Cool Uses of Mathematics in SF
rsn...@swbellnospam.net
a battle between two powerful catapults (trebuchets, if memory serves).
The antagonist makes his ballistic calculations using Roman notation
while the hero uses Arabic notation. Guess who is victorious.
Does anyone else have examples of SF gettin' jiggy wit' mathematics?
Nathan Raye
--
'Angkor Wat in Cambodia stopped being a viable alternative for a family
vacation at about the time I was in the general area riding a tank.'
- David Drake
Mark Atwood
>
> Does anyone else have examples of SF gettin' jiggy wit' mathematics?
"Luminous" by Egan, for a weird take on SF and math.
I learned kinematics when I was 13 from reading _Have Spacesuit_
--
Mark Atwood | I'm wearing black only until I find something darker.
m...@pobox.com | http://www.pobox.com/~mra
Coyu
>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>a battle between two powerful catapults (trebuchets, if memory serves).
>The antagonist makes his ballistic calculations using Roman notation
>while the hero uses Arabic notation. Guess who is victorious.
>
>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
There's a scene in Gene Wolfe's _Soldier of Arete_ where troops
calculate the height of a city wall by measuring shadows and
using the law of similar triangles -- at the time the story is set,
cutting-edge mathematics. A very nice touch.
Matt Ruff
>
> My favorite scene in Harry Harrison's novel _King and
> Emperor_ involves a battle between two powerful catapults
> (trebuchets, if memory serves). The antagonist makes his
> ballistic calculations using Roman notation while the hero
> uses Arabic notation.
Wouldn't a real Roman use an abacus to make math calculations?
> Guess who is victorious.
The Stainless Steel Rat?
> Does anyone else have examples of SF gettin' jiggy wit'
> mathematics?
Neal Stephenson's "Snow Crash" and "Cryptonomicon" both have math
interludes. And Heinlein's "Time Enough For Love" has a math trick you
can use to keep West Point upperclassman happy.
-- M. Ruff
J.B. Moreno
> My favorite scene in Harry Harrison's novel _King and Emperor_ involves
> a battle between two powerful catapults (trebuchets, if memory serves).
> The antagonist makes his ballistic calculations using Roman notation
> while the hero uses Arabic notation. Guess who is victorious.
>
> Does anyone else have examples of SF gettin' jiggy wit' mathematics?
Heh. Cool idea.
--
JBM
"Your depression will be added to my own" -- Marvin of Borg
kesi...@math.ttu.edu
: My favorite scene in Harry Harrison's novel _King and Emperor_ involves
: a battle between two powerful catapults (trebuchets, if memory serves).
: The antagonist makes his ballistic calculations using Roman notation
: while the hero uses Arabic notation. Guess who is victorious.
: Does anyone else have examples of SF gettin' jiggy wit' mathematics?
_Neverness_, by David Zindell. Sort of.
==Jake
Martin Elzen
>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>a battle between two powerful catapults (trebuchets, if memory serves).
>The antagonist makes his ballistic calculations using Roman notation
>while the hero uses Arabic notation. Guess who is victorious.
>
>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
L. Sprague de Camp's "Lest Darkness Fall" has a time traveller
introducing Arabic notation to help with book keeping. Niven wrote a
short story using an infinite series against a demon ... I'm not quite
sure what it was called, but "Limits" seems about right.
Frank van den Eijkhof
>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
A.E. van Voght wrote a story about some alien captured (on Mars?)
long ago, with a lock based on the then-current math.
An scientist from earth was tempted by the aliens to unlock this prison
using now-current math. I must confess that I never checked the story
on correctness...
Frank
Cristiano Sadun
>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>a battle between two powerful catapults (trebuchets, if memory serves).
>The antagonist makes his ballistic calculations using Roman notation
>while the hero uses Arabic notation. Guess who is victorious.
>
>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
I remember reading a very old novel by Someone :) - part of a series,
with a criminal and a police agent following him.. (Deverel and Colby of
something similar.. I read it at least ten years ago).
Well, the duo was captured into a alien parabolic mirror built
on an asteoroid and couldnt escape easily since there was no friction
on the surface. They eventually manage to do it using (if my memory
doenst fail) building a pendulum with themselves.
--
Life's something u don't get out alive..
ObjectZone - http://space.tin.it/computer/csadun
Gerry Quinn
>
>>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
>
>I remember reading a very old novel by Someone :) - part of a series,
>with a criminal and a police agent following him.. (Deverel and Colby of
>something similar.. I read it at least ten years ago).
>
>Well, the duo was captured into a alien parabolic mirror built
>on an asteoroid and couldnt escape easily since there was no friction
>on the surface. They eventually manage to do it using (if my memory
>doenst fail) building a pendulum with themselves.
>
This appeared on rec.puzzles lately, and was attributed to Asimov.
- Gerry Quinn
DaveMoore
>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>a battle between two powerful catapults (trebuchets, if memory serves).
>The antagonist makes his ballistic calculations using Roman notation
>while the hero uses Arabic notation. Guess who is victorious.
>
>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
>
>
>Nathan Raye
Check out _The Mathematical Magpie_, Clifton Fadiman, ed. A collection
of mathematically based short stories, many flat-out SF. Includes
works by Martin Gardner, Heinlein, Asimov, Clarke, and others.
Also see _Fantasia Mathematica._ These volumes have been around
forever, and are still in print.
--
Dave Moore == djm...@uh.edu == I speak for me.
In the wrong hands, sanity is a dangerous weapon.
David Empey
<Xns90EEAEE1EA9...@194.19.1.61>:
"The Men and the Mirror", Ross Rocklynne.
--
Dave Empey
What else could a millennia-spanning, reality-hopping,
transdimensional cult of genetically-perfect,
bloodthirsty superwomen want? --Kenneth Hite
David Empey
>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>a battle between two powerful catapults (trebuchets, if memory serves).
>The antagonist makes his ballistic calculations using Roman notation
>while the hero uses Arabic notation. Guess who is victorious.
Why didn't they use abacuses?
rmtodd
> L. Sprague de Camp's "Lest Darkness Fall" has a time traveller
> introducing Arabic notation to help with book keeping. Niven wrote a
> short story using an infinite series against a demon ... I'm not quite
> sure what it was called, but "Limits" seems about right.
Close. "Convergent Series" was the title.
rmtodd
Wise. As I recall, the math was complete bullshit, but like most van Vogt,
the story manages to be entertaining in spite of the fact that you know the
"science" is thoroughly bogus.
Captain Button
I'm pretty sure the story you mean is "Convergent Series".
--
"Gee, who'd a thunk it? Turns out alien superintelligence is
no match for our Earthly can-do spunk." - Jane Lane, "Daria"
Captain Button - [ but...@io.com ]
Ross Presser
As said elsewhere, the author was Ross Rocklynne, and the story was
called "The Men and the Mirror." It may have been misattributed to
Asimov because it appeared in a collection called _Before The Golden
Age_ (volume 2) edited and extensively commented on by Asimov.
--
Ross Presser * ross_p...@imtek.com
"Back stabbing is a sport best played by those that can't stand face
to face with their opponent." - Danny Taddei
David Tate
>
> Does anyone else have examples of SF gettin' jiggy wit' mathematics?
>
There are a number of excellent examples from the short fiction of
Greg Egan. Perhaps the most mathematically advanced is the story "The
Infinite Assassin", whose plot actually depends on the fact that
Cantor sets are nondenumerable, yet have measure zero. He also uses
some ideas from chaos theory in "Unstable Orbits in the Space of
Lies".
David Tate
Geoffrey A. Landis
>
> There are a number of excellent examples from the short fiction of
> Greg Egan. Perhaps the most mathematically advanced is the story "The
> Infinite Assassin", whose plot actually depends on the fact that
> Cantor sets are nondenumerable, yet have measure zero.
I have to say, that story always bothered me. In the ending, he maps a
set of non-zero measure onto a set of measure zero, which you can only
do if the resulting map has infinite density.
--
Geoffrey A. Landis
http://www.sff.net/people/geoffrey.landis
Jeff Suzuki
: Wouldn't a real Roman use an abacus to make math calculations?
Very likely. And in all probability, be faster than the person
using Arabic numerals.
It wasn't too long ago (like, 1950) that a skilled Japanese abacist
could out-calculate someone using an electric adding machine on a
_routine_ basis. It wasn't until pocket calculators came about in
the 1970s that the abacists began to lose.
Speaking of which: "Into the Comet" by Clarke. The crew uses
abaci to compute a course when their computers fail.
Jeffs
Jeff Suzuki
: L. Sprague de Camp's "Lest Darkness Fall" has a time traveller
: introducing Arabic notation to help with book keeping. Niven wrote a
: short story using an infinite series against a demon ... I'm not quite
: sure what it was called, but "Limits" seems about right.
"Convergent Series"
Jeffs
Jeff Suzuki
: I remember reading a very old novel by Someone :) - part of a series,
: with a criminal and a police agent following him.. (Deverel and Colby of
: something similar.. I read it at least ten years ago).
