HDTenEightyP/NetNews huggingface dataset:
From: clong@romulus.rutgers.edu (Chris Long)
Newsgroups: sci.math,rec.puzzles,alt.usage.english
Subject: Naming Large Numbers (Re: Negative Zero)
Date: 15 Dec 92 10:12:51 GMT
Followup-To: sci.math
Organization: Rutgers Univ., New Brunswick, N.J.

In article <1992Dec12.010711.15778@leela.cs.orst.edu>, Jeremy Smith writes:

>       Some guy in Southern California proposed a method of naming multiples 
> of ten which included a `logical' system for naming very large numbers 
> (the largest being milli-decilli-fiveillionillion).  Does anyone have a 
> reference for this.

There is other stuff I really should be doing, but ...

You are referring to John "The only man infinity fears" Candelaria
(see the 1986 _Guinness Book of World Records_.

I doubt that the following are readily available.  If there's any
real interest I can scan them in (eventually) and convert to text.

References:

Borgmann, D. "Naming the Numbers", _Word Ways_ Vol. 1,
  Num. 1 (Feb. 1968) pp. 28-31

  List from Brooks _Philosophy of Arithmetic_ (pub. 1904) reprinted.
  Largest number given is milli-millillion or 10^3,000,003.

Ondrejka, R. "Renaming the Numbers", _Word Ways_ Vol. 1,
  Num. 2 (May 1968) pp. 89-93

  Largest number given is milli-millimillillion or 10^3,000,000,003.

Candelaria, J. "Extending the Number Names", _Word Ways_ Vol. 8,
  Num. 3 (Aug. 1975) pp. 141-142

  Largest number given is nonacentillillisillillion or
  10^{3*10^{2,700,000,000}+3}.

Candelaria, J. "Extending the Number Names", _Word Ways_ Vol. 9,
  Num. 1 (Feb. 1976) p. 39

  Largest number given is centilli-millimillillion or
  10^{3*10^{300,000,000}+3}.

Ondrejka, R. "From Unillillion to Ultimillillion", _Word Ways_ Vol. 9,
  Num. 3 (Aug. 1976) pp. 177-178

  Largest number given is ultimillillion or
  10^{3*10^{3*1000^{1000}}+3}.

Candelaria, J. "A New Number Nomenclature", _Word Ways_ Vol. 16,
  Num. 2 (May 1983) p. 125-127

  Largest number given is millithreeillion or
  10^{3*10^{3*(10^{3*(10^{3*(10^6+3)}+3)}+3)}+3}, although
  millimillillionillion is mentioned.

Eckler, R. "The Only Man Infinity Fears", _Word Ways_ Vol. 19,
  Num. 4 (Nov. 1986) p. 252-254

  The largest number given is millitwoillimillionillion, which
  can't be written down in standard notation.
-- 
Chris Long, 265 Old York Rd., Bridgewater, NJ  08807-2618

"S.B., with an I.Q. of 161, failed to complete his course of study,
running away instead with his professor's wife."
H. J. Eysenck, _Know Your Own I.Q._
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Student attitudes
Organization: Purdue University Statistics Department
Date: Fri, 11 Dec 1992 22:16:31 GMT

In article <11DEC199208420044@cycvax.nscl.msu.edu> burtt@cycvax.nscl.msu.edu (BRIAN BURTT) writes:
>In article <1992Dec10.164109.17533@hubcap.clemson.edu>, jtbell@hubcap.clemson.edu (Jon Bell) writes...
>>In article <1g6gsjINNou3@agate.berkeley.edu> bh@anarres.CS.Berkeley.EDU (Brian Harvey) writes:

>>>One solution, if you could get past the politics, would be to have the
>>>math dept and the physics dept jointly design and jointly teach a course
>>>for physicists that would be both mathematically and psysically
>>>respectable.  I mean, make the calculus course and the physics course
>>>one single course, maybe with twice the usual class hours, but able to
>>>spend all of those hours on physics this week and math next week, if
>>>that turns out to be appropriate.

