Conference rusure::math

    
    The general answer is that it depends what you mean by infinity and zero.
    
    > (0)(oo)= ?

    I take it you mean - "What is zero multiplied by infinity?"
    The answer is whatever you like depending on how you define infinity
    and zero.

    eg. if x = lim (a/n) = 0, and y = lim (n) = 00    a in R
               n->00                  n->00

    then xy = lim (a) = a
              n->00

    > oo-oo=?

    Let x = lim (a+n) = 00, y = lim (b+n) = 00    a,b in R
            n->00               n->00
    then x-y = a-b, where a, b are whatever (reals) you like.

    > 1^oo=?

    1^x where x>0 is always 1. If you multiply 1 by 1 you only ever get the
    answer 1.

    > 0^0=?

    Although x^0 is usually defined as x1 = x, 0^0 is sometimes defined
    as 1 to make some formulae look nice.

    > oo^0=?

    See last - x^0 = x

    > oo/oo=?

    Let x = lim (n/a) = 00, y = lim (n/b) = 00   a,b in R-{0}
            n->00               n->00
    then x/y = b/a

    > 0/0=?

    Invert the expressions in brackets in previous.

    > oo/0=?

    Let x = lim (f(n)) = 00, y = lim (1/g(n)) = 0, where f,g are polynomial
            n->00                n->00		   functions of n.
    then x/y  = lim (f(n)g(n)) = 00
                n->00

    > 0/oo=?

    Similar argument gives = 0

    > oo^oo=?

    Well really! What do you mean by infinity? Some flavour of aleph null,
    (eg. lim (f(n)), for some polynomial function f)? Some flavour of
    aleph one (the cardinality of the reals)?

    Brian

        For limits, you can use L'Hopital's rule.
        
        Dan

    ref .1
    >> 1^oo=?
    >
    >1^x where x>0 is always 1. If you multiply 1 by 1 you only ever get the
    >answer 1.
    
    that is what i think it shopld be, but according to this text, 1^oo
    is undefined.
    along with (0)(oo),oo-oo,0^0,oo^0.
    
    iam not sure what logic author used to say 1^oo is undefined form, but
    the text is introduction to real analysis, by Robert L. Brabenec, page 155
    
    /Nasser

  lim   (1 + 1/n)^n -> e is of the form 1^oo
n -> oo

Dan

    The older meaning (pre-Cantor) of infinity in mathematics, symbolized
    with the lazy-eight (oo) symbol, meant arbitrarily large.  It only
    has meaning in the context of limits.  It is not a number.  When
    it pops out of an operation as if it were a number (for example:

		    limit x
		    x->oo

    ), the result is said to be "undefined".

    Watch out for simply finding an expression which goes "at the limit"
    to the expression you are interested in.  You need to show that any
    such expression gives the same answer, not just that one does.

				Topher

        Also, not all infinities are the same.
        For example, there are �� and ��.
        
        Dan

    >    Also, not all infinities are the same.
    >    For example, there are �� and ��.
    
    Dan, what do these symbols mean? I'v never seen them before, i looked
    up the math dictionary i have, but nothing there either?
    
    are these like special kind of infinities? 
    
    /Nasser

        It's a joke...they are supposed to represent eyes with
        various shapes of eyebrows. :-)
        
        I went back and put a smiley face in the title...
        
        Dan
        

    oh, ok Dan, you know, i did suspect something was fisshy , but it is just 
    that i believe every thing you say, even though most of times if i find
    it hard fro me to understand ;-) 

    ok, another one, what happens to cos(t) when t=oo ?

    and a history math trivia: who was the first one to give an example of a 
    real function that is continuous but not differentiable? (clue: it was
    long time ago..)

    /nasser

    
    another one, what happens to cos(t) when t=oo ?

    and a history math trivia: who was the first one to give an example of a 
    real function that is continuous but not differentiable? (clue: it was
    long time ago..)

    /nasser
    

I believe it was Weierstrass who first showed how to "construct"
an everywhere continuous, nowhere differentiable function from
the unit interval into the reals.

If not him, then it was somebody else. :-)

Dan

    ref .-1
    i think you'r right. congratulations !
    
    now , another trivia, who was the first to show an example of a continuous
    function that has no derivative at finite number of points?
    
    clue: it was long long long time ago..
    
    /nasser

RE: .10 (nasser)

>    another one, what happens to cos(t) when t=oo ?