Ross Rocklynne, "The Men and the Mirror". A wonderful puzzle story.
: Well, the duo was captured into a alien parabolic mirror built
: on an asteoroid and couldnt escape easily since there was no friction
: on the surface. They eventually manage to do it using (if my memory
: doenst fail) building a pendulum with themselves.
Jeffs
Jerry Friedman
> rsn...@swbellnospam.net wrote:
...
> > Does anyone else have examples of SF gettin' jiggy wit'
> > mathematics?
>
> Neal Stephenson's "Snow Crash" and "Cryptonomicon" both have math
> interludes. And Heinlein's "Time Enough For Love" has a math trick you
> can use to keep West Point upperclassman happy.
(Annapolis, not West Point. No doubt RAH is spinning in his grave.)
If you read "The Pragmatics of Patriotism", the second half of a
speech he gave at the Naval Academy, you can see a disappointing
scene, which he records apparently faithfully, revealing that plebes
of the '70s didn't have to meet the same
time-till-graduation-calculation standards that Heinlein's generation
dod.
--
Jerry Friedman
Steve Charlton
writes
the "Cities in Flight" series by James Blish. The whole damn thing (4
books) is based on a so far undiscovered mathematical relationship
between the force of gravity and the spin of an electron. It eventually
allows simple machines (Spindizzies) to propel whole cities to the
stars.
--
Steve Charlton |Travelling in Hyperspace is a bit like
st...@gnirekoms.freeserve.co.uk |being drunk.
hint: ^^^^^^^^^ |What's wrong with that?
reverse this |You ask a glass of water!
- Douglas Adams (sadly missed)
Steve Charlton
ld.std.com> writes
>In article <3b654039...@news.nl.net>,
>Martin Elzen <m.e...@nospam.hotmail.com> wrote:
>>
>>L. Sprague de Camp's "Lest Darkness Fall" has a time traveller
>>introducing Arabic notation to help with book keeping. Niven wrote a
>>short story using an infinite series against a demon ... I'm not quite
>>sure what it was called, but "Limits" seems about right.
>
>collection, which did NOT include "Convergent Series".
"Convergent Series" the short story lent its name to "Convergent Series"
the book; a collection of (non-KS) short stories.
Ross TenEyck
I'm a little dubious. The abacus is an excellent device for
adding and subtracting integers (or numbers of fixed precision,
which as far as the abacus is concerned are integers with a
power of ten applied.)
It's usable for multiplication and division to some fixed
precision; either by doing multiplication as repeated addition
and division as repeated subtraction, or by laying out the
multiplication/division problem on mental paper and using
the abacus to handle the addition of intermediate results.
I even saw instructions on how to take a square root on an
abacus. ("Instructions" is perhaps a bit much; the procedure
amounted to: guess the square root, square your guess, compare
to the original number, adjust your guess accordingly, repeat.)
However, unlike a reasonable calculator, or even a slide rule,
it won't do trig functions, logarithms, roots and powers (except
as noted above), or many other things that one might find useful.
This is why, pre-calculators, you generally saw abaci being used
in shops and similar places, where you needed to add and subtract
lots of numbers quickly. In scientific and engineering circles,
which typically involve more multiplying and dividing, as well as
higher math functions, the instrument of choice was the slide rule.
For ballistics, I'd take the slide rule any day.
Incidentally, now that I think about it, I'm not certain the
Romans had the abacus. For one thing, the abacus strongly implies
a place-value number system, like Arabic numerals but unlike Roman
numerals.
--
================== http://www.alumni.caltech.edu/~teneyck ==================
Ross TenEyck Seattle, WA \ Light, kindled in the furnace of hydrogen;
ten...@alumni.caltech.edu \ like smoke, sunlight carries the hot-metal
Are wa yume? Soretomo maboroshi? \ tang of Creation's forge.
Ross TenEyck
>Matt Ruff <Storyt...@worldnet.att.net> wrote in message news:<3B642E86...@worldnet.att.net>...
>>
>> interludes. And Heinlein's "Time Enough For Love" has a math trick you
>> can use to keep West Point upperclassman happy.
>(Annapolis, not West Point. No doubt RAH is spinning in his grave.)
>If you read "The Pragmatics of Patriotism", the second half of a
>speech he gave at the Naval Academy, you can see a disappointing
>scene, which he records apparently faithfully, revealing that plebes
>of the '70s didn't have to meet the same
>time-till-graduation-calculation standards that Heinlein's generation
>dod.
I always wondered... did the upperclassman typically work out what
the correct answer would be before asking the question, in order
to catch plebes who just recited a random number in the right
approximate range?
GSV Three Minds in a Can
person rmtodd <rmt...@amonduul.ecn.ou.edu> said
(spoiler ahead)
The 'logic' was along the lines of the lock being a time lock coded to
the 'ultimate prime number' which was linked to the 'Eis force' (sp?)
(phooey already!) and the 'solution' was to adjust the Eis force by an
incy-weesy bit (like adding 1) after which the prime number falls into
lots of factors, one of which happens to be 'just now', so the lock
opens.
Complete hokum, but totally on par with AEvG's comprehension of science
in general - see for instance 'The Mixed Men' (wasn't AEvG also into
LRonHubb&ard stuff?).
--
GSV Three Minds in a Can
GSV Three Minds in a Can
wonderful person David Empey <dem...@cruzio.com> said
>rsn...@swbellnospam.net wrote in <3B63C81E...@swbell.net>:
>
>>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>>a battle between two powerful catapults (trebuchets, if memory serves).
>>The antagonist makes his ballistic calculations using Roman notation
>>while the hero uses Arabic notation. Guess who is victorious.
>
>Why didn't they use abacuses?
Did the Romans =have= abacuses/(abaci?)?? I know the Chinese did, from
way back, but I'm not sure that they ever made it to Rome (well, Ancient
Rome .. you know what I mean .. 8>.).
Ian A. York
Ross TenEyck <ten...@alumnae.caltech.edu> wrote:
>Jeff Suzuki <je...@bu.edu> writes:
>>Matt Ruff <Storyt...@worldnet.att.net> wrote:
>
>>: Wouldn't a real Roman use an abacus to make math calculations?
>
>>Very likely. And in all probability, be faster than the person
>>using Arabic numerals.
>
>adding and subtracting integers (or numbers of fixed precision,
>which as far as the abacus is concerned are integers with a
>power of ten applied.)
A guy with an abacus beat Richard Feynman on adding and multiplication,
tied on division, and came up a little short on a cube root (of 1729.03,
if you're interested). ("Surely You're Joking, Mr. Feynman!")
Mind you, Feynman was doing it in his head, but that's still pretty
impressive.
Ian
--
Ian York (iay...@panix.com) <http://www.panix.com/~iayork/>
"-but as he was a York, I am rather inclined to suppose him a
very respectable Man." -Jane Austen, The History of England
Chris Byler
"Convergent Series", iirc. Which is ironic - if you assume that the
time required for the demon to disappear and reappear is proportional
to the distance (he moves at the speed of light, for example), then
the time series _would_ be convergent and the demon would escape the
trap in finite time. Fortunately that doesn't happen in the story.
--
Chris Byler cby...@vt.edu
Kubera: "It occurred to me that Sam would be the number one suspect,
except for the fact that he was dead."
Sam: "I had assumed that to be sufficient defense against detection."
-- Roger Zelazny, _Lord of Light_
David Tate
> David Tate wrote:
> >
> > There are a number of excellent examples from the short fiction of
> > Greg Egan. Perhaps the most mathematically advanced is the story "The
> > Infinite Assassin", whose plot actually depends on the fact that
> > Cantor sets are nondenumerable, yet have measure zero.
>
> I have to say, that story always bothered me. In the ending, he maps a
> set of non-zero measure onto a set of measure zero, which you can only
> do if the resulting map has infinite density.
Well... you can put any set of cardinality aleph-1, like (say) a
bounded closed interval in R(n), into 1-1 correspondence with any
Cantor set extracted from (say) that piece of R(n). That's a 1-1
mapping between a set of nonzero measure and a proper subset of
measure zero. That's what's so weird about it. Yes, the Cantor set
is dense in the parent set, but it still has measure zero -- which was
the point of the climactic act of the story.
Is there something I'm missing here? Are you saying that a dense set
of zero measure would have sufficed for Our Hero's purposes?
David Tate
Richard Horton
wrote:
I believe it's "The Men and the Mirror" (or maybe "The Men in the
Mirror") by Ross Rocklynne. The Asimov attribution may result from
him having anthologized it in _Before the Golden Age_, as my vague
memory tells me he did.
--
Rich Horton | Stable Email: mailto://richard...@sff.net
Home Page: http://www.sff.net/people/richard.horton
Also visit SF Site (http://www.sfsite.com) and Tangent Online (http://www.tangentonline.com)
Timothy A. McDaniel
>...
Infinite *sequence*. "A series is an infinite sum ...". If the demon
had been caught in an infinite series, he would have gotten larger and
larger, though perhaps converging to a finite limit depending on the
exact series.
