From: trebla@vex.net (Albert Y.C. Lai)
Newsgroups: alt.algebra.help
Date: Sat, 17 May 1997 15:23:42 -0400
Organization: InterLog Internet Services
In article ,
R Clanchy 1 wrote:
>Prime and composite numbers form a convenient dichotomy until the
>problem with 1 arises. When I first met prime numbers we were all
>allowed to happily believe that 1 was prime. The slight hiccup with the
>prime factors of 6 being 2 * 3 and not 1 * 2 * 3 was easily enough
>accepted as the 1 was not necessary.
These all point to the necessity to introduce a third class, "units",
and we will have a trichotomy instead: units, primes, composites.
Intuitively, units are what you can throw in or take away from a
factorization without feeling uncomfortable, like 2*3 = 1*2*3.
Allow me to introduce negative numbers. Then there are several
factorizations of 6: 2*3, (-2)*(-3), (-1)*2*(-1)*3, (-1)*2*(-3), ...
You do not feel uncomfortable about that; they still look strikingly
similar to you. Why, 1 and -1 do not introduce "new values", do they?
The formal definition of "unit" is --- think about it --- anything
admitting a reciprocal in the number system you use. We are confining
ourselves to natural numbers or integers. No one has a reciprocal...
except 1 and -1, of course. What a coincidence.
The coincidence looks even more stunning when you think about it.
Why do you think that 2*3 and (-2)*(-3) look alike? Because you can
cancel out the -1's. Why can you cancel them out? Why, because
precisely -1 has a reciprocal!
It is pretty clear that it is insensible to talk about factorizations
of 1 and -1. Well, it is sensible, but you can't ask for uniqueness.
If you don't think it is because they are units, consider this.
Working in the rational numbers, factorize 3/4.
You can't, right? Well, you can, but you feel that your leg is pulled.
3/4 = 3/4 = 3 * 1/4 = 3/2 * 1/2 = 5 * 3/20 = ... They do not look
alike. Which one should you prefer, or is it just a skill-testing
question?
The problem is that 3/4 is a unit (its reciprocal, 4/3, is still a
rational number), so its factorizations cannot be unique. 3/4 = 4/3 *
3/4 * 3/4 = 4/3 * 3/4 * 4/3 * 3/4 * 3/4 = ... And there are many more
units in the rational numbers, so that you can synthesize a "different"
factorization by introducing units. For example, I came up with 5 *
3/20 by considering that 3/4 = 5 * 1/5 * 3/4, knowing that 5 is a unit.
Finally, there is no prime --- basic building block --- in the rational
numbers, because everyone is a unit.
Recently, someone in this newsgroup considered subsets of rational
numbers in which unique factorizations make sense. An example is the
integers augmented with the rational number 1/2. Let's call it R.
Every member of R can be expressed in a compact form.
R = { k/(2^n) | k is integer, n is natural }
(My natural numbers include 0.)
Anyway, it can be shown that R admits unique factorizations. I will
not prove that (but I can), instead I will show an example of
factorizations in R. Before we begin, let us recognize the units. 1
and -1 are always units. 2, -2, 1/2, and -1/2 are the new kids,
obviously. In fact +/- 2^n and +/- (1/2)^n, for each natural n, are.
There are no more:
Suppose k/2^n is a unit, then it has a reciprocal j/2^m, so that
k/2^n * j/2^m = 1 multiply by 2^n and 2^m
k * j = 2^(n+m)
where k and j are ordinary integers. Unique factorization in the
integers implies that k must be a power of 2, times 1 or -1 (units
of integers). Thus k/2^n must be also a power of 2, times 1 or -1.
So in R we cannot ask for a unique factorization of 4, because it is a
unit: 4 = 2*2 = 1*4 = 16 * 1/8 = ... Contrast this to the integer case
4 = 2 * 2 = (-2) * (-2).
However, 3 is still a prime in R. That 3 = 3/2 * 2 is not a problem
because 3 = -3 * -1 has never been a problem. 2, 1/2, and -1 are
units. This also shows that 3/2 is as prime as 3, just like -3 is as
prime as 3. If you are still with me, 6 is also a prime, since 6 = 3*2
where 2 is just a unit.
We call 3/2 and 6 associates of 3. (Or 6 and 3 are associates of 3/2,
etc.) Formally, two numbers are associates iff by multiplying a unit
to one you can get another.
Let us factor 9. We know that 3*3 is ok. So is 3/2 * 6, so is 3/4 *
12, etc. Even 3 * 6 * 1/2 will do. We still say that the
factorization is unique, because the only differences are ordering and
units. The primes used in one version are no more than associates of
the primes used in another version.
The definition of "prime" must be revised to allow for the new changes.
The revised version is still convenient to use in the case of naturals
and integers.
1. A prime is a non-unit, and the only way to split it into x*y is to
have x a unit or y a unit. Another way to say the same thing: a
non-unit, and its only divisors are its associates and units.