    It is "indeterminant" -- which means you get different values depending
    on how you take the limits.

					Topher

          <<< Note 1641.12 by STAR::ABBASI "i^(-i) = SQRT(exp(PI))" >>>
                        -< another math history trivia >-

    ref .-1
    i think you'r right. congratulations !
    
>    now , another trivia, who was the first to show an example of a continuous
>    function that has no derivative at finite number of points?
>    
>    clue: it was long long long time ago..
>    
>    /nasser

Is this also supposed to be a joke? How about a broken line?

Oh by the way, Nasser, who gave the first example of nowhere continuous
function. (It was not so long ago and it's almost trivial.) 

now we can extend the history lesson...who gave the first example of 
nowhere continuous function of finite measure? (OK, OK, you can restrict
yourself to a finite interval.)

Mladen

P.S. All real functions, of course...

    ref .-1
>Is this also supposed to be a joke? How about a broken line?
    
    is this a joke?
    you mean a broken line like this:  --------o      o------      ?
                                               --------
    
    but this is not continouse. 
    
    >Oh by the way, Nasser, who gave the first example of nowhere continuous
    >function. (It was not so long ago and it's almost trivial.)
    
    if it where not long ago, then it cant be trivial , else it would not
    have taken long time to come up with. 
    
    >now we can extend the history lesson...who gave the first example of
    >nowhere continuous function of finite measure? (OK, OK, you can
    >restrict yourself to a finite interval.)
    
    i'll medidate over these questions later tonite.
                                      
    thanks for the lesson.
    regards,
    /nasser
    

    
>    is this a joke?
>    you mean a broken line like this:  --------o      o------      ?
>                                               --------
>    
>    but this is not continouse. 

    No, "zig-zag", like this  ----
                                   \    /
                                    \  /
                                     \/
    for example...    
    The function that describes it, a finite set of connected linear
    segments is not differentiable at any of the "sharp" corners.

    
>    if it where not long ago, then it cant be trivial , else it would not
>    have taken long time to come up with. 

    Either that, or people worked on problems one could use for something.

Mladen    

    since we are on the subject:
    
    imagin all continuouse functions in a bag, what is the probability
    of picking a function out of the bag that is differentiable at
    a one point at least?
    
     answer..
    
    
        almost Zero !
    
    is'nt this amazing?

    ref .14
    >Oh by the way, Nasser, who gave the first example of nowhere continuous
    >function. (It was not so long ago and it's almost trivial.)
    
    I dont know who did, but this is an example of nowhere continouse
    function:
    
    since between every two rational points there is a rational point,
    and between any two irrational points there is a rational point
    (proof is trivial, see below), then define a function so that when x is an 
    irrational point to have value PI , and when x is rational point
    to have a value of PI/SQRT(2) .(why not?)
    
    proof:
    let a,b be two rational points, let |a-b|< epsilon, for any epsilon>0,
    then we can find find a delta such that delta*PI is a point that lies
    between a and b, choose delta as small as you want to make this true.
    
    now let a,b be any rational points, let |a-b|< epsilon, for any
    epsilon>0, we could always find a rational between them by expressing
    a and b in decimal points, and cut of the decimal when it first
    changes.
    
    ie. let a=3.123456789etc..
            b=3.123456788etc..
    then  a point between a and b that is rational is
            c=3.12345678
    
    /nasser

    >ie. let a=3.123456789etc..
    >        b=3.123456788etc..
    >then  a point between a and b that is rational is
    >        c=3.12345678
    that is not right offcourse, one need to chop off all decimals from
    the larger rational number right after a change is found between
    the two rational numbers.
    so c= 3.123456789
    not   3.123456788
    
    i think this looks right
    /nasser

	yeah, you are right, in fact, if you take the real numbers and
	define a function to have value "1" on Q (rationals), and value
	"0" on R\Q (elsewhere), you have your answer. This is obviously
	the cardinal function for the set of rationals. Now you can do
	all sorts of things with it.

	Mladen

First of all, why is .0 0 lines ?  Is that a joke too ?

Here's a slightly different example of undefined infinity.  Most people
have heard this old puzzle:

	Two trains are on a collision course 100 miles apart, each travelling
	100 miles per hour, and a fly is travelling at 200 mph from the lefthand
	train to the righthand one and back again, until finally it gets
	squished between them.  How far did the fly fly ?

The question that brings up infinity:

	Which direction is the fly facing when it gets squished ?

