This is it, the place to come and praise or discuss those shiney black
and magnificent numbers.
    Their curves, their angels, the sheer magnificence of them all. 
    Everything from using calculators to obscure refrences to that PBS
sensation "Math Net".
    Reverse Polish notation? No problem
    Advanced calculus? No problem
    Geometric proofs? No problem

    Get your fork and dig in.

132 responses total.

By the way, while I respect math and all its glory, I am positivley
incompetent when it comes to its application.

I won't contribute any answers in this item, but I probably will ask
some question <maybe>.

I'm a real engineer when it comes to mathematics.  Unless someone can show me a
practical application, I'm not that interested in it.

I don't tend to embrace a lot of "wish I would have" type
thinking but I do wish I'd consistently used algebra 
so that it would be there now when I need it.

An example - just last Saturday, while waiting at the airport,
I was trying to figure out the time difference between Ann Arbor
and London, England, as well as the flight time simply by using
the known departure and arrival times both ways.  The plane
left DTW at 7:25 p.m., local time, arriving Heathrow at 7:30 a.m.,
London time.  Returning, it left London at 10:50 a.m., London time,
arriving Ann Arbor, 1:50 p.m., local time.  Now, I would have
had to think this out long-hand, if you will.  Whereas John
took pen to paper and had the answer in moments.  There was
a bit of rounding that had to take place in consideration
of the jet stream but algebra worked in a very practical 
application.

I wish I hadn't let my mathematics skills get so rusty.

When I cxan remember what to do, I kinda like algebra, my problem is
that I am usually just plain confused by it.

Re #3: It was a simple problem of two linear equations in two
unknowns. I've taught that stuff often enough that I can do it
in my sleep, almost.

This one had the slight extra complication that it was a
Diophantine equation in one of the variables (i.e. the solution
had to be a whole number), since you don't have fractional
differences in time zones (except maybe in places like Nova
Scotia).

 i*pi
e     + 1 = 0

(I want to attribute to Euler, but I keep getting the feeling that that's
off.  Do you recall, John?)

This response has been erased.

Re #6: I believe it's called Euler's identity, but I could be
misremembering. The equation is interesting in that it connects
five fundamental mathematical constants: 0, 1, e, pi, i.

Anyone ever tried placing 8 queens(ministers or whatever you call) on a chess
board without their powers clashing each other? I guess its an Artificial
Intelligenc(AI) question.

Not really AI. I (and many other people as well) have written
programs to generate and display all such configurations. It
turns out that there are 12 essentially distinct solutions.
(Where by "essentially distinct" I mean that one is not simply
a rotation or reflection of another.)

See Edsger Dijkstra's essay in the book _Structured Programming_
by Dijkstra, Hoare, and Dahl for an excellent discussion of this
problem and its algorithmic solution.

Re:  #8
Alas, no.  Rudin (_Real & Complex_, 3rd ed., p. 2) uses the title "Euler's
                            it  
identity" for the equation e   = cos t + i sin t.  (Meanwhile, Ahlfors
calls that formula "Euler's formula" in _Complex Analysis_, 3rd ed., p. 42.) 
The source I first saw it in (I want to say Rudin, but can't find it in
either _Principles_ or _Real & Complex_) noted the property you did and
attributed the formula to one of the old masters.  

But I now think you're right that it's Euler. 

Re: #7  Carlos is taking a summer term at Cambridge.

Yep, it takes 7 hours to fly to London and 8 hours to
fly back, against the jet stream.

Re #11: Gad, people are still reading Ahlfors' _Complex
Analysis_. That was the textbook when I took complex analysis
in college (the class was taught by Ahlfors as well).

Re: #13
This was Ahlfors *3rd ed.* - (c)1979 - I took it in '86.  You probably used
the original - (c)1953.  (But the 2nd ed. was (c)1966, so you may have been
a guinea pig for an early draft of it or some such...)  

Looking in the Columbia Encyclopedia (5th ed.) under Euler, I found the
formula directly attributed to him with the 5-big-numbers property noted.
But I don't think that's where I originally saw it.  <sigh>  Perhaps my
memory hails from some untracable & nearly forgotten lecture of math 
classes past... 

I think I had a problem similar to that last year.  It was a trick question.

Re #14: Yes, it was the 1st edition. Glad it's made it to a
third edition. Of course, it's a classic textbook.

And, too, math doesn't change nearly as much as, say, physics, which requires
a new book (depending on the specific topic) every year.  It's a pain to sell
back physics books.  I've never been able to sell one back, because they are
always getting a new one.

What level of math/physics are you referring to? Undergraduate physics
hardly changes at all, except for a little decoration with mention of
recent discoveries. Most changes in the selected textbook occur because
of changes in the instructors, who have personal preferences. Perhaps what
you observe is due more to more frequent changes in who teaches physics
than in who teaches math?

This response has been erased.

Books change more often because of fads, trends, etc. in how the subject is
taught.  America's it's-gotta-be-new-&-improved culture requires that the
education establishment come up with different ways to teach the same old 
subject every few years to keep their jobs.  The metric system  gets shoved
in, then fades.  History has to be re-written to prove that all white males
are evil.  Every such change requires new textbooks.

Of my 4 undergrad physics classes so far, only one (Electricity and Magnitism)
had not changed in the previous year. (Granted, I took mechanics in HS, snd
that hasn't changed either)  

(I cannot resist obsesrving that neither electricity or magnetism, or
mechanics, has changed since Maxwell....)

They haven't even changed before Maxwell...  Only our understanding of
them has ;-)

touche....

It *does* strike me as rather interesting that while math laws and physics
never change, and never will, the text books teaching their governance have
to be replaced on a regular basis.