--
Tim McDaniel is tm...@jump.net; if that fail,
tm...@us.ibm.com is my work account.
"To join the Clueless Club, send a followup to this message quoting everything
up to and including this sig!" -- Jukka....@hut.fi (Jukka Korpela)
Timothy A. McDaniel
Cristiano Sadun <crs...@tin.it> wrote:
>They eventually manage to do it using (if my memory
>doenst fail) building a pendulum with themselves.
Actually, in the story, they pass a cord between them, push off so as
to spin up around the center of the cord, then release the cord at the
apex of their trajectory. One person flies over the edge of the
mirror at that point, and the other person flies off after one trip
across the mirror.
Alas! yet another story ruined by the brutal facts of Newtonian
mechanics -- in this case, conservation of angular momentum. The
"push off so as to spin up" is the "here a miracle occurs" step.
John Thomas
designs a house as a model of an unfolded hypercube. They're in
California. There's an earthquake and the building folds up _through a
fourth spatial dimension_. It looks like a plain cube on the building
lot.
There's another earthquake and the building folds into nothing. Not
rubble, but gone.
----------
There's another story, much more recent and not by Heinlein, where
someone in orbit figures when a jettisoned item will return to the
space station by using epicycles. The story was in Analog. It might be
the same story that had jokes that started "A mathematician, an
physicist, and an engineer..."
JT
David Empey
<PhzcJ5lu...@quik.freeuk.net>:
>Bitstring <90EE63F13dgem...@cnews.newsguy.com>, from the
>wonderful person David Empey <dem...@cruzio.com> said
>>rsn...@swbellnospam.net wrote in <3B63C81E...@swbell.net>:
>>
>>>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>>>a battle between two powerful catapults (trebuchets, if memory serves).
>>>The antagonist makes his ballistic calculations using Roman notation
>>>while the hero uses Arabic notation. Guess who is victorious.
>>
>>Why didn't they use abacuses?
>
>Did the Romans =have= abacuses/(abaci?)??
My memory says yes, and a Google search for abacus Romans turns
up Web pages that agree. Apparently they were counters
sliding in a grooved board.
www.m-w.com says either plural form is correct; they list
abaci first.
>I know the Chinese did, from
>way back, but I'm not sure that they ever made it to Rome (well, Ancient
>Rome .. you know what I mean .. 8>.).
One of these Web pages says the Chinese got it from the Romans!
David Empey
>
>A guy with an abacus beat Richard Feynman on adding and multiplication,
>tied on division, and came up a little short on a cube root (of 1729.03,
>if you're interested). ("Surely You're Joking, Mr. Feynman!")
Are you sure it was 1729.03? I thought it was someting like 27.03.
As I recall, Feynman started to explain to the abacus operator how he
figured out the cube root by saying "The cube root of <largest cube
less than the number in question> is <whatever it was>", at which
point the abacus operator stopped to figure it out on his abacus,
which surprised Feynman, who thought the cube root should have
been obvious. Now I can believe that Feynman would have thought
it obvious that the cube root of 27 is 3, but it seems unlikely
he'd think it obvious that the cube root of 1728 is 12.
>
>Mind you, Feynman was doing it in his head, but that's still pretty
>impressive.
>
>Ian
--
Nancy Lebovitz
>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
>a battle between two powerful catapults (trebuchets, if memory serves).
>The antagonist makes his ballistic calculations using Roman notation
>while the hero uses Arabic notation. Guess who is victorious.
>
>
about sentient (?) numerical sequences in suns.
Just barely sf: Margaret Ball's _Bridge to the Sky_ has a cathedral
architect trying to get a chance to learn enough algebra to make
his work easier.
--
Nancy Lebovitz na...@netaxs.com www.nancybuttons.com
Doug Palmer
<st...@nospam.freeserve.co.uk> wrote:
> I can't believe it! It's been more than 24 hours and nobodies mentioned
> the "Cities in Flight" series by James Blish.
The bit I remember about it is that they meet some aliens and there's a
discussion about the difficulties of adapting to each other's notation.
(From memory, eg., the aliens use a 'D' symbol to indicate that something
is a constant.)
As a side thought, are there any stories about encounters with aliens
that have explored a completely different region of the space of
mathematics to humans? "Sorry, now explain this concept of 'number'
again? It looks like a sort of wierd mapping onto a Fxchil group to me,
but it makes my antennae twist just to think about it."
--
Doug Palmer do...@charvolant.org http://www.charvolant.org/~doug
Ian A. York
David Empey <dem...@cruzio.com> wrote:
>iay...@panix.com (Ian A. York) wrote in <9k4tqc$q2s$1...@news.panix.com>:
>>
>>A guy with an abacus beat Richard Feynman on adding and multiplication,
>>tied on division, and came up a little short on a cube root (of 1729.03,
>>if you're interested). ("Surely You're Joking, Mr. Feynman!")
>
>Are you sure it was 1729.03? I thought it was someting like 27.03.
"He writes down a number on some paper--any old number--and I still
remember it: 1729.03."
>As I recall, Feynman started to explain to the abacus operator how he
>figured out the cube root by saying "The cube root of <largest cube
>less than the number in question> is <whatever it was>", at which
>point the abacus operator stopped to figure it out on his abacus,
>which surprised Feynman, who thought the cube root should have
>been obvious. Now I can believe that Feynman would have thought
>it obvious that the cube root of 27 is 3, but it seems unlikely
>he'd think it obvious that the cube root of 1728 is 12.
I started to explain that it was an approximate method, and had to do
with the percentage of error. "Suppose you had given me 28. Now the
cube root of 27 is 3 ... "
He picks up his abacus: zzzzzzzzzzzzzzzzzzzzz-- "Oh yes, he says."
I think you actually remembered it much better than I did; I just looked
it up. (After spending fifteen minutes looking for the damn mis-shelved
book, and I never did find the other one.)
Ian A. York
Timothy A. McDaniel <tm...@jump.net> wrote:
>
>Alas! yet another story ruined by the brutal facts of Newtonian
>mechanics -- in this case, conservation of angular momentum. The
>"push off so as to spin up" is the "here a miracle occurs" step.
Ah. I've been feeling vaguely uneasy about that story for years, though
never clearly enough to try to work out where it lost me. Thanks.
Jens Kilian
> I even saw instructions on how to take a square root on an
> abacus. ("Instructions" is perhaps a bit much; the procedure
> amounted to: guess the square root, square your guess, compare
> to the original number, adjust your guess accordingly, repeat.)
That's essentially how a numerical square root algorithm works ;-)
I've seen a description of taking a square root by hand, similar
to long division, but couldn't make head or tail of it[1].
Jens.
[1] Part of the problem was that I read it in an old schoolbook, and apparently
the method of long division taught in German schools has changed since
the time it was printed; at least the notation was unfamiliar.
--
mailto:j...@acm.org phone:+49-7031-464-7698 (TELNET 778-7698)
http://www.bawue.de/~jjk/ fax:+49-7031-464-7351
PGP: 06 04 1C 35 7B DC 1F 26 As the air to a bird, or the sea to a fish,
0x555DA8B5 BB A2 F0 66 77 75 E1 08 so is contempt to the contemptible. [Blake]
Jens Kilian
> L. Sprague de Camp's "Lest Darkness Fall" has a time traveller
> introducing Arabic notation to help with book keeping. Niven wrote a
> short story using an infinite series against a demon ... I'm not quite
> sure what it was called, but "Limits" seems about right.
_Convergent Series_
Jens Kilian
> Did the Romans =have= abacuses/(abaci?)??
_Abacus_ *is* a Latin word...
David Allsopp
<G...@quik.freeuk.com> writes
There is only One True Reference Work on topics such as these: "The
Universal History Of Numbers" by Georges Ifrah et. al.[1] Apparently
the Romans typically used calculating tables (with sand or wax), but did
have an abacus that was just about equivalent to the classical Chinese
one. For some reason, use of counting tables survived the fall of Rome,
but abaci didn't.
Incidentally, referring to the story, no-one even in Roman times
actually calculated using Roman numerals. They used calculating tables,
which, at a pinch, you can draw in mud. There was a long apprenticeship
in learning how to use them. The thing about Arabic (really Indian, of
course) numerals and positional notation is that they make it *possible*
to calculate quickly on paper. On the other hand, doing advanced
arithmetic was high mathematics (degree-level equivalent) up to about
the Renaissance. On the gripping hand, how many people remember
(indeed, were ever taught) how to do long multiplication and long
division by hand?
[1] Drooling fanboy plug for this book. It's *awesome*, however you
want to interpret the word.
--
David Allsopp Houston, this is Tranquillity Base.
Remove SPAM to email me The Eagle has landed.
Coyu
>Are you sure it was 1729.03? I thought it was someting like 27.03.