/Eric

    >First of all, why is .0 0 lines ?  Is that a joke too ?

    first of all, math people dont joke.

    second, the note is either deleted by mistake, or something weird
    happened that deleted it.
    I dont I think I deleted it, no reason, but may be I did, I dont know.

    about the fly: this sounds similar to the problem of finding how long the 
    coast line is?

    but I think what will happen is that the fly will get dizzy long time
    before the trains hit each other, and it will fall to the ground
    and wont be squished. but that is based on my intuition only.

    /Nasser
    I spell checked

    .21
    
>	Two trains are on a collision course 100 miles apart, each travelling
>	100 miles per hour, and a fly is travelling at 200 mph from the lefthand
>	train to the righthand one and back again, until finally it gets
>	squished between them.  How far did the fly fly ?
>
>The question that brings up infinity:
>
>	Which direction is the fly facing when it gets squished ?
    
    
              100
     +-------------------------------+
     |->100 m/h           100 m/h <--|
     
     |->200 m/h

the total time before the crash occurs is offcours 1/2 hour.
s=d/t, t= 100/(combined speed) = 100/200 = 1/2 hour.

the time the fly takes, is offcours must be the same, i.e. 1/2.
the time the fly takes is also shown to be sum(n=1..oo) 1/3^n

i.e, sum(n=1..oo) 1/3^n = 1/2

the above applies when the speeds are given as the above diagram

I worked it out, though for general case, and used the above as special
case:

              A1
     +-------------------------------+
     |->V1 m/h           V2   m/h <--|
     |->U m/h (the fly)

this is a general case, the distance covered by the fly is

           A_1 U      A_2 U    A_3 U            A_n U
        --------- + -------- + ------ + .... + -------
           V2+U      V1+U       V2+U            V_n+U

where V_n = V2 if n is odd, or V1 if n is even

now, A_n is given by recursive formula 

                  A_n = A_n-1  - ( V1 t_n-1 + V2 t_n-1)
and t_n is given by
                         A_n       A_n
                  t_n = ------  or ----    depending if n is odd or even
                         V1+U      V2+U

the index n above means after n 'flips' by the fly, to see how this works
apply to special case: (V1=V2=100 m/h,  U=200 m/h)

after one flight by the fly (n=1) :

                               A_1        100
                        t_1 = -------- = ----- = 1/3 hour
                              100+200     300

                              
                        A_2 = A_1 - ( 100 t_1 + 100 t_1)
                            = 100 - ( 200/3) = 100/3

                               A_2       100/3 
               so,      t_2 = ------- = ------- = 100/900 = 1/9 hour
                               V2+U     100+200

continue the process, we see the sequence 1/3,1/9,1/27,....1/3^n
so total time after n 'flips' is the sum of the sequence, i.e the series

sum(n=1..oo) 1/3^n

we see offcourse this converges by ratio test, and the limit is 1/2 .

to see the distant covered by the fly , lets do few iterations:

             A_1 U    100 * 200
n=1,  d_1 = ------- = --------- = 200/3 miles  = 66.666..
            V2+U      100+200

            A_2 U     100/3 * 200
n=2,  d_2 = ------- = ------------ = 300/9  = 33.33..
            V1+U      100+200


so, the sequence is (200/3, 300/9, ....., (100*n)+100
                                          -----------  )
                                             3^n

so after n flips, the fly travles  

                  (sum=n=1..oo) 100/3^n + sum(n=1..oo) 100*n/3^n
this series is convergent by the ratio test.

example after n=1    => 66.66
              n=2    => 100
              n=3    => 114.81
              n=4    => 120.98  miles

so, the answer to the question how long a distance the fly travels, is
answered by asking to what accuracy to want it?

    to answer, to what direction is the fly facing when it get squashed,
    is hard for me to see. since as n->oo, it all depends if n was odd
    or even when n gets to oo !?!
    
    ok, may the fly get killed while it is turning around? and not facing
    either train? humm.. Iam starting to get a headach thinking of this
    part, so iam off to sleep, i'll worry about the direction of the fly 
    more tommorrow.
    
/Nasser
I spelled checked

    
    

    The fly travels at 200 mph for � hour, therefore it traverses 100
    miles.  This is only if we are dealing with mathematical ideals.
    
    In the real world, the fly will traverse a somewhat lessor distance as
    it takes time to reverse direction.  Or perhaps the average speed is
    maintained despite direction changes.

