# Tirade (offtopic) "It's been done"

Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

So 9*x = 9 => x = 1

Also, I found the link to the video speech by William McDonough.
http://www.npr.org/programs/npc/020424.wmcdonough.html.
but I haven’t read the book. It was also reviewed a few days ago at
4

cheers,

Kris

“Robert Krten” <nospam88@parse.com> wrote in message

Sean Boudreau <> seanb@qnx.com> > wrote:

Or Antoine de Saint Exupery.

Or, as discussed at the Ottawa QNX Users’ Group > > we decided this
was somewhat akin to Einstein’s famous quote (paraphrased) “You should
make things as simple as possible, but no simpler” >

Cheers,
-RK

Chris Wiebe <> cwiebe@qnx.com> > wrote:
: Nah, he’s just ripping off Michelangelo… >

## :> Geez Kris, I’ve never heard something so ‘elegant’ come out of your mouth. :> In fact, I still haven’t. But your fingers–wow–they’ve got something to :> say to the world! \

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at > www.parse.com> .
Email my initials at parse dot com.

Kris Warkentin <kewarken@qnx.com> wrote:

Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

So 9*x = 9 => x = 1

Also, I found the link to the video speech by William McDonough.
http://www.npr.org/programs/npc/020424.wmcdonough.html> .
but I haven’t read the book. It was also reviewed a few days ago at
4

Kewl! Thanks for the elegant proof and book ref.

Cheers,
-RK

cheers,

Kris

“Robert Krten” <> nospam88@parse.com> > wrote in message
Sean Boudreau <> seanb@qnx.com> > wrote:

Or Antoine de Saint Exupery.

Or, as discussed at the Ottawa QNX Users’ Group > > we decided this
was somewhat akin to Einstein’s famous quote (paraphrased) “You should
make things as simple as possible, but no simpler” >

Cheers,
-RK

Chris Wiebe <> cwiebe@qnx.com> > wrote:
: Nah, he’s just ripping off Michelangelo… >

## :> Geez Kris, I’ve never heard something so ‘elegant’ come out of your mouth. :> In fact, I still haven’t. But your fingers–wow–they’ve got something to :> say to the world! \

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at > www.parse.com> .
Email my initials at parse dot com.

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at www.parse.com.
Email my initials at parse dot com.

“Robert Krten” <nospam88@parse.com> wrote in message

Sean Boudreau <> seanb@qnx.com> > wrote:

Or Antoine de Saint Exupery.

Or, as discussed at the Ottawa QNX Users’ Group > > we decided this
was somewhat akin to Einstein’s famous quote (paraphrased) “You should
make things as simple as possible, but no simpler” >

In some ways it’s related to Occam’s Razor: always choose the simplest
explanation that fits the facts. Or, to paraphrase, choose the simplest
implementation that solves the problem.

Kris

Cheers,
-RK

Chris Wiebe <> cwiebe@qnx.com> > wrote:
: Nah, he’s just ripping off Michelangelo… >

## :> Geez Kris, I’ve never heard something so ‘elegant’ come out of your mouth. :> In fact, I still haven’t. But your fingers–wow–they’ve got something to :> say to the world! \

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at > www.parse.com> .
Email my initials at parse dot com.

“Kris Warkentin” <kewarken@qnx.com> wrote in message

Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

It’s not a real proof because that’s not entirely true:
9.999… - .9999… != 9

9.99999…

• 0.99999…

= ~9 (but not exactly, because you have to round at some point instead of
checking the next value which is 9, not quite 0). Sorry to disappoint.

Jerry

Jerry,

I’m shocked and saddened that you would show such a poor understanding of
infinity.

9.9999… - 0.9999 is NOT approximately 1. It is precisely 1. Consider.
For each 9 to the right of the decimal in the first term, is there a 9 in
the second term to correspond to it? So at what point do they diverge?

cheers,

Kris

“Jerry Chappell” <jchappell@cyberus.ca> wrote in message

“Kris Warkentin” <> kewarken@qnx.com> > wrote in message
Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

It’s not a real proof because that’s not entirely true:
9.999… - .9999… != 9

9.99999…

• 0.99999…

= ~9 (but not exactly, because you have to round at some point instead of
checking the next value which is 9, not quite 0). Sorry to disappoint.

