topics: |  mathematics

Why should -1 x -1 equal +1?

Some time before Zubin was six, he became interested in negative numbers,
and shortly thereafter he started asking me why -1 x -1 should be = +1.
Until then, I had thought that this result must be completely rationally
derivable - but I found that there was no way I could justify, let alone
prove, this "identity.  I asked many people I respected about this, but never
got a really clear answer.  I realized that this is not a trivial statement
by any means - but this is far from obvious to any school child.

This book I had purchased (for $5 from the Dover and Hudson used books store
in downtown Albany) in 1996, a few years after Zubin.  Much later, I looked
it up and found this question being raised in its first sentence.  However,
it isn't addressed until p.55.  Courant's answer however, is couched in his
formalist position and didn't help me with Zubin.  See notes from p.55 below
for an alternative suggestion from Martin Gardner.

This book is of course a classic.  It provides excellent mathematical
treatment leading to formal methods, but also traces the intuition behind
some of these (like the five axioms of arithmetic).  What was important for
me was the discussion of the intellectual histories of some of these
problems, presented more reliably (but perhaps less
colourfully) than in Eric Bell's Men of Mathematics.  Starting with
what is a number, takes some 60 pages to discuss the genesis of rational
numbers, leading to ideas of the infinite, algebras, and finally addresses
issues in geometry and topology before coming to limits and moving on to
calculus, it outlines the basic mathematics and also some of their histories
and how they were clarified through intense debate and discussion, often
lasting for centuries.

Today, formalism appears to be somewhat weaker than in the days of logical
positivism, the time of Courant.  What is the role of formalism in the
future of mathematics?  The answer to this is not clear.  The very idea of
"rigour in proof" and the creative processes by which proofs are arrived at,
have been under challenge since Kuhn, but most impressively in Imre Lakatos'
brilliant Proofs and Refutations, which is actually the history of a
proof, presented as a dialogue (or a drama) between a teacher and several
students.  But the basic human processes underlying proof remains.  Perhaps,
in the end, only that is a proof which you can convince many others about.
But there are rules that you can use in convincing, and these rules may
involve notions of rigour.

Excerpts

The count systems of many languages show remnants of the use of other bases,
notably twelve and twenty.  e.g. in English and German, the words for eleven
and twelve are not constructed on decimal principles but are linguistically
independent.
Similarly, the construct of quatre-vingt for 80 and _quatre-vingt-dix for 90
in french may reflect some ancient 20-based system.
   [this does not appear to be the case for numbers in the Indian languages,
   which are mostly decimal.
   Japanese: the older "counting" numerals in Japanese hitotsu=ichi (1),
   futatsu=ni (2), mittsu=san (3)... mutsu = roku (6), seem to converge at 7,
   nanatsu=nana; (yatsu=hachi,ya=8); etc.  Perhaps this reflects some ancient
   6-based system?].

p.7-9 - interesting exploration of 6-based and binary systems.  Every
schoolkid should do these.

Induction proof of (1+p)^n >= 1+np for p> -1 and n>0; introduction to induction.

Distribution of Primes 25

Prime Generating functions:
    F(n) = 2^2^n + 1 seems to yield primes (Fermat).
      3, 5, 17, 257, 65537
But in 1732 Euler discovered that 2^(2⁵) + 1 is composite = 641 x 5700417.
	[How does one do this factorization?]
In fact, to date it has not been proved that any of the numbers > F(4) are
prime. p.26

     [Mersenne numbers, where F(n) = 2^p, p is prime, is another series
     often yielding primes. Not discussed by Courant/ Robbins. ]

F(n) = n² - n + 41 : generates primes upto n<41
F(n) = n² - 79n + 1601 : generates primes upto n<80

The prime number theorem - that the density of primes #primes / N goes as
1/ln(n) [conjectured by Gauss] - is one of the "most remarkable discoveries in
the whole of mathematics ... for it is surprising that two mathematical
concepts which seem so unrelated should in fact be so intimately connected "
- p.28-29

It took a hundred years before analysis was developed to the point where
Hadamard and separately de la Vallee' Poussin gave a proof in 1896.  This
proof rested on simplifications and other ideas of v. Mangoldt and Landau and
earlier by Riemann.
[This proof was subsequently simplified by N. Wiener.  But it is still not
simple enough to be included in Graham, Knuth Pataschnik. ]

Two unsolved problems with Prime numbers

A. Goldback conjecture:
Goldbach (1690-1764) has no significance in mathematics except for this
problem. proposed in a letter to Euler in 1742.  He observed that for every
case he tried, any even number, except 2, could be represented as the sum of
two primes.   The empirical evidence favouring this conjecture is "thoroughly
convincing".  The problem is that primes are defined in terms of
multiplication, whereas the conjecture involves addition.  p.30.

   [Originally, Goldbach had said: "at least it seems that every number that
   is greater than 2 is the sum of three primes".  Note that here Goldbach
   considered the number 1 to be a prime, a convention that is no longer
   followed. As re-expressed by Euler, an equivalent form of this conjecture
   (called the "strong" or "binary" Goldbach conjecture) asserts that all
   positive even integers >=4 can be expressed as the sum of two primes.
		- http://mathworld.wolfram.com/GoldbachConjecture.html ]

Towards the proof:  In 1931, promising Russian mathematician Schnirelamann
showed that every +ve integer can be repr as the sum of no more than 300,000
primes.  Though this result seems ludicrous in comparison with the original
goal, nevertheless it was a first step in that direction.  More recently,
Vinogradoff, based on Hardy, Littlewood and their great Indian collaborator
Ramanujan, has been able to reduce the number from 300K to 4.