What's the last thing that crosses the fly's mind ?

Answer:

	His rear end.

    ref .24
    you know, what you say makes sense, so where the heck did I do wrong?
    I made the fly travel about 30 miles more than it should?
    
    back to the drawing board ;-(
    
    /Nasser
    I did not spell check

    re .22
    
>>    >First of all, why is .0 0 lines ?  Is that a joke too ?
>>
>>    first of all, math people dont joke.
    
    That's simply not true.  Were you joking when you said that?
    
    Dan

    RE: .26
    
    The first change of direction takes place, as you say, at 1/3 hours -
    the fly having travelled 66 2/3 miles.  The second change of direction
    takes place at 4/9 hours - the fly having then travelled 88 8/9 miles.
    
    In general the Nth change of direction takes place at time T(N), where
    T(N) is given by the following recursive formula;
    
    		T(1) = 1/3
    		T(N) = 4*T(N-1)/3
    
    The distance D(N) travelled by the fly at the Nth change of direction
    is given by the following formula;
    
    		D(N) = 200*T(N)
    
    Since no earthly fly can attain a speed of 200 mph on its own; I submit
    that the assisting agent need not switch the direction of the fly at
    all, so the fly will be heading in its original direction at the moment
    of its demise.

    re .27
    
    /nasser jokes in every note.  Grin and bear it.
    
    Dick

    ref .26
    Thanks , I thought I must have had the error in the recursive equation
    for figuring the distance after n flips, i did not have the time to sit
    and look at this again, but i still kick my head on the wall whenever I 
    see one, of why I did not see that the final distance cant be more than 
    100 miles.
    i got buried in details and did not see the simple thing ;-(
	
    later,
    /Nasser
    I spelled checked
    

    RE: .28, .30
    
    Seems I've botched the formula also.  The corrected one (double
    checked) is as follow;
    
    		T(1) = 1/3
    		T(N) = T(N-1) + 1/3^N

I disagree with the claim about which direction the fly is facing.  Since
the fly is flying faster than the trains are moving, the fly *always* has
some time, although ever diminishing, to turn around and start flying back
again, before the trains hit.

This is why I classified this under the topic of "infinity".  Microseconds
before the collision, the fly is changing direction "quite often".

Do you agree ?

/Eric

re .32
    
>This is why I classified this under the topic of "infinity".  Microseconds
>before the collision, the fly is changing direction "quite often".
>
>Do you agree ?
    
    My intuition agrees with you, Eric.
    
    This seems similar to Topic 1330 where mathematical analysis gives
    consistent answers to a physically impossible situation.

    to change directions, the fly must come to full stop first, which means
    it must reduce accelerations from 200 m/h to zero and then increase
    accelerations from zero to 200 m/h in zero time. this is impossible.

    but assume it not not impossible, then it is possible to find what
    direction the fly is facing when it dies, all what we have to assume
    is that the fly has a certain length, and that the fly on a horizontal 
    axis, hence at time of death, the fly is like this

        
    
                this thing is the fly    
     ))))))))            |             ((((((  <- smoke from train
              \          |            /
             ------+     v     +--------
                   |           |
      --->Train     \    \    /    <---- train
                     | ''''  |
             -o--o---+ ''''  +--o---o---
              o  o       /      o   o
           ====================================    <--- rail for train
                     |-------|
                        Y

    distance Y is known, which is the length of the fly full stretched,
    so, since we know how long it will take for the trains to be Y distance 
    away from each other, call this time Z, we then refer to our recursive 
    equations that tell us after n flips by the fly, m units of time has 
    elapsed, so what we do is keep increasing n until we get to time value 
    that is just larger than the time Z, and then take the n values of 1
    less than it, then see if n is odd or even, or whatever convention you
    choose, but the point is that we can find how many flips the fly have
    done before the trains are distance Z apart.

    this solves the direction of the fly problem.

    /Nasser
    I spell checked

    
    

    Why did the spherical cow swish its tail?

    To get rid of the point flies...

    Sorry!

>    Why did the spherical cow swish its tail?
    
    To draw a Zeno in the air?

I've been assuming the fly has direction but ZERO length for the purpose of
simplifying the problem.  And yes, I've been assuming the fly is capable
of reversing direction instantaneously.

Side riddle:

	Four stiff standers, six dilly danders, and a flip-flop

You get to ask only yes-no questions.