Jerry

If I remember my freshman calculus correctly, the fallacy in your
proof lies in assuming that 0.999… exists and is a real number.

I think a sufficient proof would be to state that since each
term of the sequence {0.9, 0.99, 0.999, …} is obviously less than
or equal to 1, and since the sequence is monotonic, therefore
it must converge to 1.

Kris Warkentin <kewarken@qnx.com> wrote:

Jerry,

I’m shocked and saddened that you would show such a poor understanding of
infinity. >

9.9999… - 0.9999 is NOT approximately 1. It is precisely 1. Consider.
For each 9 to the right of the decimal in the first term, is there a 9 in
the second term to correspond to it? So at what point do they diverge?

cheers,

Kris

“Jerry Chappell” <> jchappell@cyberus.ca> > wrote in message

“Kris Warkentin” <> kewarken@qnx.com> > wrote in message
Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

It’s not a real proof because that’s not entirely true:
9.999… - .9999… != 9

9.99999…

• 0.99999…

= ~9 (but not exactly, because you have to round at some point instead of
checking the next value which is 9, not quite 0). Sorry to disappoint.

Jerry

That is very intrestring. In Russia the first part will be opposite. It
sounds like:

“Keep quite idiot, people will think you are smart!”

-Misha.

The following is in original language:
“??? ???, ?? ??? ???”
I hope I didn’t make to many mistakes in it ;-(.

“Kris Warkentin” <kewarken@qnx.com> wrote in message

Yeah. I knew I had heard it somewhere before. I probably should have
said,
“Someone once said…” but it ruined the flow of the prose.

Perhaps now I should make another quote:

"It is better to keep your mouth closed and let people think you are a
fool
than to open it and remove all doubt. "
Mark Twain (1835 - 1910)

cheers,

Kris

“Sean Boudreau” <> seanb@qnx.com> > wrote in message

Or Antoine de Saint Exupery.

Chris Wiebe <> cwiebe@qnx.com> > wrote:
: Nah, he’s just ripping off Michelangelo… >

:> Geez Kris, I’ve never heard something so ‘elegant’ come out of your
mouth.
:> In fact, I still haven’t. But your fingers–wow–they’ve got
something
to
:> say to the world!

\

I don’t think there is necessarily a fallacy in the proof since a fallacy
would imply that the proof is invalid. It’s just that it looks a lot
sillier if you take into account that 0.9999… is equal to 1.
ie.

Set x = 1

Then 10x - x = 9x
But 10x = 10
So 10
x - x = 10 - 1 = 9

So 9*x = 9 => x = 1

So actually, it’s a tautology. I’m good at those.

The problem lies in convincing people that repeating the 9’s to infinity
does not result in something that is APPROXIMATELY 1 but rather PRECISELY 1.
The third step is handy because subtraction is something people understand
readily. The obvious argument (which Jerry made) is, “Well, it must be
rounded somewhere” but this is easily refuted by showing that each 9 in the
first term has a corresponding 9 in the second. You are correct that your
proof is sufficient but, when someone doesn’t understand, your proof is no
more informative than just saying, “0.99999… = 1 because the nines go
forever”.

from a circle with infinite diameter” Infinity can be tough to wrap your

Kris

“Angela Lin” <alin@qnx.com> wrote in message

If I remember my freshman calculus correctly, the fallacy in your
proof lies in assuming that 0.999… exists and is a real number.

I think a sufficient proof would be to state that since each
term of the sequence {0.9, 0.99, 0.999, …} is obviously less than
or equal to 1, and since the sequence is monotonic, therefore
it must converge to 1.

Kris Warkentin <> kewarken@qnx.com> > wrote:
Jerry,

I’m shocked and saddened that you would show such a poor understanding
of
infinity. >

9.9999… - 0.9999 is NOT approximately 1. It is precisely 1.
Consider.
For each 9 to the right of the decimal in the first term, is there a 9
in
the second term to correspond to it? So at what point do they diverge?

cheers,

Kris

“Jerry Chappell” <> jchappell@cyberus.ca> > wrote in message

“Kris Warkentin” <> kewarken@qnx.com> > wrote in message
Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

It’s not a real proof because that’s not entirely true:
9.999… - .9999… != 9

9.99999…

• 0.99999…

= ~9 (but not exactly, because you have to round at some point instead
of
checking the next value which is 9, not quite 0). Sorry to disappoint.