>An idea for a college/university large enough to offer several calculus
>sections:
>	Split the calculus sequence into three parts:
>	a) a sequence oriented to physical scientists and engineers that
>targets the math useful to them-- this is essentially freshman calculus
>as we know it
>	b) a sequence oriented to economists, business students, statisticians,
>etc., which give them calculus without relying on "physics" problems
>that are what calc textbooks normally have
>        c) a sequence oriented to mathematicians, that covers the theoretical
>aspects of calculus and offers a very early introduction to proofs and 
>developing maturity of mathematical thought.
>	Before I get flamed:  I not advocating either insulating scientists
>totally from mathematical theory, or mathematicians from the applications
>of their work.  I just think the classes should be a little more relevant
>to those who take them.
>	I am a student, not an educator, so I'm no expert and welcome
>comments and criticism.

This is too much what is done, and it has exascerbated the problem of the
students in the applied fields learning "relevant" computations, but no
concepts.  

The concepts in the calculus course are those of limit, derivative, and
integral; all else consists of consequences of these, and manipulation.
What the students acquire now in whichever calculus course they take is
only the manipulations, unless it is the mathematics honors (NOT advanced
placement) course, and not always then.  These manipulative skills are
quickly lost, and nothing is left.

It does no good to know how to calculate a derivative or an integral
if one does not know what it means.  And it does harm to formulate an
applied problem in a certain manner because those are the ones which
one knows how to solve.  Will an engineering student formulate something
using a differential equation whose solution is unknown?  More likely,
terms will be left out to make the equation standard.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Newsgroups: sci.math,rec.games.abstract
From: neil@progress.COM (Neil Galarneau)
Subject: Re: Game of pentominos
Organization: Progress Software Corp.
Date: Wed, 16 Dec 1992 18:26:11 GMT

torbenm@diku.dk (Torben AEgidius Mogensen) writes:

>martel@marvin.mr.sintef.no (Paulo Martel) writes:

>>After several tries I gave up a combinatorial analysis of the game of
>>pentominos. Would someone point me to a reference, or briefly explain
>>how one could compute the total number of solutions for a grid of a
>>given size (6x10, 5x12, 4x15, 3x20). 

>I saw a paper once that reported the number of solutions to each of
>these rectangle sizes. It used a heavily optimized machine code
>program to exhaustively search for all solutions. I remember that for
>the 3x20 case there are only two solutions barring reflections and
>rotations. These are quite easy to find by hand. The number of
>solutions for the 6x10 case was quite large, but I don't recall the
>number. I also don't recall the title or author of the paper.

>	Torben Mogensen (torbenm@diku.dk)

Ah... Good old pentominoes.

Several years ago, motivated by a passage in one of Clarke's novels, several
of us programmed pentominoes.

It is not very hard to solve it, it is a lot harder to solve it efficiently.

The programming was a lot of fun.

There are about 2100 solutions to the 6x10 case.

A slow program will take 30 seconds per solution, a fast one can take less
than 5 seconds per solution.

We didn't try other board sizes.


Neil
neil@progress.com
From: werdna@cco.caltech.edu (Andrew Tong)
Newsgroups: sci.math
Subject: Re: Would Pi repeat, were it expressed in a base other than 10?
Date: 11 Dec 1992 19:08:43 GMT
Organization: California Institute of Technology, Pasadena
NNTP-Posting-Host: sandman.caltech.edu

G.Joly@cs.ucl.ac.uk (Gordon Joly) writes:


>That is say non-decimal.

Intuitively (not saying much, that is), I'd say yes, but that it would re-
quire a transcendental base....
From: pehanna@vnet.ibm.com (Phil Hanna)
Date: Mon, 14 Dec 92 10:42:11 EST
Newsgroups: sci.math
Subject: Math trivia puzzle

Is it Benoit Mandlebrot?

Phil Hanna
pehanna@vnet.ibm.com
Newsgroups: sci.math,rec.puzzles
From: converse@cs.uchicago.edu (Tim Converse)
Subject: Re: Naming Large Numbers (Re: Negative Zero)
Organization: University of Chicago Computer Science
	<1992Dec15.210004.2556@hobbes.kzoo.edu>
	<1992Dec17.144306.10885@vax.oxford.ac.uk>
	<1992Dec18.104930.4583@sol.ctr.columbia.edu>
Date: Fri, 18 Dec 1992 15:51:07 GMT

In article <1992Dec18.104930.4583@sol.ctr.columbia.edu> slb22@cunixb.cc.columbia.edu (Seth "the Lesser") writes:
>wilcox@vax.oxford.ac.uk writes:

>>         Theoretically speaking, one can. Assuming one
>>can write down pi primitive recursively (I think one can) one can even write
>>your number down primitive recursively, which is as near practicality as
>>mathematicians ever get.