>As I recall, Feynman started to explain to the abacus operator how he
>figured out the cube root by saying "The cube root of <largest cube
>less than the number in question> is <whatever it was>", at which
>point the abacus operator stopped to figure it out on his abacus,
>which surprised Feynman, who thought the cube root should have
>been obvious. Now I can believe that Feynman would have thought
>it obvious that the cube root of 27 is 3, but it seems unlikely
>he'd think it obvious that the cube root of 1728 is 12.
I'm no Feynman, but I know that one. 1728 is the number of cubic inches
in a cubic foot.
Robert Shaw
"Jeff Suzuki" <je...@bu.edu> wrote in message
news:9k4jmm$3cm$2...@news3.bu.edu...
> Matt Ruff <Storyt...@worldnet.att.net> wrote:
>
> : Wouldn't a real Roman use an abacus to make math calculations?
>
> Very likely. And in all probability, be faster than the person
> using Arabic numerals.
But this story is taking place in Viking England, with the
clerk using roman numerals and the vikings arabic.
Was the abacus known in England in the 800's?
--
Matter is fundamentally lazy:- It always takes the path of least effort
Matter is fundamentally stupid:- It tries every other path first.
That is the heart of physics - The rest is details.- Robert Shaw
David Tate
>(wasn't AEvG also into LRonHubb&ard stuff?).
Not that I know of, but he *was* the foremost convert to the
quintessential 1950's equivalent, General Semantics. That's where all
the "Null-A" stuff came from.
David Tate
Nancy Lebovitz
David Allsopp <d...@tqSPAMbase.demon.co.uk> wrote:
>
>
>Incidentally, referring to the story, no-one even in Roman times
>actually calculated using Roman numerals. They used calculating tables,
>which, at a pinch, you can draw in mud. There was a long apprenticeship
>in learning how to use them. The thing about Arabic (really Indian, of
>course) numerals and positional notation is that they make it *possible*
>to calculate quickly on paper. On the other hand, doing advanced
>arithmetic was high mathematics (degree-level equivalent) up to about
>the Renaissance. On the gripping hand, how many people remember
>(indeed, were ever taught) how to do long multiplication and long
>division by hand?
>
do multiplication with sub-totals. I was also taught how to do
long division, and I can still do it. I'm 48, and I don't think
calculators were common till well after I was out of high school--
there should be people younger than I am who can do long division.
I may have re-invented an elementary form of the calculating table.
I found it was easier to do big long division problems if I made a
little table what the divisor times 2, 3, 4, 5, 6, 7, 8, and 9 was.
I also got around the need to memorize trigonometry equations by solving
problems in terms of the sides of the triangle.
I've seen but never memorized that method of finding square roots. Iirc,
it was rather like long division, but you took the digits of the number
you were trying to find the square root of by pairs and looked for
their square root.
OBSF: Asimov's "The Feeling of Power" (about the unfortunate rediscovery
of arithmetic), of course.
of
Nancy Lebovitz
wasn't especially nonsensical, though van Vogt was, um, exuberant in
his portrayal of GS as he was of science.
I don't think Heinlein ever mentioned General Semantics by name, but
he mentioned now and again that the word is *not* the thing, and that's
very GSish.
I haven't studied Scientology[1], but it's my impression that there was
at least a little legitimate mind/body stuff in it, and I've heard
that there's a breakaway group that tries to pull the good stuff out
and study it without being a cult.
[1] I've actually used the word on-line. Nothing happened. I've seen
other people use the word on-line, and nothing happened to them, either.
I think someone's stopped grepping.
Vondermort. Vondermort. Vondermort.
Daniel McRae
wrote:
>ten...@alumnae.caltech.edu (Ross TenEyck) writes:
>> I even saw instructions on how to take a square root on an
>> abacus. ("Instructions" is perhaps a bit much; the procedure
>> amounted to: guess the square root, square your guess, compare
>> to the original number, adjust your guess accordingly, repeat.)
>
>That's essentially how a numerical square root algorithm works ;-)
>I've seen a description of taking a square root by hand, similar
>to long division, but couldn't make head or tail of it[1].
>
> Jens.
>
>[1] Part of the problem was that I read it in an old schoolbook, and apparently
> the method of long division taught in German schools has changed since
> the time it was printed; at least the notation was unfamiliar.
Personally I always use the Newton-Raphson method: For an equation of
f(x)=0, a root may be found by iterating:
x_(n+1) = x_n - f(x_n)/f'(x_n)
So, for a square root, the function is f(x)=x^2 - y, where y is the
number you want to find the root of, and so you iterate:
x_(n+1) = x_n - (x_n^2 - y)/(2*x_n).
Start with a reasonable initial guess, and it gets pretty close pretty
fast. I think calculators use y/2 for an initial guess.
Lets see: take two for an example. Initial guess of 1:
n x_n
0 1
1 1.5
2 1.416666
3 1.41421569
4 1.414213562
And there is the square root of two! Not to hard, even on paper.
Daniel
(.student reversed in email to prevent spam)
Arthur Kimes
] A guy with an abacus beat Richard Feynman on adding and
multiplication,
] tied on division, and came up a little short on a cube root (of
1729.03,
] if you're interested). ("Surely You're Joking, Mr. Feynman!")
]
] Mind you, Feynman was doing it in his head, but that's still pretty
] impressive.
That made it into the movie "Infinity" (with Matthew Broderick
playing Feynman). The hysterical part about the scene was when the
shop-keeper pulled out a smaller abacus and laid it next to the big
abacus so he could calculate more decimal places in the cube root
challenge.
Where IS Everybody?
http://setiathome.ssl.berkeley.edu/
Jeff Suzuki
: Incidentally, now that I think about it, I'm not certain the
: Romans had the abacus. For one thing, the abacus strongly implies
: a place-value number system, like Arabic numerals but unlike Roman
: numerals.
The Romans definitely had the abacus; see Menninger's
_Number Words and Number Symbols_ for an example of a pocket calculator.
Incidentally, calculator <-- calculus = pebble, in Latin.
The abacus is in fact _better_ suited for Roman notation than Arabic
notation. It's one of the reasons it took five hundred years before
Arabic numerals finally won out in the west: the abacus works on
Roman numerals, but new methods of computation had to be invented
for Arabic numerals.
Jeffs
Daniel McRae
(Daniel McRae) wrote:
>Personally I always use the Newton-Raphson method: For an equation of
>f(x)=0, a root may be found by iterating:
>x_(n+1) = x_n - f(x_n)/f'(x_n)
>So, for a square root, the function is f(x)=x^2 - y, where y is the
>number you want to find the root of, and so you iterate:
>x_(n+1) = x_n - (x_n^2 - y)/(2*x_n).
>Start with a reasonable initial guess, and it gets pretty close pretty
>fast. I think calculators use y/2 for an initial guess.
>Lets see: take two for an example. Initial guess of 1:
>n x_n
>0 1
>1 1.5
>2 1.416666
>3 1.41421569
>4 1.414213562
>And there is the square root of two! Not to hard, even on paper.
I meant "too", not "to" in that last sentence, and I also meant to add
"at least for a few significant figures" on to the end of the
sentence. No, I did not calculate all those decimal places on paper!
Ulrich Elsner
>I've seen but never memorized that method of finding square roots. Iirc,
>it was rather like long division, but you took the digits of the number
>you were trying to find the square root of by pairs and looked for
>their square root.
The 'classical' method (I seem to remember that is was known to the
Babylonians already) to calculate the square root of k is:
1. Take a guess, say x.
2. calculate (1/2)*(x + k/x). This is your new guess.
3. repeat a few times.
If your first guess is not too bad, 'few' means two or three.
So, the only hard part is the division.
This is just the so-called Newton method for finding the zeros of
x^2-k.
Ulrich
Jeff Suzuki
:> Matt Ruff <Storyt...@worldnet.att.net> wrote:
:>
:> : Wouldn't a real Roman use an abacus to make math calculations?
:>
:> Very likely. And in all probability, be faster than the person
:> using Arabic numerals.
: But this story is taking place in Viking England, with the
: clerk using roman numerals and the vikings arabic.
: Was the abacus known in England in the 800's?
It was known to the Romans, so I don't see why not.
Here's another etymology: one form of the abacus used during the middle
ages was a "counting board", essentially a table with lines laid across
it (see the illustration "Typus Arithmetic" in Boyer's _A History of
Mathematics_). This is essentially an abacus without beads. You put
markers into the right spaces to indicate prices, and totaled up
a customer's purchase.
On the _counter_.
Jeffs
Jeff Suzuki
: In article <3b660afc...@news.vt.edu>,
: Chris Byler <cby...@REMOVE-TO-REPLY.vt.edu> wrote:
:>
:>"Convergent Series", iirc. Which is ironic - if you assume that the
:>time required for the demon to disappear and reappear is proportional
:>to the distance (he moves at the speed of light, for example), then
:>the time series _would_ be convergent and the demon would escape the
:>trap in finite time. Fortunately that doesn't happen in the story.
: Would he indeed "escape"? He would reach zero size in finite time,
: but what happens after that?