Jerry

Well, if people don’t understand my proof, it’s because I left out the
chapters of the textbook where we proved the theorems whose names I can’t
remember anymore (Cauchy was probably involved) and constructed the
real numbers.

My proof sketch basically says that, as you add 9’s, the difference
between 1 and 0.999… becomes infinitesimally small. Then, according
to the definition of the real numbers, two real numbers are the same
if the difference between them is infinitesimally small.

If you look at its construction, a real number isn’t really an exact
quantity, it’s the limit of an approximation.

I think you’re treading on dangerous territory when you treat limits
like a number. You can’t subtract 0.999… from anything if it doesn’t
exist. Just because we can represent it doesn’t mean it exists.
“A circle of infinite diameter” doesn’t exist. A line isn’t a circle,
but it can be approximated by really big circles.

Anyway, I’ve probably just approached the limit of what can be
considered “interesting.”

Kris Warkentin <kewarken@qnx.com> wrote:

The problem lies in convincing people that repeating the 9’s to infinity
does not result in something that is APPROXIMATELY 1 but rather PRECISELY 1.
The third step is handy because subtraction is something people understand
readily. The obvious argument (which Jerry made) is, “Well, it must be
rounded somewhere” but this is easily refuted by showing that each 9 in the
first term has a corresponding 9 in the second. You are correct that your
proof is sufficient but, when someone doesn’t understand, your proof is no
more informative than just saying, “0.99999… = 1 because the nines go
forever”.

from a circle with infinite diameter” Infinity can be tough to wrap your

Kris

“Angela Lin” <> alin@qnx.com> > wrote in message
If I remember my freshman calculus correctly, the fallacy in your
proof lies in assuming that 0.999… exists and is a real number.

I think a sufficient proof would be to state that since each
term of the sequence {0.9, 0.99, 0.999, …} is obviously less than
or equal to 1, and since the sequence is monotonic, therefore
it must converge to 1.

Kris Warkentin <> kewarken@qnx.com> > wrote:
Jerry,

I’m shocked and saddened that you would show such a poor understanding
of
infinity. >

9.9999… - 0.9999 is NOT approximately 1. It is precisely 1.
Consider.
For each 9 to the right of the decimal in the first term, is there a 9
in
the second term to correspond to it? So at what point do they diverge?

cheers,

Kris

“Jerry Chappell” <> jchappell@cyberus.ca> > wrote in message

“Kris Warkentin” <> kewarken@qnx.com> > wrote in message
Coincidentally, our discussion about how to prove 0.99999… == 1 was

Set x = 0.99999999…

Then 10x - x = 9x
But 10x = 9.9999999…
So 10
x - x = 9.999… - .999… = 9

It’s not a real proof because that’s not entirely true:
9.999… - .9999… != 9

9.99999…

• 0.99999…

= ~9 (but not exactly, because you have to round at some point instead
of
checking the next value which is 9, not quite 0). Sorry to disappoint.

Jerry

I think my point has been proven… thanks guys.
“Doug Rixmann” <rixmannd@rdsdata.com> wrote in message

Great insight Robert.

I think there’s another result of “It’s Been Done”:

As a relative newcomer to the software industry, my feeling is that
because
“it’s been done”, developers generally try to find new ways to do the same
thing making very simple code very complex. I’m a believer in the
philosophy
of KISS (Keep It Simple Stupid) or as our politically correct Cub Scout
leader in our company says KISMIF (Keep It Simple, Make It Fun).

We all want to be innovators but sometimes simple is the way to go.

Doug

“Robert Krten” <> nospam88@parse.com> > wrote in message

Remember the Simpson’s episode with Homer on top of Moe’s bar with his
band?
Limo pulls up, George Harrison pokes his head out and says, “It’s been
done”.

Well, that’s kind of the way I feel about the current state of affairs
in
the software field.