Direct vs Indirect proof:

Vinogradov's proof was an indirect proof - there exists some N s.t. all
numbers n> N are composable of 4 primes, but his proof does not permit us to
appraise N.  What he really proved was that the assumption that infinitely
many integers cannot be decomposed into at most 4 primes leads to an
abusrdity.  Here we have a good example of the difference between indirect
and direct proofs. p.31

B. That there are infinitely many prime pairs p, p+2.  Not even a step has
been taken in this direction. p.31

Negative numbers: Rule of Signs 55

Rule of signs: -1 x -1 = +1

When we introduce negative numbers, we choose (-1)(-1) = +1 to preserve the
law of distribution - a (b+c) = ab + ac.

Otherwise, if we have -1 x -1  = -1, then we get -1 (1-1) = -1 -1 = -2.

It took a long time for mathematicians to realize that the "rule of signs",
together with all the other definitions governing negative integers and
fractions cannot be "proved".  They are created by us to attain freedom of
operation while preserving the fundamental laws of arithmetic.  What can -
and must - be proved is that on the basis of these definitions, the
commutative, associative, and distributive laws of arithmetic are preserved.
Even the great Euler resorted to a thoroughly unconvincing argument to show
that -1 x -1  "must" be equal to +1.   For he reasoned, it must be either -1
or +1, and cannot be -1 since -1 = +1 (-1).

	[As for Zubin, clarly I do not find this satisfactory.  Why should the
	law of distribution have primacy over our intuitions?  In contrast to
	the box based method Courant adopts on p. 3 to provide the intuition
	for the three "laws", this argument completely lacks any
	intuitive correlation. In other words, I can't take a box of marbles
	and demonstrate the distribution property for negative numbers. So the
	really clear answer to Zubin's question remains a problem.

	However, the fact that even Euler tried to prove this in vain, tells
	me that at least I am in good company.  ]

[For another take on this, here's how Martin Gardner had described it:

"What does it mean to multiply a negative by a negative"?  This is a major
sticking point in arithmetic for many people.  The best explanation I have
seen is by Martin Gardner: Consider a large auditorium filled with two kinds
of people, good people, and bad people.  I define "addition" to mean
"sending people to the auditorium".  I define "subtraction" to mean "calling
people out of the auditorium."  I define "positive" to mean "good" (as in
"good people", and "negative to mean "bad".  Adding positive numbers means
sending some good people to the auditorium; Adding negative numbers means
sending some bad people in, which decreases the net goodness.  Subtracting a
positive number means calling out some good people; subtracting -ve num is
calling out bad people (goodness goes up).  Thus, adding a -ve num is just
like subtracting a positive.  Multiplication is like repeated addition.
Minus three times minus five?  Call out five bad people.  Do this three
times.  Result?  Net goodness increases by 15.  When I tried this out on
6-year-old Daniel Derbyshire, he said, "What if you call for the bad people
to come out and they won't come?  A moral philosopher in the making!"
- p. 367-368,  Derbyshire, John; Prime Obsession

Monotone sequences and Limits p.297

Euler series e -

	a(n) = 1 + 1/1! + 1/2! + ... 1/n!

Now n! > 2^n-1, so 1/n! < 1/2^n-1, so

	a(n) < 1 + 1/1 + 1/2 + 1/2² +/... 1/2^n-1
	       = 1-(1/2)^n / (1 -1/2)  < 3

Thus a(n) is bounded from above.  This limit is e.
   [Why it should have such properties like being of relevance in the prime
   number theorem is not discussed - perhaps this is not quite understood
   even...]

The (epsilon,delta) definition of limit is the result of more than a hundred
years of trial and error, and embodies in a few words the result of persistent
effort to put this concept on a sound mathematical basis.   ... In the 17th
c. mathematicisna accepted the concept of a quantity x changing and moving in
a continuous flow towards a limiting value x1.  Associated with this primary
flow of time or of a quantity they considered a secondary value u = f(u) that
followed the motion of x.  The problem was to attach a precise meaning to the
idea that f(x) tends to or approaches a fixed value a as x moves to x1.

From the time of Zeno, the intuitive physical or metaphysical notion of the
continuous has eluded exact mathematical formulation...  Cauchy's achievement
was to realize that, as far as mathematical concepts are concerned, any
reference to a prior intuitive idea of continuous motion may and even must be
omitted.  ... His definition is static it does not presuppose the intuitive
idea of motion.  Only such a static definition makes precise mathematical
analysis possible and disposes of Zeno's paradoxes as far as mathematical
science is concerned.  p.305-06

NOTE: can we then have a similar notion to define "pile of sand"?  How about
  - there exists a bucket s.t. if we remove a pile with this bucket, then the
  result is no longer a "pile"?  But the result is not deterministic.  We
  would have to say - if we remove a bucket, then x% of respondents would no
  longer consider it a pile.  But then, the result, of adding back the bucket
  is not the same as taking it away - persistence of human concepts
  etc... sigh!]

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