>One can't.  Pi is not a computable number.  This was proven by Alan Turing, I
>think.

>Seth L. Blumberg

	No -- Turing proved that non-computable numbers exist.  Pi
is not one of them, since it is easy to write a program that will
keep cranking out digits of pi for as long as you like.  It's true
that this program will never stop (is that what you're thinking of?),
but that is equally true of a program that computes 1/3.

	What Turing meant by numbers that aren't computable is that there
doesn't exist a program that will keep generating digits of the number
correctly.  Numbers like this must exist, simply because programs are 
countable and the reals are not.

	Tim Converse




--
-------------------------------------------------------------------------------
Tim Converse  U. of Chicago CS Dept.  converse@cs.uchicago.edu  (312) 702-8584
Newsgroups: sci.math
Subject: Re: Alleged shortage of mathematicians (was Re: M
From: kubo@kovalevskaia.harvard.edu (Tal Kubo)
Date: 18 Dec 92 14:25:13 EST
Organization: Dept. of Math, Harvard Univ.

In article <1992Dec17.183124.10150@alchemy.chem.utoronto.ca>
mroussel@alchemy.chem.utoronto.ca (Marc Roussel) writes: 
>In article <1992Dec17.001527.18598@husc3.harvard.edu>
kubo@boucher.harvard.edu (Tal Kubo) writes:
>>  [academia ahead of industry in chemistry]
>
>     I think you had better drop this example and think up another one.
>  [+ opposing view from practicing chemist]

All right.  I'd rather not get entangled here in a discussion of industrial
vs. academic chemistry. It might be of interest on sci.chem, though.

>>Publications aside, I see this as part of a more general problem with
>>private employment, where you miss out on a lot of opportunities to meet
>>people outside your own backyard.  The mechanisms for doing so are much
>>better developed in academia (visiting scholars, colloquia, sabbatical
>>leave, conferences, travel grants, etc).  Employers are gradually becoming
>>more enlightened in this respect, though.
>
>     You must not go to the right conferences.  At SIAM conferences, a
>large proportion of the participants and an even larger proportion of
>the general attendance consists of industrial mathematicians.  This was
>true even at the recent Conference on Dynamical Systems, a field which
>shows much promise but is hardly a mainstay of industrial mathematics at
>this point.

I didn't say there are no means of professional contact, just that these
are better developed in academia.  How easy would it be for someone in
industry to spend a sabbatical year working at a rival firm, for example?

>     Since I'm not a mathematician, it's difficult for me to comment
>with any authority on this matter, but doesn't the very existence of
>SIAM shed some doubt on this part of your argument?  SIAM is supported
>in part by direct corporate sponsorships.  Some companies clearly
>think that fostering contacts among industrial mathematicians is
>important.

This is part of the aforementioned partial enlightenment among 
certain employers. But in general it doesn't go far enough.

-Tal    kubo@math.harvard.edu
From: arodgers@dcs.qmw.ac.uk (Angus H Rodgers)
Newsgroups: sci.math
Subject: Re: Negative Zero
Date: 12 Dec 92 15:53:14 GMT
Organization: Computer Science Dept, QMW, University of London

In <1992Dec12.010711.15778@leela.cs.orst.edu> 
jeremy@atlantis.CS.ORST.EDU ((-| Jeremy |-) Smith) writes:

> 3              three
> 2              two
> 1              one
> 0              zero
>-0              negative zero
>-1              negative one
>-2              negative two
>-3              negative three

Er, what is n + (-0)  (e.g. for n = 0, or n = -0) ?
--
Gus Rodgers,  Dept. of Computer Science, | 
Queen Mary & Westfield College, Mile End | 
Road, London, England.   +44 71 975 5241 | 
E-mail (JANET):   arodgers@dcs.qmw.ac.uk | Post in haste, repent at leisure.
Xref: sparky sci.math:16932 misc.education:5271
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math,misc.education
Subject: Re: The Continuum Hypothesis:  Must it be {True or False}, or Not?
Date: 15 Dec 92 05:20:26 GMT
Distribution: usa
Organization: M.I.T. Artificial Intelligence Lab.