2. A prime is a non-unit, and whenever it divides into some u*t it must
divide into u or it must divide into t.
(We must also add that they are non-zero's.)
It has been taken for granted that the two are equivalent. In number
systems admitting unique factorizations, the two are really equivalent.
But there are bizzare number systems in which factorizations are really
not unique, and there the two definitions are different. In this case,
the first definition is called "irredicible", the second "prime".
Educationists, (like, the education authorities), in their infinite
wisdom, decree that we will teach only the prime-composite dichotomy,
not the unit-prime-composite trichotomy. In their holy vision, the
only "useful" number systems are the naturals, the integers, the
rationals, the reals, and questionably the complexes. In these
systems, either there are almost no units, or almost the whole system
is units, so it is not "educational" to introduce an "unnecessary"
term.
Many of us have played "war games": you know, using tokens to represent
armies or ships or airplanes, on a board representing a 2D map of the
battlefield. The discretization of the battlefield is unorthodox:
regular hexagons are used to tile the map. One day, a friend on
the phone posed the problem of computerizing this hexagonal board.
Within half a minute, I recalled something I learned from university
algebra classes.
Augment the integers with the complex number -1/3 + i/2. You recognize
that it is one of the most important roots of the polynomial x^3 - 1.
Let w = -1/3 + i/2. Pretend that w is the Greek letter omega. :)
Consider the number system
W = { b + c*w | b, c are integers }
Try to plot this lattice on a paper or computer screen. You will see
that you get the centres of the hexagons you need. Each cell on the
board can be identified with its centre; the centre obviously comes
with a Cartesian coordinate (x,y) or a complex number x+iy, but is more
conveniently expressed as b + wc, where b and c are humble integers.
Thus the data structure emerges: to name a cell, use a tuple **
representing b+wc; to store the entire board, use a 2D array
Board(1..100, 1..100) or something, where Board(i,j) stores the cell
*** at i+wj.
We still need to move from a cell to another. Pretend that you are at
cell <0,0>. It has six neighbours in the six directions: 1 = <1,0>,
1+w = <1,1>, w = <0,1>, -1 = <-1,0>, -w-1 = <-1,-1>, -w = <0,-1>. I
named them in the counterclockwise order. By translation, if you are
at any cell, adding one of those six numbers to your present position
will move you by one cell in the corresponding direction. For example,
suppose you are at b+wc = *** now, then (b+1)+w(c+1) = ****
will move you in the 60-degrees direction, because you are adding 1+w.
A computer programmer can easily take these as basic rules of
operations and codify them into a program of war game. More
computation is done than manipulating rectangular grids, but it works
elegantly. And so much for educationists deciding what we need to
know.
BTW, unique factorization makes sense in W.
>The idea seemed to be to find the
>set of numbers which could not be formed by multiplying smaller numbers
>together. 1 fits into this category. As I understood it then, the name
>prime implies that these numbers are needed to start with, once they are
>in place all the composite numbers can be formed from them. (I simply
>don't understand it now.)
>
>There seems to be no simple, reasonable explanation. I would be
>grateful if anyone can convince me (within the realm of the natural
>numbers only).
Oops, ok. Finding numbers not reducible to smaller numbers is not
quite the same as finding numbers needed to form larger numbers. I
guess the traditional definition of "prime" tries to capture the
latter. Sure, 1 cannot be split into smaller numbers, but it never
gives you any larger number either. Units are a sore point in any
study of factorization. More reason to give them a name --- unit ---
because they are a sore point, no matter how rarely they occur.
>Also, why is zero not a natural number, (even if it was rather an
>afterthought in the big bang theory of mathematics)? The football
>scores, amongst many other non-negative concepts are so peculiar without
>it.
It is just convention. But lately you find that this convention is
inconvenient. I support 0 to be included into the naturals, as do a
number of computer science books. The ancient Greeks even excluded 1
from "numbers": "number" started with 2. This is still reminiscent in
English and probably other European languages. I have a number of
wives. What is your first impression? Your basic instinct tells you
that I must have at least 2 eh? As if 1 is not "a number" eh? And if
I now tell you, I am still single actually, you will royally object:
"you can't say you have a number of wives when you have none!" So
there.
It also depends on the topic. If you just want to talk about prime
factorizations, you would kill to get 0 out of the way. It only annoys
you.
>Another annoying question someone asked me relates to the convention of
>labelling the transformation of the point A as A' which I was always
>taught to read as "A prime". This is simply an overloading of the word
>prime but as my questioner pointed out, it can be a source of confusion
>for pupils. Does anyone else read "A prime" for A' and if so, is there
>a good reason for it?
I read it as A primed, or A prime if I am lazy. If the word "prime" is
needed for other purposes in the discussion, I will call it A dashed,
or A dash. To be exotic, you could call it A apostrophe too. :)
**