: FOr the remainder of the 24 hour period, the demon must appear in whatever
: the protagonist had drawn the pentagram. Apparently the demon is
: incapable of erasing the pentagram, at least for that 24 hour period,
: but the pentagram is drawn on his own belly. So, the demon "converges" to
: zero size in finite time, but keeps shrinking for 24 hours. Paradox!
: (We are assuming that the "finite time" is less than the time remaining
: in the protagonist's 24 hour grace period; that is justified, given the
: time the first few "terms of the series" seem to take, and the fact that
: the size ratio would be significantly less than one.)
: Now, let me suggest an alternative ending: The demon's thought processes,
: magical reappearances, etc. do NOT speed up indefinately, the time series
: does NOT converge, and at the end of 24 hours a very pissed off demon,
: shrunk to unimaginably small size, is released from the pentagram and
: free to restore himself to his usual size.
: Or else he gets swallowed by a quantum fluctuation before the 24 hours are
: up.
: --
: pci...@antiabuseworld.std.com | "Evolution is a theory that accounts
: Please remove "antiabuse" | for variety, not superiority."
: when replying. Thank you. | -- Joan Pontius
Jeff Suzuki
: As a side thought, are there any stories about encounters with aliens
: that have explored a completely different region of the space of
: mathematics to humans? "Sorry, now explain this concept of 'number'
: again? It looks like a sort of wierd mapping onto a Fxchil group to me,
: but it makes my antennae twist just to think about it."
It seems to, er, alien an idea. The closest I can think of is "Green
Thumb", by Simak, where Simak suggests there are ideas more fundamental
than mathematics.
One of the STrek fan fiction pieces, "The Sleeping God", suggests a human
mutation capable of using multi-valued logic routinely (i.e., allowing
statements to have truth values besides "True", "False"). However,
such would include standard logic as a subset, so such a hypothetical
entity would be able to understand _us_.
Jeffs
Robert Carnegie
> Bitstring <90EE63F13dgem...@cnews.newsguy.com>, from the
> wonderful person David Empey <dem...@cruzio.com> said
> >rsn...@swbellnospam.net wrote in <3B63C81E...@swbell.net>:
> >
> >>My favorite scene in Harry Harrison's novel _King and Emperor_ involves
> >>a battle between two powerful catapults (trebuchets, if memory serves).
> >>The antagonist makes his ballistic calculations using Roman notation
> >>while the hero uses Arabic notation. Guess who is victorious.
> >
> >Why didn't they use abacuses?
>
> Did the Romans =have= abacuses/(abaci?)?? I know the Chinese did, from
> way back, but I'm not sure that they ever made it to Rome (well, Ancient
> Rome .. you know what I mean .. 8>.).
http://www.dotpoint.com/xnumber/pic_roman_abacus.htm
http://www.gol.com/abacus/abacus.html
claim so, and there's at least one picture. Doesn't prove anything,
I suppose, being on the Internet and all.
I also searched Google.com for roman+trebuchet and unfortunately
got a whole lot of articles about fonts...however,
http://members.iinet.net.au/~rmine/seemore.html describes
the trebuchet as "medieval", and also features Greek and Roman
machinery as "Not Trebuchets".
Having checked on what _King and Emperor_ is actually about,
it appears that actual Romans weren't involved anyway??
That page from www.gol.com asserts that the abacus fell into _disuse_
when Arabic notation was adopted and allowed long multiplication, etc.,
on paper (checkable?) which puts - um - Byzantine Romans? - ahead of
the rest of Western Europe. I think.
Geoffrey A. Landis
>
> Well... you can put any set of cardinality aleph-1, like (say) a
> bounded closed interval in R(n), into 1-1 correspondence with any
> Cantor set extracted from (say) that piece of R(n). That's a 1-1
> mapping between a set of nonzero measure and a proper subset of
> measure zero.
Hmmm-- okay, obviously I should think about it some more.
> That's what's so weird about it. Yes, the Cantor set
> is dense in the parent set, but it still has measure zero -- which was
> the point of the climactic act of the story.
Yes, that sure is weird.
Paul Ciszek wrote:
:>
:> "Banned from Aleph"
:> lyrics by Kevin Wald
:> ttto "Bannde from Argo" by Leslie Fish
:>...
Say, cool song.
--
Geoffrey A. Landis
http://www.sff.net/people/geoffrey.landis
Ross TenEyck
>> I even saw instructions on how to take a square root on an
>> abacus. ("Instructions" is perhaps a bit much; the procedure
>> amounted to: guess the square root, square your guess, compare
>> to the original number, adjust your guess accordingly, repeat.)
>That's essentially how a numerical square root algorithm works ;-)
>I've seen a description of taking a square root by hand, similar
>to long division, but couldn't make head or tail of it[1].
They actually taught us the successive-approximation method in
school. I have seen the proper procedure for taking a square
root by hand -- I think it was in one of Asimov's non-fiction
books -- but I don't remember it.
Hmm... a quick google turns up:
http://forum.swarthmore.edu/dr.math/problems/tim3.30.98.html
I'll have to study that to grok it fully...
--
================== http://www.alumni.caltech.edu/~teneyck ==================
Ross TenEyck Seattle, WA \ Light, kindled in the furnace of hydrogen;
ten...@alumni.caltech.edu \ like smoke, sunlight carries the hot-metal
Are wa yume? Soretomo maboroshi? \ tang of Creation's forge.
Ross TenEyck
Eh? Explain, please... the way I learned an abacus, it was a
straightforward mapping of a place-value number system: the
first column represented ones, the next columns tens, and so
on. How is that better suited to Roman numerals than Arabic
numerals?
Mark Atwood
> ten...@alumnae.caltech.edu (Ross TenEyck) writes:
> > I even saw instructions on how to take a square root on an
> > abacus. ("Instructions" is perhaps a bit much; the procedure
> > amounted to: guess the square root, square your guess, compare
> > to the original number, adjust your guess accordingly, repeat.)
>
> That's essentially how a numerical square root algorithm works ;-)
Back before I had instant access to a calculator with a root button, I
would just eyeball it, and then iterate Newton's Method 2 or 3 times.
--
Mark Atwood | I'm wearing black only until I find something darker.
m...@pobox.com | http://www.pobox.com/~mra
Joe Pfeiffer
>
> The abacus is in fact _better_ suited for Roman notation than Arabic
> notation. It's one of the reasons it took five hundred years before
> Arabic numerals finally won out in the west: the abacus works on
> Roman numerals, but new methods of computation had to be invented
> for Arabic numerals.
I don't follow -- I can see where an abacus would in some sense be
more necessary for roman numerals than for a place-number system (in
the sense that translating the numbers to put them on the abacus would
put them in a form where they could reasonably be manipulated). But I
sure don't see why it would actually be better suited; putting numbers
from a place-number system on the abacus involves much less
translation.
It always seemed to me that in using an abacus, the Romans came within
inches of moving to a place-number system; all it would have take
would have been for some bright clerk to write down a representation
of the beads rather than translating back to Roman, and they'd have
been there.
--
Joseph J. Pfeiffer, Jr., Ph.D. Phone -- (505) 646-1605
Department of Computer Science FAX -- (505) 646-1002
New Mexico State University http://www.cs.nmsu.edu/~pfeiffer
SWNMRSEF: http://www.nmsu.edu/~scifair
Avram Grumer
na...@unix1.netaxs.com (Nancy Lebovitz) wrote:
> [1] I've actually used the word on-line. Nothing happened. I've seen
> other people use the word on-line, and nothing happened to them,
> either. I think someone's stopped grepping.
>
> Vondermort. Vondermort. Vondermort.
I don't know what that last bit means. For a moment I was trying to
figure out if Harry Potter's arch-nemesis is a Scientologist.
--
Avram Grumer | av...@grumer.org | http://www.PigsAndFishes.org
Robert Shaw
> Robert Shaw <Rob...@shavian.fsnet.co.uk> wrote:
>
> :> Matt Ruff <Storyt...@worldnet.att.net> wrote:
> :>
> :> : Wouldn't a real Roman use an abacus to make math calculations?
> :>
> :> Very likely. And in all probability, be faster than the person
> :> using Arabic numerals.
>
> : But this story is taking place in Viking England, with the
> : clerk using roman numerals and the vikings arabic.
>
> : Was the abacus known in England in the 800's?
>
> It was known to the Romans, so I don't see why not.
>
I doubt the early anglo-saxons had abaci when they arrived,
so the abacus would have to have been reintroduced by the
church, possible but not obviously inevitable.
> Here's another etymology: one form of the abacus used during the middle
> ages was a "counting board", essentially a table with lines laid across
> it (see the illustration "Typus Arithmetic" in Boyer's _A History of
> Mathematics_). This is essentially an abacus without beads. You put
> markers into the right spaces to indicate prices, and totaled up
> a customer's purchase.
>
> On the _counter_.
The medieval English treasury used tally rods and checked cloths for
its accounting. I'm not sure if they officially used abaci, though
they will have known of them.