If there were really such a thing as a set, it might make sense
to ask whether the continuum hypothesis is tru or false.

Starting with the early paradoxes and continuing with the independence
proofs, we have learned that despite our gut feelings to the contrary,
there really is no such concept as "set". We have some properties of
sets that we agree on and beyond that we lack the power to make the
concept more precise, although each individual might have certain
opinions one way or the other as to which are the common notions.

It is like the situation with numbers. After millenia of subtle 
introspection, we now know that there is no such concept as number.
We have a number of algebraic systems some of which, by convention,
we refer to as number systems, the elements of those systems being
called numbers (rational numbers, whole numbers, Cayley numbers,
p-adic numbers, complex numbers, real, cardinal numbers and
ordinal numbers numbers). But it purely
a matter of convention which are considered numbers and which not.
It is not the case that one can combine any two numbers using the
operations of arithmetic, since, e.g. there is no canonical way to add
real numbers and p-adic numbers.

In the face of this, it makes no sense to ask whether the square root of
-1 is a number, since the concept of number does not exist. But in
a particular algebraic system it might make sense to ask whether
it contains a square root of -1.

It is the same with sets. It does not make sense to ask whether one
or all sets have a certain property. It only makes sense to ask whether
a given universe has a certain property or whether a given element
of a given universe has a certain property or whether all universes do.

Meanwhile, the concept of a universe will also evolve...

Allan Adler
ara@altdorf.ai.mit.edu
Xref: sparky sci.math:17051 comp.edu:2206 misc.education:5333
From: pcg@aber.ac.uk (Piercarlo Grandi)
Newsgroups: sci.math,comp.edu,misc.education
Subject: Re: Integration  Was: Re: Student attitudes
Date: 14 Dec 92 17:51:29 GMT
	<1992Dec10.212140.5483@massey.ac.nz> <Bz3MMq.5At@mentor.cc.purdue.edu>
Reply-To: pcg@aber.ac.uk (Piercarlo Grandi)
Organization: Prifysgol Cymru, Aberystwyth

On 11 Dec 92 14:23:13 GMT, hrubin@pop.stat.purdue.edu (Herman Rubin) said:

(Scott Dorsey) writes:

Scott> Unfortunately, in most introductory calculus classes, there is no
Scott> attempt at providing a real concept of what integrals and
Scott> derivatives are.  It was in my third semester of calculus that I
Scott> learned that an integral was really the area under a curve, and
Scott> by that time we were doing multiple integration.

hrubin> It is surprising that this wasn't reinforced, most books contain
hrubin> this fallacy.  The area under a curve is a good elementary
hrubin> example of an integral - it is NOT what an integral is.

Most mathematics books are so ludicrously written that no wonder maths
is seen as a dreaded subject. Just notation is usually so needlessly
cryptic.

hrubin> An integral is the best introduced by giving a variety of
hrubin> examples illustrating the idea of slicing up the region of
hrubin> integration into pieces such that the function is approximately
hrubin> constant on [most of] the pieces. Then each piece is just a
hrubin> multiplication. In that sense an integral is a generalisation of
hrubin> multiplication - if a problem reduces to multiplication when a
hrubin> function is constant, then integration is appropriate.

I am not a mathematician, but I cannot keep my trap shout on this. The
usual way I explain integrals is:

	Integration is a purely formal exercise in which a formula is
	transformed into another formula, using particularly odd rules.

	It so happens that the new formula, in several useful and
	usually simple cases, describes what we intuitively perceive as
	the area or volume delimited in some way by the first formula,
	if it seen as describing a geometrical shape.

	That's why the particular transformation called integration has
	been invented and defined that particular way; but having
	defined integration in that particular way largely for that
	purpose, mathematicians found out such definition had many other
	fairly nonintuitive properties.