At the high undergraduate & graduate level (at least in math), most of the
educational fashion experts are out of their depth, and its common to see
standard texts run 10-15 years between minor-revision editions.  I doubt
that selling more has anything to do with the revisions - I've seen prof.'s
teach classes from an older edition than what the class has, and far more
texts are kept for future reference by students at that level.

This response has been erased.

Hhhhm. A *lot* of the books we review are reissues of books published
thirty years ago, and a lot more are revisions of books which are
anywhere from twenty to ten years old.  OF course, there are books
which come out every year or tow with new revisions, but that's 
uncommon (we usually get every %$!@ book that isn't too basic, so
I see a lot of math textbooks)...

Re: #27
I went through (mainly theoretical) math from about '85 to '89.  Sub-senior
level texts certainly turned over faster, even in math.  Computer-oriented
math texts seemed to be replaced by a new generation about every 10 weeks 
for a while...

hells bells, newtonian mechanics hasn't changed in 300 years! amazing,
it's still 'new' to students. 
  
i did get a teacher to try teaching  math  as a language, translatable
to/from english. about a year later, she said she was ahving much
better success with er students as a result. i smiled.

Re #30:  That's a great idea, teaching people to translate between math and
English.  I saw a low of college freshmen who had no clue how to do that.

Doesn't anyone have a problem?  I suppose I could dig some out, but most of
my good ones have been used up in previous math items.

Ok, all I have at present are logic problems, but here goes.  I'll go through 
a chapter in Raymond Smullyan's book "The Lady or the Tiger?", which has a lot
of good stuff in it.

-------------------

Many of you are familiar with Frank Stockton's story "The Lady or the Tiger?", 
in which the prisoner must choose between two rooms, one of which contains a 
lady and the other a tiger.  If he chooses the former, he marries the lady; if 
he chooses the latter, he (probably) gets eaten by the tiger.

The king of a certain land had also read the story, and it gave him an idea.  
"Just the perfect way to try my prisoners!" he said one day to his minister.  
"Only, I won't leave it to chance; I'll have signs on the doors of the rooms, 
and in each case I'll tell the prisoner certain facts about the signs.  If the 
prisoner is clever and can reason logically, he'll save his life - and win a 
nice bride to boot!"

"Excellent idea!" said the minister.

THE TRIALS OF THE FIRST DAY

On the first day, there were three trials.  In all three, the king explained 
to the prisoner that each of the two rooms contained either a lady or a tiger, 
but it could be that there were tigers in both rooms, or ladies in both rooms, 
or then again, maybe one room contained a lady and the other room a tiger.

The First Trial

   "Suppose both rooms contain tigers," asked the prisoner.  "What do I do 
then?"
   "That's your hard luck!" replied the king.
   "Suppose both rooms contain ladies?" asked the prisoner.
   "Then, obviously, that's your good luck," replied the king.  "Surely you 
could have guessed the answer to that!"
   "Well, suppose one room contains a lady and the other a tiger, what happens 
then?" asked the prisoner.
   "In that case, it makes quite a difference which room you choose, doesn't 
it?"
   "How do I know which room to choose?" asked the prisoner.
   The king pointed to the signs on the doors of the rooms:


         I                           II
IN THIS ROOM THERE          IN ONE OF THESE ROOMS
IS A LADY, AND IN           THERE IS A LADY, AND
  THE OTHER ROOM            IN ONE OF THESE ROOMS
 THERE IS A TIGER             THERE IS A TIGER

   "Is it true, what the signs say?" asked the prisoner.
   "One of them is true," replied the king, "but the other one is false."
   If you were the prisoner, which door would you open (assuming, of course, 
that you preferred the lady to the tiger)?

----------------------

(Of course I expect a full explanation of your answer!)

Well, I => II, so if I were true, so would II. Hence if one of
them is true and the other false, I must be the false one. So
II must have the lady, and I'd go with that.

  John's logic is hard to argue with..  Can we proceed to the second day's
test?

Very nice, John.

The Second Trial

And so, the first prisoner saved his life and made off with the lady.  The
signs on the doors were then changed, and new occupants of the rooms were
selected accordingly.  This time the signs read as follows:

         I                     II
AT LEAST ONE OF THESE     A TIGER IS IN
ROOMS CONTAINS A LADY     THE OTHER ROOM

   "Are the statements on the signs true?" asked the second prisoner.
   "They are either both true or both false", replied the king.
   Which room should the prisoner pick?

II    cause if I is true then their is a lady in one room and if I is
true the II is also true

if I is false then both rooms must contain a tiger and if I is false
then II is also false, meaning that I doesn't have a tiger in it, which
would be impossible, because I being false says so.

Right:
   Both false => (I) both rooms contain tigers and (II) room I does not
contain a tiger, two statements which are clearly in conflict.  So they can't
both be false, which means (according to the king) they must both be true.
   Both true => (I) either there is a lady in room I or there is a lady in
room II, and (II) there is not a lady in room I.  So there must be a lady in
room II.

The Third Trial

In this trial, the king explained that, again, the signs were either both true
or both false.  Here are the signs:

          I                       II
 EITHER A TIGER IS IN        A LADY IS IN
THIS ROOM OR A LADY IS      THE OTHER ROOM
  IN THE OTHER ROOM

   Does the first room contain a lady or a tiger?  What about the other
room?

BTW the symbol "=>" that remmers used in #33 and I used in #37 means
"implies".  So "A => B" can be read "A implies B", or in other words,
"If A is true then B must be true as well".

- Backtalk version 1.3.30 - Copyright 1996-2006, Jan Wolter and Steve Weiss