Remember when an “operating system” was something to fear and hold in
awe?
How did those guys do it? What made it work? How did their memory
system
work? Nowadays, it’s “been done”. POSIX says “this is the set of
functions
you will have”. QSSL, to their credit, has done an excellent job in
implementing
the POSIX specs, leaving very little to the imagination. Sure, QNX 4
and
Neutrino
are really “clean” implementations of operating systems, taking tons of
talent
and so on to do, but… “it’s been done”.

The separation between filesystem, TCP/IP stack, device drivers, and
core
OS
is again so well done that it leaves little to “wonder” at.

Remember compilers? This was something that was almost “godlike” in its
ability to “understand” C code and generate low level machine code.

There are tons of other examples.

The database guys haven’t progressed – a database is still not some
kind
of
wonderful, futuristic AI-like replacement for memory, it’s just a set of
indexes
into files. Wheee… “it’s been done”.

The point of this tirade (and I did warn you-all it would be one > > )
is that I’m really kinda bored – what’s going to be
the “next” exciting thing in the software field that you can hold in
awe
and terror (like OS’s and compilers used to be)?

About the only thing that jumps out at me is the field of AI – and this
is
because of two things: my own general ignorance of the state-of-the-art
in
the field, and my perception that it’s all just the same old crap – do
they
still use LISP? Are they any closer to making a truly artificial
intelligence?
I don’t need something that looks and acts like a human – if it is
truly
“artificial” it could have its own way of looking at the world and
interacting
with it – that’s fine. It doesn’t need to love, or express emotion, or
be
able to compose music (necessarily). It just needs to be able to do
something
that shows its intelligence.

Anyway, I’m done. Anyone have any ideas on what would be “fun”? (apart
from
restoring old computers, of course > > )

## Cheers, -RK

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at > www.parse.com> .
Email my initials at parse dot com.

“Bored now.”
-Willow

“Kris Warkentin” <kewarken@qnx.com> wrote in message

“Robert Krten” <> nospam88@parse.com> > wrote in message
Sean Boudreau <> seanb@qnx.com> > wrote:

Or Antoine de Saint Exupery.

Or, as discussed at the Ottawa QNX Users’ Group > > we decided this
was somewhat akin to Einstein’s famous quote (paraphrased) “You should
make things as simple as possible, but no simpler” >

In some ways it’s related to Occam’s Razor: always choose the simplest
explanation that fits the facts. Or, to paraphrase, choose the simplest
implementation that solves the problem.

That makes me thing of something I heard from a great drummer. He
said one needs to become an expert in order to master simplicity.

What I understood from that is simplicity becomes elegant when it’s
made be choice instead of by lack of knowledge. Hence in the case
of drumming, many drummers will only do bass drum kick every beat for
the whole duration of the song not because it fits the song best, but
because it’s all they can do.

Kris

Cheers,
-RK

Chris Wiebe <> cwiebe@qnx.com> > wrote:
: Nah, he’s just ripping off Michelangelo… >

## :> Geez Kris, I’ve never heard something so ‘elegant’ come out of your mouth. :> In fact, I still haven’t. But your fingers–wow–they’ve got something to :> say to the world! \

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at > www.parse.com> .
Email my initials at parse dot com.

“Mario Charest” postmaster@l127.0.0.1 wrote in message

That makes me thing of something I heard from a great drummer. He
said one needs to become an expert in order to master simplicity.

What I understood from that is simplicity becomes elegant when it’s
made be choice instead of by lack of knowledge. Hence in the case
of drumming, many drummers will only do bass drum kick every beat for
the whole duration of the song not because it fits the song best, but
because it’s all they can do.

I don’t know if it’s the same drummer but Bill Bruford once said, “The art
is to conceal the art”. This was a guy who could tap 4/4 with one hand, 6/8
with another, 13/8 with his left foot and a syncopated 3/4 swing with his
right. (I don’t remember the details exactly but I saw him do something
like that on a talk show and needed to pick my jaw up off the floor).

Kris

Kris Warkentin <kewarken@qnx.com> wrote:

“Bored now.”
-Willow

You forgot to rip off the flesh of the poster and set them on fire.