Incidentally, I am one of those that think that infinitesimal quantities
are a bad idea.
--
Piercarlo Grandi, Dept of CS, PC/UW@Aberystwyth <pcg@aber.ac.uk>
  E l'italiano cantava, cantava. E le sue disperate invocazioni giunsero
  alle orecchie del suo divino protettore, il dio della barzelletta
Xref: sparky sci.math:17234 rec.puzzles:8067
From: clong@romulus.rutgers.edu (Chris Long)
Newsgroups: sci.math,rec.puzzles
Subject: Re: Integral Puzzle (SPOILER)
Date: 20 Dec 92 16:00:29 GMT
Followup-To: sci.math
Organization: Rutgers Univ., New Brunswick, N.J.

In article <1992Dec19.011244.2780@Csli.Stanford.EDU>, Yuzuru Hiraga writes:

> Let f be any function such that:
> 	* for 0<=x<=1; 0<=f(x)<= a

> 	  /1
> 	* |  f(x)dx = A
> 	  /0
> where a and A are given constants (obviously, A<=a).

> What are the maximum and minimum values that

> 	/1       2
> 	|  {f(x)}  dx
> 	/0

> can take?

Let I = \int_0^1 f(x)^2 dx; the fact that I >= A^2 follows immediately
from the Cauchy-Schwarz inequality.  For the other (more interesting)
direction, consider that if f(x) is non-negative, then so is a-f(x),
and so (f(x)+a-f(x))^2 >= f(x)^2+(a-f(x))^2, hence \int_0^1 a^2 dx >=
\int_0^1 f(x)^2 dx + \int_0^1 (a-f(x))^2 dx ==> a^2 >= I + a^2 - 2aA
+ I ==> I <= aA.  To finish, such functions exist.  For the first case,
consider f(x)=A.  For the second, consider f(x)=a for 0 <= x <= A/a,
f(x)=0 elsewhere.
-- 
Chris Long, 265 Old York Rd., Bridgewater, NJ  08807-2618

"S.B., with an I.Q. of 161, failed to complete his course of study,
running away instead with his professor's wife."
H. J. Eysenck, _Know Your Own I.Q._
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Philosophy of Pi
Organization: Computer Science Department,  Stanford University.
Date: Fri, 11 Dec 1992 20:05:38 GMT

In article <COLUMBUS.92Dec11123510@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>In article <1992Dec11.155846.10861@ee.ubc.ca> bdow@ee.ubc.ca (DOW BRUCE
>PAUL) writes: 
>
>   I'm looking for some opinions on Pi. Lately I have been thinking about it
>   a lot. Does anybody know why Pi equals what it does? Why is it an irrational
>   number? Why is it between 3 and 4 in magnitude? Does it stem from the circle
>   alone? What is it about the circle that gives Pi its value? I know Pi can
>   be derived without using the circle (differential equations, infinite series).
>   What determines the value of Pi? Does it have something to do with time and
>   space in a three-dimensional universe? Would Pi have a different value in
>   another universe? Or is it a constant in any universe obeying the laws of 
>   Euclidean geometry? [...]
>
>Boy, are you asking for it in this newsgroup!  But I won't flame; here are
>three answers.
>
>1.  Why does Pi equal what it does?  Well, what do you mean by Pi?  The use
>    of (quick, count on my fingers!) the 14th letter of the Greek alphabet
>    to represent the ratio of circumference of the circle to its diameter
>    is due to Jones, a contemporary of Newton.  (You wouldn't ask, "Why is
>    a robin a bird?", would you?  Maybe, "Why do English speaking people
>    call that feathered thing over there a `robin'?"--- but that's
>    for a different newsgroup.)
>
>2.  Pi was first proved irrational by Lambert, I forget when. [...]
>
>3.  Would Pi have a different value in a different universe? [...]

I think the last formulation does the most justice to the spirit of
this question.  So let's imagine a universe where circles are square.
Then pi used to compute the perimeter would vary between 2.828 (2
sqrt(2)) and 4 depending on where you measured the diameter, and if
used to compute the area would vary between 2 and 4.  The arithmetic
mean of these four quantities is 3.207107 while their geometric mean is
3.084422.  The arithmetic and geometric means of *those* two quantities
is in turn 3.145764 and 3.145166, and after one more iteration they
have converged to 3.145465, within .1% of pi for round circles.

