraction is 0.909090(90)r
Sum of fractions is 0.999999(99)r

A little re-org on the original expression

(1 + 10)/11 yields 11/11 is 1

Did I miss part of this thread?

Regards,
Dave

Bob Ammerman wrote:
{Quote hidden}

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All,

I have sort of followed this thread for some time now, and there
have been some good arguments both ways concerning this issue.

But, out of all the arguments and all the explanations, I still
have to wonder what all the fuss is about.   Each person on this
list is going to handle the situation as he or she sees fit, and
according to their own rules.  And I would also venture to say that
no amount of discussion is going to sway anyone from their current
way of thinking to the other persons viewpoint.  We're all human,
and if we were perfectly honest with ourselves, we each think that
we're right.  That's just human nature and ego.   So, therefore,
why discuss it any further?  I mean it just seems to me that this
is a fruitless pursuit.  You can discuss it till the snow comes in
June, and all you will have done was used up a lot of time.  The
basic issue will still be there, more or less evenly divided between
the two camps.

I'm not trying to dictate what you should and shouldn't do.  Just
pointing out the fact that I doubt that anyones mind will be changed
drastically from what they think and believe right now about how to
handle math problems involving this issue.

I know my mind hasn't been changed.  I would still handle these
problems the way I was taught, and have handled them for years.

Anyway, that's my take on the situation.   You may continue (or not).
I'm done ranting now.

Regards,

Jim

On Fri, 08 June 2001, David Minkler wrote:

{Quote hidden}

jimjpes.com

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Ok, if 0.(9) == 1 then 1.(9) == 2 ...., also -0.(9) = -1 etc ? I think
that this makes a whole class of numbers. This is too important to skip.
Why did I not learn this ? Hmm.

Peter

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>  I have sort of followed this thread for some time now, and there
>  have been some good arguments both ways concerning this issue.
>
>  But, out of all the arguments and all the explanations, I still
>  have to wonder what all the fuss is about.   Each person on this
>  list is going to handle the situation as he or she sees fit, and
>  according to their own rules.  And I would also venture to say that
>  no amount of discussion is going to sway anyone from their current
>  way of thinking to the other persons viewpoint.

This is Maths (or Math :-) )

There is ONE correct answer .
There are an infinite number repeating of wrong answers.
If the people without the right answer cannot be swayed by correct logic to
accept the right answer then they are in trouble. If people have been taught
wrongly then they need to be able to see that this is so, or else.

regards,

Russell McMahon

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> Ok, if 0.(9) == 1 then 1.(9) == 2 ...., also -0.(9) = -1 etc ? I think
> that this makes a whole class of numbers. This is too important to skip.

Ah.
0.999(9) + 0.999(9)
= 1.999(9)8

with the 8 occurring at infinity.

Ans 6 x 0.999(9)
= 5.999(9)4
with the 4 at infinity - arguably noy equal to 6.
I know I am doing violence to the proper use of handling repeating numbers
but the apparently intuitive result is interesting.

NB be well aware that 0.(9) = 42/42 !!! (to any base (except, perhaps,
zero)).

Russell "all your base are belong to us" McMahon

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> 0.999(9) + 0.999(9)
> = 1.999(9)8

Where did you get the 8 from? There is a 9 after EVERY 9, so each 9 + 9 gets
a carry from the two 9's after them. To say that there is a last 9 (at
inifinity) is denying the 'unendiness'. If such a reasoning (an identifiable
'last 9') were accepted al the beautifull reasonings about infinities (e.g.
Cantors diagonal argument) would be invalid.

I guess most of you know the funny aspects of unendiness (of the first kind,
aleph-0):

A hotel has infinite rooms, all of which are occupied. One more guest
arrives, and he gets room 0. The guest in room 0 moves to room 1, 1 moves to
2, etcettera. Now all the old guests + the one new guest have a room.
[ aleph0 + 1 = aleph0 ]
[ aleph0 + n = aleph0 for each n in N ]

Two hotels A and B, each with infinite rooms, all occupied, merge. Hotel A
is sold. The guest from rooms A(n) (for all n) move to the rooms B(2*n). The
guest in rooms Bn move to room B(2*n+1). All guest from both hotels now have
a room in hotel B.
[ 2 * aleph0 = aleph0 ]
[ n * aleph0 = aleph0 for each n in N ]

As an excercise for the reader: show that infinite hotels, each with
infinite rooms, can merge without loss of capacity. Note that it is not
enough to repeat the previous argument ad infinitum, because aleph0 is not a
member of N.
[ aleph0 * aleph0 = aleph0 ]

But there are limits. Cantors' diagonal argument shows that the number of
rational numbers is larger than aleph-0.