Cheers,
-RK

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at www.parse.com.
Email my initials at parse dot com.

That’s funny that you mention this book … I’ve had “A New Kind of Science”
on backorder since well over a year ago, for Kim. It came in a few weeks
ago. Very interesting read, but a bit bias on impressive nature of cellular
automata and chaotic systems. Kim seemed happy with it … she only had to
wait for 15 months for her present to arrive (!) Never believe Amazon.com
when it says a new book is now available. Check to make sure it’s actually
been written, first.

-bill

“Kris Warkentin” <kewarken@qnx.com> wrote in message

If I may recommend a book that is an incredibly readable overview of some
of
the leading edges in computer science research, “The Computation Beauty of
Nature” by Gary William Flake is one of my favorites.

Another interesting thing is Steven Wolfram’s new book, “A New Kind of
Science”. It’s a 1200 page stump composed of 20 years of his research (he
got a PhD from MIT at the age of 20, got the MacArthur Foundation genius
award, wrote Mathematica…) which is basically theorizing that all of
nature and the universe might be reducable to cellular automata. That is,
there is no grand, unifying, horrendously complicated equation that
returns
‘42’ but rather a beautiful interaction by many types of simple automata.
(note - I haven’t read this one >

I know this is not related to software directly but I like the fact that
computers are opening up the wonders of the universe for us.

Kris

“Robert Krten” <> nospam88@parse.com> > wrote in message

Remember the Simpson’s episode with Homer on top of Moe’s bar with his
band?
Limo pulls up, George Harrison pokes his head out and says, “It’s been
done”.

Well, that’s kind of the way I feel about the current state of affairs
in
the software field.

Remember when an “operating system” was something to fear and hold in
awe?
How did those guys do it? What made it work? How did their memory
system
work? Nowadays, it’s “been done”. POSIX says “this is the set of
functions
you will have”. QSSL, to their credit, has done an excellent job in
implementing
the POSIX specs, leaving very little to the imagination. Sure, QNX 4
and
Neutrino
are really “clean” implementations of operating systems, taking tons of
talent
and so on to do, but… “it’s been done”.

The separation between filesystem, TCP/IP stack, device drivers, and
core
OS
is again so well done that it leaves little to “wonder” at.

Remember compilers? This was something that was almost “godlike” in its
ability to “understand” C code and generate low level machine code.

There are tons of other examples.

The database guys haven’t progressed – a database is still not some
kind
of
wonderful, futuristic AI-like replacement for memory, it’s just a set of
indexes
into files. Wheee… “it’s been done”.

The point of this tirade (and I did warn you-all it would be one > > )
is that I’m really kinda bored – what’s going to be
the “next” exciting thing in the software field that you can hold in
awe
and terror (like OS’s and compilers used to be)?

About the only thing that jumps out at me is the field of AI – and this
is
because of two things: my own general ignorance of the state-of-the-art
in
the field, and my perception that it’s all just the same old crap – do
they
still use LISP? Are they any closer to making a truly artificial
intelligence?
I don’t need something that looks and acts like a human – if it is
truly
“artificial” it could have its own way of looking at the world and
interacting
with it – that’s fine. It doesn’t need to love, or express emotion, or
be
able to compose music (necessarily). It just needs to be able to do
something
that shows its intelligence.

Anyway, I’m done. Anyone have any ideas on what would be “fun”? (apart
from
restoring old computers, of course > > )

## Cheers, -RK

Robert Krten, PARSE Software Devices +1 613 599 8316.
Realtime Systems Architecture, Books, Video-based and Instructor-led
Training and Consulting at > www.parse.com> .
Email my initials at parse dot com.

On Thu, 6 Jun 2002 11:24:09 -0400, “Kris Warkentin” <kewarken@qnx.com>
wrote:

There’s a bunch of things you can aim for in your code - readability,
efficiency (time and space), reliability, maintainability and the ever
elusive ‘elegance’.