So square wheels should work very well on average.  :-)

Any volunteers to compute pi for octagons or 16-agons?
-- 
Vaughan Pratt				All knowledge resides in the going odds
Newsgroups: sci.math
From: ags@seaman.cc.purdue.edu (Dave Seaman)
Subject: Re: Function Terminology
Organization: Purdue University
Date: Sat, 12 Dec 1992 18:09:48 GMT

In article <Bz48w1.G5C@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>>In article <1gaq3tINNg9q@uwm.edu> radcliff@csd4.csd.uwm.edu (David G Radcliffe) writes:
>>>Suppose I have a function f: A --> B, and C is a subset of B which
>>>contains the image set of f.  I define a function g: A --> C by
>>>setting g(a) = f(a) for all a in A.  Usually, f and g can be considered 
>>>as the same function, but sometimes the distinction is important.

>The function is exactly the same.  A function, in whatever foundational
>system is used, is something which takes arguments in a domain and
>operates on them.  The image set depends only on f and A.  Which
>superset of the range is used does not affect the function.

The point is that that *sometimes* the distinction is important. A definition
that I have seen more than once is:

	A function f is a triple (A,B,GAMMA), where

		1. A is a set, called the domain of f,
		2. B is a set, called the range of f, and
		3. GAMMA is a subset of AxB, called the graph of f,
		   with the property ... (you know the rest).

--
Dave Seaman
ags@seaman.cc.purdue.edu
From: doner@Aero.org (John Doner)
Newsgroups: sci.math
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Date: 16 Dec 1992 00:24:53 GMT
Organization: The Aerospace Corporation
Distribution: world
NNTP-Posting-Host: armadillo.aero.org

In article <1glr2qINN2vg@usenet.INS.CWRU.Edu>, fau@po.CWRU.Edu (Francis A. Uy) writes:
|> 
|> I can't believe I forgot the proof of this one.
|>  Subject line says it all.

First tell us what you want to use for a definition of e.
Xref: sparky sci.math:16963 rec.puzzles:7936 alt.usage.english:9542
From: J.Theodore.Schuerzinger@dartmouth.edu (J. Theodore Schuerzinger)
Newsgroups: sci.math,rec.puzzles,alt.usage.english
Subject: Naming Large Numbers (Re: Negative Zero)
Date: 15 Dec 92 16:23:24 GMT
 <Dec.15.05.12.50.1992.13518@romulus.rutgers.edu>
Organization: Dartmouth College, Hanover, NH
X-Posted-From: InterNews1.0b1@newshost.dartmouth.edu

In article <Dec.15.05.12.50.1992.13518@romulus.rutgers.edu>
clong@romulus.rutgers.edu (Chris Long) writes:

> Eckler, R. "The Only Man Infinity Fears", _Word Ways_ Vol. 19,
>   Num. 4 (Nov. 1986) p. 252-254
> 
>   The largest number given is millitwoillimillionillion, which
>   can't be written down in standard notation.

Why can't it be written down in standard notation?  Unless it's
irrational or transcendental, one should be able to write it down.  It
would seem to me that this number is just ten to some 'kazillion' (my
word for a R-E-A-L-L-Y big number) power.


--Ted Schuerzinger
email: .zed@Dartmouth.EDU
"I should have known it would be bad vodka when all the label said was
'Russian Vodka'."
Newsgroups: sci.math
From: hiraga@Csli.Stanford.EDU (Yuzuru Hiraga)
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Organization: Stanford University CSLI
Date: Fri, 18 Dec 1992 01:13:31 GMT

In article <1992Dec18.010543.11995@Csli.Stanford.EDU> hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:
>Given e = (1 + 1/n)^n, prove that (e^x)' = e^x;
>which boils down to lim(h->0) (e^h - 1)/h = 1.

Sorry, the first eq. is obviously e = lim(n->oo) (1+1/n)^n.

-YH
Xref: sparky sci.math:16907 sci.physics:21162 sci.chem:5404 sci.bio:4575