I really loved this subject way back at university. Sadly they also teached
me a lot of less interesting subjects, like algebra....

Wouter

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Russell,

I believe you are correct in stating that there is only ONE CORRECT answer.
And you are correct in stating that there are a number of repeating WRONG
answers.   But that still brings me back to my original question of what is
the
fuss all about.   Intheory, there is only one correct answer, but in
practice, how
many time do you need an absolute answer to the particular question?  Once?
Twice?  Fifty?  My whole point I guess is that an absolute answer is not
needed
in most situations.  An approximation is generally sufficient.  And if you
did
sway everyone to the one correct way, what exactly would you have
accomplished?   That everyone would be guaranteed to be correct, if and when
the sisuation were to present itself in each of thos persons lives, whereby
an
absolute answewr to this question was needed.  It seems to me that you (and
others) are trying to teach something to someone that either doesn't want to
be
taught that particular lesson, or doesn't care that muxh about it because
the
opportunities to use it in lide are few and far between.

But not that this is my opinion only.  I do not and cannot speak for others.
This is just the impression I get from reading the thread.   Maybe I'm
missing the
point altogether, but it seems to me you are wasting your time.   However,
it is
your time.   Therefore, you are free to pursue whatever endeavors you
choose,
no matter what my or others opinions are.

Have fun abd good luck.

Regards,

Jim

{Original Message removed}
Russell McMahon wrote:

> >  according to their own rules.  And I would also venture to say that
> >  no amount of discussion is going to sway anyone from their current
> >  way of thinking to the other persons viewpoint.
>
> This is Maths (or Math :-) )
>
> There is ONE correct answer .
> There are an infinite number repeating of wrong answers.
> If the people without the right answer cannot be swayed by correct logic to
> accept the right answer then they are in trouble.

But you're forgetting one thing, the "right" answers
are traditonally found by those pioneers who DARED
to criticse the "correct" viewpoint and push the
science to a NEW correct viewpoint.

If everone blindly accepted all the "truths" then
we are stuck with the current technology. I know my
math training may be less than many people here,
but what I have been seeing is the more highly
trained people are the ones too quick to say
"well that is right whether you like it or not".

History has shown us that many of the advances in
science (and maths) have been made by the people
willing to challenge the fashionable "correct"
way of looking at it.

Personally I would rather see a thousand smart
people arguing about 0.9999 (repeating) than
100 smart people patting each other on the back
because they can prove the current correct answer
is correct. Prove that the current answer is correct
and you have achieved nothing, prove it is wrong
and you are great. :o)
-Roman

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But but but but... :)

The statement in question is incontrovertible, unless
one wishes to demonstrate that he does not understand
the nature of mathematical proof. There is nothing to
debate - It's a matter of definition.

Oh well, I give up. (I know - It's about time...)

best regards anyway,

Jack

Roman Black wrote:

> Personally I would rather see a thousand smart
> people arguing about 0.9999 (repeating) than
> 100 smart people patting each other on the back
> because they can prove the current correct answer
> is correct. Prove that the current answer is correct
> and you have achieved nothing, prove it is wrong
> and you are great. :o)

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> But you're forgetting one thing, the "right" answers
> are traditonally found by those pioneers who DARED
> to criticse the "correct" viewpoint and push the
> science to a NEW correct viewpoint.

Imho, those were the ones who found the right *questions*. The answers
were usually found much later by someone else.

Peter

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> Personally I would rather see a thousand smart
> people arguing about 0.9999 (repeating) than
> 100 smart people patting each other on the back

Shouldn't that be 99.999... smart people?

dak

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