I think elegance is one of the coolest concepts in programming, the closest
to art. It’s not even simple to define. I heard it once defined as, “the
shortest length of code to accomplish the desired task”. By that standard
though, some pretty obfuscated perl code would be considered elegant. Maybe
it’s like the old saw, “I don’t know art but I know what I like.” Quicksort
is elegant. Dijkstra’s algorithm is elegant. I think microkernel
architectures are elegant. Does the beauty lie in simplicity of form and
utility of function? Perhaps our job is complete not when there’s nothing
more to add but when there’s nothing left to take away.

Kris

In this exchange of quotations let me recall another one - on
not so elusive elegance (from EWD709):

“Later I learned that for the kind of effectiveness that I
call it ‘mathematical elegance’, or ‘elegance’ for short. I also
discovered that the term is much more ‘technical’ that most
mathematicians suspect, much more in the sense that even among
mathematicians of very different brands there exist a much greater
consensus about what is a really elegant argument than they themselves
seemed to be aware of. Show any mathematician a really elegant
argument that is new to him: at the moment it becomes his intellectual
property, he starts to laugh!”

ako

“Angela Lin” <alin@qnx.com> wrote in message

Well, if people don’t understand my proof, it’s because I left out the
chapters of the textbook where we proved the theorems whose names I can’t
remember anymore (Cauchy was probably involved) and constructed the
real numbers. >

My proof sketch basically says that, as you add 9’s, the difference
between 1 and 0.999… becomes infinitesimally small. Then, according
to the definition of the real numbers, two real numbers are the same
if the difference between them is infinitesimally small.

If you look at its construction, a real number isn’t really an exact
quantity, it’s the limit of an approximation.

I think you’re treading on dangerous territory when you treat limits
like a number. You can’t subtract 0.999… from anything if it doesn’t
exist. Just because we can represent it doesn’t mean it exists.
“A circle of infinite diameter” doesn’t exist. A line isn’t a circle,
but it can be approximated by really big circles.

…on the other hand, if we take for example a light ray, we can find a
really big circle due to the curvature of basically linear object… “A
circle isn’t a line, but it can be approximated by really big lines”

Anyway, I’ve probably just approached the limit of what can be
considered “interesting.” >

// wbr

“Angela Lin” <alin@qnx.com> wrote in message

I think you’re treading on dangerous territory when you treat limits
like a number. You can’t subtract 0.999… from anything if it doesn’t
exist. Just because we can represent it doesn’t mean it exists.

I think I disagree with your assertion that 0.9999… doesn’t exist. It’s
just a different way of writing ‘1’. All the ‘proof’ did was illustrate
that equivalence of the two values by substituting that value for 1 and
showing how it drops out in the end.

“A circle of infinite diameter” doesn’t exist. A line isn’t a circle,
but it can be approximated by really big circles.

Something I thought of this morning - a circle of infinite diameter exists,
just not in reality. These are all mathematical concepts, just like in
calculus when you take the limit as some value approaches zero (or infinity
as in the case of the many nines). So, a segment of a circle is still a
straight line if you let the diameter approach infinity. If you had a
formula which described the curvature of a line taken from a circle, I would
bet that the diameter would exist somewhere in the denominator causing the
curvature to approach zero as the diameter approached infinity. Either way,
it’s not an approximation. If we were still in school, some professor would
probably make us come up with a proof of this…good thing we don’t have to
do those type of things anymore.

I guess I lied…I wasn’t bored with the subject yet. It was just Willow
who was bored.

Kris

Kris Warkentin <kewarken@qnx.com> wrote in article <adqea2\$a8h\$1@nntp.qnx.com>…

from a circle with infinite diameter” Infinity can be tough to wrap your

## Also good definition from my school: The parallel lines are the lines which are crossed at infinitly far point

Eduard.
ed1k at ukr dot net

Misha Nefedov <mnefedov@qnx.com> wrote in article <adqcfi\$4hm\$1@inn.qnx.com>…

That is very intrestring. In Russia the first part will be opposite. It
sounds like:

“Keep quite idiot, people will think you are smart!”

-Misha.

The following is in original language:
“??? ???, ?? ??? ???”

Misha,
Out of curiosity… What is those ???

## Never wrote russians letters here, just experiment: “ðÏ ËÏÍÁÎÄÅ shutdown ÎÁÓÔÕÐÉÌÏ ÕÔÒÏ…”

Eduard.
ed1k at ukr dot net