Jerome Bigge
wrote:
Still can do it. Private school education however.... My wife,
who went to public schools, has a hard time determining which
is larger, 1/2 or 1/3rd. Differences in the quality of education.
>[1] Drooling fanboy plug for this book. It's *awesome*, however you
>want to interpret the word.
Jerome Bigge
NRA Life Member
Supporter of National Health Insurance
CompTIA A+ Certified Computer Technician
Author of the "Warlady" & "Wartime" series.
Download at "http://members.tripod.com/~jbigge"
Konrad Gaertner
>
[snip]
>
> There is only One True Reference Work on topics such as these: "The
> Universal History Of Numbers" by Georges Ifrah et. al.[1] Apparently
> the Romans typically used calculating tables (with sand or wax), but did
> have an abacus that was just about equivalent to the classical Chinese
> one. For some reason, use of counting tables survived the fall of Rome,
> but abaci didn't.
I picked up the sequel (UH of Computers) from the library, barely got
through the summary of the UHoN before returning it as unreadable.
Note that I like mathematics, and a lot of these topics really
interested me, yet the author managed to make it so mind-numbingly
boring that I couldn't bring myself to finish it. He seems to be a
sort of anti-Asimov.
Of course, the fact it took five (5) people to translate it into
English is probably part of the problem.
> Incidentally, referring to the story, no-one even in Roman times
> actually calculated using Roman numerals. They used calculating tables,
> which, at a pinch, you can draw in mud. There was a long apprenticeship
> in learning how to use them. The thing about Arabic (really Indian, of
> course) numerals and positional notation is that they make it *possible*
> to calculate quickly on paper. On the other hand, doing advanced
> arithmetic was high mathematics (degree-level equivalent) up to about
> the Renaissance. On the gripping hand, how many people remember
> (indeed, were ever taught) how to do long multiplication and long
> division by hand?
I learned long division and multiplication in school (in the '80s).
I also learned synthetic division (for factoring large polynomials).
I've found that doing calculations on paper is generally faster than
hunting through my desk for my pocket calculator.
And often sometimes what you need is quotient and remainder or you're
dealing with some base other than ten.
--KG
Konrad Gaertner
>
[snip]
>
> As a side thought, are there any stories about encounters with aliens
> that have explored a completely different region of the space of
> mathematics to humans? "Sorry, now explain this concept of 'number'
> again? It looks like a sort of wierd mapping onto a Fxchil group to me,
> but it makes my antennae twist just to think about it."
Not exactly mathematics, but the Free Entities in Hickman's _Lemurian
Stone_ have difficulty with the concept of 'time', and the Jenoine in
Brust's Dragaera books seem to have a radically different view of
'space' (discussed in _Issola_).
--KG
kesi...@math.ttu.edu
: Well... you can put any set of cardinality aleph-1, like (say) a
: bounded closed interval in R(n), into 1-1 correspondence with any
: Cantor set extracted from (say) that piece of R(n). That's a 1-1
By ``aleph-1,'' you really mean ``C,'' right?
==Jake
David Empey
mg.aol.com>:
>David Empey wrote:
>
>>been obvious. Now I can believe that Feynman would have thought
>>it obvious that the cube root of 27 is 3, but it seems unlikely
>>he'd think it obvious that the cube root of 1728 is 12.
>
>I'm no Feynman, but I know that one. 1728 is the number of cubic inches
>in a cubic foot.
Sure; I bet lots of Feynman fans, or sf fans, would know this.
But would you expect J. Random Abacus Operator to know it?
--
Dave Empey
What else could a millennia-spanning, reality-hopping,
transdimensional cult of genetically-perfect,
bloodthirsty superwomen want? --Kenneth Hite
kesi...@math.ttu.edu
: "Convergent Series", iirc. Which is ironic - if you assume that the
: time required for the demon to disappear and reappear is proportional
: to the distance (he moves at the speed of light, for example), then
: the time series _would_ be convergent and the demon would escape the
: trap in finite time. Fortunately that doesn't happen in the story.
From my recollection of the story, I'd think it would take finite time
(bounded away from zero) between manifestations for the demon to locate
the new pentagram.
But it's been several years.
==Jake
kesi...@math.ttu.edu
: As a side thought, are there any stories about encounters with aliens
: that have explored a completely different region of the space of
: mathematics to humans? "Sorry, now explain this concept of 'number'
: again? It looks like a sort of wierd mapping onto a Fxchil group to me,
: but it makes my antennae twist just to think about it."
_Anvil_of_Stars_ by Greg Bear springs to mind, but the aliens'[0]
math is not hideously different. As I recall, they don't really
have a concept of ``integer.''
==Jake
[0] The ones that show up about half/two thirds of the way through the
text, not the ones that built the ship or are the target thereof.
David Empey
>I'm not sure how what long multiplication is--I was taught how to
>do multiplication with sub-totals. I was also taught how to do
>long division, and I can still do it. I'm 48, and I don't think
>calculators were common till well after I was out of high school--
>there should be people younger than I am who can do long division.
>
Present. *And* I can do square roots, too! Granted, they didn't
teach me that one in class.
Nancy Lebovitz
Avram Grumer <av...@grumer.org> wrote:
>In article <9k6efc$i...@netaxs.com>,
> na...@unix1.netaxs.com (Nancy Lebovitz) wrote:
>
>> [1] I've actually used the word on-line. Nothing happened. I've seen
>> other people use the word on-line, and nothing happened to them,
>> either. I think someone's stopped grepping.
>>
>> Vondermort. Vondermort. Vondermort.
>
>I don't know what that last bit means. For a moment I was trying to
>figure out if Harry Potter's arch-nemesis is a Scientologist.
>
to say Vondermort's name.
David Allsopp
<na...@unix1.netaxs.com> writes
>In article <$GAhvgB5...@tqbase.demon.co.uk>,
>David Allsopp <d...@tqSPAMbase.demon.co.uk> wrote:
>>
>>
>>Incidentally, referring to the story, no-one even in Roman times
>>actually calculated using Roman numerals. They used calculating tables,
>>which, at a pinch, you can draw in mud. There was a long apprenticeship
>>in learning how to use them. The thing about Arabic (really Indian, of
>>course) numerals and positional notation is that they make it *possible*
>>to calculate quickly on paper. On the other hand, doing advanced
>>arithmetic was high mathematics (degree-level equivalent) up to about
>>the Renaissance. On the gripping hand, how many people remember
>>(indeed, were ever taught) how to do long multiplication and long
>>division by hand?
>>
>do multiplication with sub-totals. I was also taught how to do
>long division, and I can still do it. I'm 48, and I don't think
>calculators were common till well after I was out of high school--
>there should be people younger than I am who can do long division.
You're in the right age bracket to remember such archaisms :-). I'm 40,
and when we did our exams at 16, ours was the last year *not* allowed
calculators in exams -- we were still on log tables. I'm not sure
whether modern syllabuses still teach pencil-and-paper arithmetic, but
I'll bet that if they do, far fewer people remember it now that
calculators are so ubiquitous.
> [snip]
>OBSF: Asimov's "The Feeling of Power" (about the unfortunate rediscovery
>of arithmetic), of course.
The scary thing is, you can almost see this happening. I sometimes
amuse myself by trying to add up (say) a restaurant bill faster than the
cashier putting it into the till, and often get very odd looks when I
do. And I'm not a lightning calculator by any means.
--
David Allsopp Houston, this is Tranquillity Base.
Remove SPAM to email me The Eagle has landed.
Mark Atwood
> to calculate quickly on paper. On the other hand, doing advanced
> arithmetic was high mathematics (degree-level equivalent) up to about
> the Renaissance. On the gripping hand, how many people remember
> (indeed, were ever taught) how to do long multiplication and long
> division by hand?
I was. I can. Often faster that I can pull out a calculator.
In fact, on the the things *wrong* with my public education was that
we sepnt TOO MUCH time drilling long multiplication and long division
after most of the students had it down cold.
But NOOOOO, we had to keep at it for 4 frikken YEARS...
Coyu
>>>been obvious. Now I can believe that Feynman would have thought
>>>it obvious that the cube root of 27 is 3, but it seems unlikely
>>>he'd think it obvious that the cube root of 1728 is 12.
>>
>>I'm no Feynman, but I know that one. 1728 is the number of cubic inches
>>in a cubic foot.
>
>Sure; I bet lots of Feynman fans, or sf fans, would know this.
[snort] Doubt that. A lot of casual unthinking innumeracy in SF.
>But would you expect J. Random Abacus Operator to know it?
If J. had to do unit conversions within the English system of measurements
on a regular basis, sure. 231 cubic inches to the gallon, and 1760 yards
to the mile.
Feynman would certainly have had to.
It's like Hardy being impressed with Ramanujan saying that 1729
is the smallest number that can be represented as two cubes
in two different ways... yeah, except:
1000 + 729 = 1728 + 1
is kind of a red flag to anyone who plays with numbers regularly.
ObSF: 42
Timothy A. McDaniel
Arthur Kimes <ar...@yahoo.com> wrote:
> That made it into the movie "Infinity" (with Matthew Broderick
>playing Feynman).
I thought you were joking, so I checked IMDB to debunk it.
Omygawd.
http://us.imdb.com/Title?0116635
Only 5.8 stars out of 10, and some non-glowing reviews explain why,
but yes, Matthew Broderick as Richard Feynman.
And I thought my bogglemeter had broken when the Russian Duma declared
independence from the Soviet Union.
--
Tim McDaniel is tm...@jump.net; if that fail,
tm...@us.ibm.com is my work account.
"To join the Clueless Club, send a followup to this message quoting everything
up to and including this sig!" -- Jukka....@hut.fi (Jukka Korpela)
Ross Presser
> I don't think Heinlein ever mentioned General Semantics by name,
> but he mentioned now and again that the word is *not* the thing,
> and that's very GSish.
"If This Goes On--" and "Methusaleh's Children" both mention
semanticists and give them great praise (okay, not quite so great in
MC) without naming GS specifically.
--
Ross Presser * ross_p...@imtek.com
"Back stabbing is a sport best played by those that can't stand face
to face with their opponent." - Danny Taddei
David Empey
<9k6j15$248$1...@narses.hrz.tu-chemnitz.de>:
>According to Nancy Lebovitz <na...@unix1.netaxs.com>:
>>I've seen but never memorized that method of finding square roots. Iirc,
>>it was rather like long division, but you took the digits of the number
>>you were trying to find the square root of by pairs and looked for
>>their square root.
More or less, though it's a bit more complicated than that.
>
>The 'classical' method (I seem to remember that is was known to the
>Babylonians already) to calculate the square root of k is:
>
>1. Take a guess, say x.
>2. calculate (1/2)*(x + k/x). This is your new guess.
>3. repeat a few times.
>
>If your first guess is not too bad, 'few' means two or three.
Depending on how many digits of accuracy you want.
Isrt this converges fairly quickly. I think I worked out
once that this was just as fast as the method Nancy mentions
above.
>So, the only hard part is the division.
>
>This is just the so-called Newton method for finding the zeros of
>x^2-k.
>
>
>
>Ulrich
Bertil Jonell
Frank van den Eijkhof <fran...@cs.uu.nl> wrote:
>On Sun, 29 Jul 2001 02:23:59 -0600, rsn...@swbellnospam.net wrote:
>
>>Does anyone else have examples of SF gettin' jiggy wit' mathematics?
>
>A.E. van Voght wrote a story about some alien captured (on Mars?)
>long ago, with a lock based on the then-current math.
>An scientist from earth was tempted by the aliens to unlock this prison
>using now-current math. I must confess that I never checked the story
>on correctness...
I remember that story. It had a very excellent Lovecraftian feeling
to it. Much more than what is often labelled as 'Mythos'.
>Frank
-bertil-
--
"It can be shown that for any nutty theory, beyond-the-fringe political view or
strange religion there exists a proponent on the Net. The proof is left as an
exercise for your kill-file."
Ross TenEyck
>I'm not sure how what long multiplication is--I was taught how to
>do multiplication with sub-totals. I was also taught how to do
>long division, and I can still do it. I'm 48, and I don't think
>calculators were common till well after I was out of high school--
>there should be people younger than I am who can do long division.
I was taught long division, certainly -- I think that was a little
before calculators became ubiquitous.
Are they no longer teaching long division in school? Appalling
if true...
David Allsopp
<pfei...@cs.nmsu.edu> writes
>Jeff Suzuki <je...@bu.edu> writes:
>>
>> The abacus is in fact _better_ suited for Roman notation than Arabic
>> notation. It's one of the reasons it took five hundred years before
>> Arabic numerals finally won out in the west: the abacus works on
>> Roman numerals, but new methods of computation had to be invented
>> for Arabic numerals.
>
>I don't follow -- I can see where an abacus would in some sense be
>more necessary for roman numerals than for a place-number system (in
>the sense that translating the numbers to put them on the abacus would
>put them in a form where they could reasonably be manipulated). But I
>sure don't see why it would actually be better suited; putting numbers
>from a place-number system on the abacus involves much less
>translation.
>
>It always seemed to me that in using an abacus, the Romans came within
>inches of moving to a place-number system; all it would have take
>would have been for some bright clerk to write down a representation
>of the beads rather than translating back to Roman, and they'd have
>been there.
The Romans were actually at least two steps short: they didn't have a
single abstract figure for each digit, and they didn't have a zero.
Consider the number 2062, which on a per-digit basis comes out as
"MMVIII", which is somewhat ambiguous. It seems that there's a big
conceptual gap between using Roman-style numbers, where 2062 is MMLXII
(two thousands, fifty, ten and two) and real positional notation. It took
a few generations of very bright Indian mathematicians to make that jump.
And I'd like to plug again my source for this, "The Universal History Of
Numbers" by Georges Ifrah et. al.
David Allsopp
<kgae...@worldnet.att.net> writes
>David Allsopp wrote:
>>
>[snip]
>>
>> There is only One True Reference Work on topics such as these: "The
>> Universal History Of Numbers" by Georges Ifrah et. al.[1] Apparently
>> the Romans typically used calculating tables (with sand or wax), but did
>> have an abacus that was just about equivalent to the classical Chinese
>> one. For some reason, use of counting tables survived the fall of Rome,
>> but abaci didn't.
>
>I picked up the sequel (UH of Computers) from the library, barely got
>through the summary of the UHoN before returning it as unreadable.
>Note that I like mathematics, and a lot of these topics really
>interested me, yet the author managed to make it so mind-numbingly
>boring that I couldn't bring myself to finish it. He seems to be a
>sort of anti-Asimov.
>
>Of course, the fact it took five (5) people to translate it into
>English is probably part of the problem.
It certainly starts slowly, and even after skipping I know far more
about medieval finger-counting methods that I really need to. However,
if you persevere, the sheer weight of scholarship drags you in, and the
whole thing becomes fascinating. I'd recommend it unhesitatingly, but
with a warning to skip where necessary.
John F Carr
GSV Three Minds in a Can <G...@quik.freeuk.com> wrote:
>>Wise. As I recall, the math was complete bullshit, but like most van Vogt,
>>the story manages to be entertaining in spite of the fact that you know the
>>"science" is thoroughly bogus.
>
>(spoiler ahead)
>The 'logic' was along the lines of the lock being a time lock coded to
>the 'ultimate prime number' which was linked to the 'Eis force' (sp?)
>(phooey already!) and the 'solution' was to adjust the Eis force by an
>incy-weesy bit (like adding 1) after which the prime number falls into
>lots of factors, one of which happens to be 'just now', so the lock
>opens.
The description is not much worse than the description of
Greg Egan's "Luminous", a story which I liked.
--
John Carr (j...@mit.edu)
Joe Pfeiffer
>
> The Romans were actually at least two steps short: they didn't have a
> single abstract figure for each digit, and they didn't have a zero.
> Consider the number 2062, which on a per-digit basis comes out as
> "MMVIII", which is somewhat ambiguous. It seems that there's a big
> conceptual gap between using Roman-style numbers, where 2062 is MMLXII
> (two thousands, fifty, ten and two) and real positional notation. It took
> a few generations of very bright Indian mathematicians to make that jump.
My guess is that my hypothetical bright clerk would have left some
kind of gap to mark the columns where there were no beads... now that
I think about it, if he'd done that, and had had the additional
insight to use a compact symbol for the number of items in each
column, it could still have been generations before anybody twigged to
the idea that his laziness was actually an equivalent (and handier)
notation to roman numerals.
> And I'd like to plug again my source for this, "The Universal History Of
> Numbers" by Georges Ifrah et. al.
The few people on Amazon who've reviewed it sure gave it high marks --
I want it. Hmmmm, sales rank is 40,546.... but it's popular in
Kentucky universities!
Jeff Suzuki
: Eh? Explain, please... the way I learned an abacus, it was a
: straightforward mapping of a place-value number system: the
: first column represented ones, the next columns tens, and so
: on. How is that better suited to Roman numerals than Arabic
: numerals?
The quick answer is that numbers are written in Roman in the
same way they are represented on the abacus whereas this is
_not_ true for numbers in Arabic.
Let's say you have a standard Roman era abacus: two beads above
the bar, five beads below (identical to a Chinese suan pan). The
Japanese abacus is split 1/4 (that is, 1 above, 4 below), but the
principle below is the same.
Now, take a number like eight. To represent that on an abacus, you
need to know eight is five plus three, so it's one bead above
(the five) plus three beads below (the three), which I'll write as
1/3 (one above/three below).
Now look at the number in Roman notation: V III. It is far more
obvious that this is represented by a one/three split, than it is
that "8" is a 1/3 split. In the former case, you _look_ at the number
and know how it splits. In the latter case, you have to remember
"8 = 5 + 3", and you need to remember a different rule for "7",
whereas the Roman calculator need only note that the number is "V II".
There are other advantages to Roman numerals (one of the reasons they
survived the coming of Arabic numerals for five hundred years). Here's
one communications theorists should enjoy: they're robust.
If you're in a hurry, 500 and 5000 look very similar; D and (V) do
not ((V) being the closest approximation I can get to the Roman 5000
symbol), and heaven help you if you're trying to read someone's hastily
scrawled 1497 (or is it 7941?) MCCCCLXXXXVII is hard to mistake for
anything else, and even if you don't read the number, you can start
to enter it on the abacus: that's a 0/1, 0/4, 1/4, 1/2.
(NB: The use of subtractive notation, IX for nine, was rarely used by
the Romans, and did not become common until after the invention of
printing. IX meant one and ten, or eleven)
Jeffs
Jeff Suzuki
: The Romans were actually at least two steps short: they didn't have a
: single abstract figure for each digit, and they didn't have a zero.
: Consider the number 2062, which on a per-digit basis comes out as
: "MMVIII", which is somewhat ambiguous.
What's ambiguous about it? It's two thousand, five and three.
But I'm not sure why you would translate 2062 as MMVIII: 2062 would
be MMLXII, which is clearly "two thousand, fifty, ten and two."
Jeffs
Jeff Suzuki
: The medieval English treasury used tally rods and checked cloths for
: its accounting. I'm not sure if they officially used abaci, though
: they will have known of them.
Possibly not abaci, as some people have pointed out. But the counter
board was used throughout Europe during the Middle Ages, so it seems
that even the English would have known about it.
Jeffs
Jorj Strumolo
JS> What's ambiguous about it? It's two thousand, five and three.
> But I'm not sure why you would translate 2062 as MMVIII
Because David is expanding on his two previous points, that the
Romans lacked unique symbols for each number, and lacked a zero,
so that if they tried to do the equivalent of "2062" using their
symbols for numbers, they'd get "MM VIII". Which I'm not sure
they would. Wouldn't a true attempt at columns notation be
"II VIII", that is, dropping the special symbol for thousand?
mstemper - emis . com
>Doug Palmer <do...@charvolant.org> wrote:
>
>: As a side thought, are there any stories about encounters with aliens
>: that have explored a completely different region of the space of
>: mathematics to humans? "Sorry, now explain this concept of 'number'
>: again? It looks like a sort of wierd mapping onto a Fxchil group to me,
>: but it makes my antennae twist just to think about it."
>
>_Anvil_of_Stars_ by Greg Bear springs to mind, but the aliens'[0]
>math is not hideously different. As I recall, they don't really
>have a concept of ``integer.''
That sounds just as painful to WSOD as does the Sundiver universe, where
the galactic society doesn't accept the concept of "real number", only
finite rational approximations.
--
Michael F. Stemper
#include <Standard_Disclaimer>
COFFEE.SYS not found. Abort, Retry, Fail?
James Nicoll
Cristiano Sadun <crs...@tin.it> wrote:
>
>I remember reading a very old novel by Someone :) - part of a series,
>with a criminal and a police agent following him.. (Deverel and Colby of
>something similar.. I read it at least ten years ago).
>
>Well, the duo was captured into a alien parabolic mirror built
>on an asteoroid and couldnt escape easily since there was no friction
>on the surface. They eventually manage to do it using (if my memory
>doenst fail) building a pendulum with themselves.
Ross Rocklyne (sp?), "The Men in the Mirror", which I have
in the collection of that name. He was also in one of the Dangerous
Vision collections, and the idea of a guy whose career peaked an
amazing -thirty- -years- -before- writing a modern sf story amazed
me when I was a teenager, thirty years previous being rough contemporary
with the days when George Washington fought the dinosaurs as far as
I was concerned. Heck, my parents were _kids_ back then.
James Nicoll
James Nicoll
>Cristiano Sadun <crs...@tin.it> wrote:
>
>: I remember reading a very old novel by Someone :) - part of a series,
>: with a criminal and a police agent following him.. (Deverel and Colby of
>: something similar.. I read it at least ten years ago).
>
Although in the collection, didn't Rocklynne admit the RPMs
of the pair worked out a bit higher than he guessed for the story?
Ross TenEyck
>: Eh? Explain, please... the way I learned an abacus, it was a
>: straightforward mapping of a place-value number system: the
>: first column represented ones, the next columns tens, and so
>: on. How is that better suited to Roman numerals than Arabic
>: numerals?
>The quick answer is that numbers are written in Roman in the
>same way they are represented on the abacus whereas this is
>_not_ true for numbers in Arabic.
>Let's say you have a standard Roman era abacus: two beads above
>the bar, five beads below (identical to a Chinese suan pan). The
>Japanese abacus is split 1/4 (that is, 1 above, 4 below), but the
>principle below is the same.
>Now, take a number like eight. To represent that on an abacus, you
>need to know eight is five plus three, so it's one bead above
>(the five) plus three beads below (the three), which I'll write as
>1/3 (one above/three below).
>Now look at the number in Roman notation: V III. It is far more
>obvious that this is represented by a one/three split, than it is
>that "8" is a 1/3 split. In the former case, you _look_ at the number
>and know how it splits. In the latter case, you have to remember
>"8 = 5 + 3", and you need to remember a different rule for "7",
>whereas the Roman calculator need only note that the number is "V II".
Eh... I don't buy it. I mean, consider a number like, oh, 98.
In Arabic, you instantly set up 8 in the first column and 9 in
the second. In Roman, you look at LXXXXVIII, and you have to
remember that an L is five in the second column, and an X is
a one in the second column, and a V is five in the _first_
column, while a I is one on the first column... in Arabic, an
8 is an 8, no matter which column it is.
And really, after a couple of days you will memorize all the five/one
splits. Heck, there're only four: 6, 7, 8 and 9. How hard is
that? Fewer items to remember than unique symbols for 1 and 5
in every power of ten up to whatever your computation limit is.
I still say that the abacus is naturally linked to a place-value
system.
In fact -- pardon me while I have a moment of insight -- it's so
closely linked that the Romans didn't have a great *need* for Arabic
numerals. Their notation was clunky, but they didn't use their
own notation for computation -- they used the abacus or a functionally
equivalent device that gave them all the computational advantages of
the place-value notation, without actually using it. They only used
the Roman numerals for recording the results.
It's as though they turned all the numbers into machine language
to operate on them, then translated back into human-readable form
at the end. The translation to and from machine language would
have been more convenient using Arabic notation, but it wasn't
considered the hard part of the process, so they didn't feel the
need to fix it.
Heh.
Konrad Gaertner
>
[snip]
>
> In fact -- pardon me while I have a moment of insight -- it's so
> closely linked that the Romans didn't have a great *need* for Arabic
> numerals. Their notation was clunky, but they didn't use their
> own notation for computation -- they used the abacus or a functionally
> equivalent device that gave them all the computational advantages of
> the place-value notation, without actually using it. They only used
> the Roman numerals for recording the results.
>
> It's as though they turned all the numbers into machine language
> to operate on them, then translated back into human-readable form
> at the end. The translation to and from machine language would
> have been more convenient using Arabic notation, but it wasn't
> considered the hard part of the process, so they didn't feel the
> need to fix it.
The problem with this system, of course, is that you can't check for
calculation errors, since none of the intermediate results are
recorded.
--KG
Joe Slater
>>: As a side thought, are there any stories about encounters with aliens
>>: that have explored a completely different region of the space of
>>: mathematics to humans? "Sorry, now explain this concept of 'number'
>>: again? It looks like a sort of wierd mapping onto a Fxchil group to me,
>>: but it makes my antennae twist just to think about it."
>In article <9k6ue...@enews3.newsguy.com>, kesi...@math.ttu.edu writes:
>>_Anvil_of_Stars_ by Greg Bear springs to mind, but the aliens'[0]
>>math is not hideously different. As I recall, they don't really
>>have a concept of ``integer.''
mstemper @ siemens - emis . com (Michael Stemper) wrote:
>That sounds just as painful to WSOD as does the Sundiver universe, where
>the galactic society doesn't accept the concept of "real number", only
>finite rational approximations.
I think Fred Pohl's Heechee use a different notation involving powers
of primes. That's not really a different sort of mathematics, though,
unless that notation lets you do cool things. And I don't think it
does.
--
Joe Slater was but a low-grade paranoiac, whose fantastic notions must
have come from the crude hereditary folk-tales which circulated in even
the most decadent of communities.
_Beyond the Wall of Sleep_ by H P Lovecraft
Coyu
>That sounds just as painful to WSOD as does the Sundiver universe, where
>the galactic society doesn't accept the concept of "real number", only
>finite rational approximations.
Why not? Some human philosophers don't.
GSV Three Minds in a Can
wonderful person Jens Kilian <Jens_...@agilent.com> said
>GSV Three Minds in a Can <G...@quik.freeuk.com> writes:
>> Did the Romans =have= abacuses/(abaci?)??
>
>_Abacus_ *is* a Latin word...
True, but that proves nothing, since Latin (along with Greek) is
regularly used (by scientists) to label and describe all sorts of things
the Romans had never conceived of.
--