Versatile Numbers Part II
WLauritzen
Richard Friedberg, in 1968, (An Adventurer's Guide to Number Theory)
implied
that Pythagorus, around 600 BC, knew these three classes. Friedberg also
suggested that these classes developed because the Egyptians never wrote
fractions such as 11/12. Instead they would write it as 1/2 + 1/3 +
1/12,
never
putting anything but a "1" in the numerator. Also they never used the
same
denominator more than once. As a result all the perfect numbers can be
split
up "perfectly." Six can be split into 1/2 + 1/3 + 1/6 or 6/6. However,
10
can be split up only into 1/5 + 1/2 or 7/10. It's a "deficient" number.
Twelve can be split up
into 1/2 + 1/3 + 1/4 + 1/6 or 14/12. Twelve is "abundant."
If Frieberg was correct, since we don't use Egyptian fractions anymore,
these numbers and their names are an anachronism. I would suggest the
name
"factor sum" for "perfect number." In our OWN number system, using OUR
fractions, perfect numbers are not so perfect.
I must say I don't like the name "abundant number." To the uninitiated,
it
implies simply a large number. I believe "antiprime" or "versatile" to be
much more descriptive. I call these "abundant" numbers "flexible"
numbers.
Continuing my library search, I located an astonishing article published
in
1915
(Proceedings of the London Mathematical Society, Vol 14) in which the
noted
Indian mathematician Srinivasa Ramanujan analyzes what he calls "Highly
Composite Numbers."
Although Ramanujan studied mathematics extensively in India, his only
exposure to modern European mathematics (of his time) was one book on
mathematics. He single-handedly re-derived much of modern (1915)
mathematics
(and a good deal more) by himself. His story was a fascinating but tragic
one.
Here's his definition of a "highly composite number": "I define a highly
composite number as a number whose number of divisors [factors] exceed
that
of all its predecessors." This class of number is exactly what I call a
versatile number.
(In mathematical language: the number n is called highly composite
if d(m) < d(n) for all m < n where d(n) is the number of divisors
[factors] of n.)
The term "highly composite" is descriptive to someone
trained in mathematics, however, I believe the terms "versatile" and
"antiprime" are more descriptive, and should be used in order to
communicate to the largest number of people the character and usefuless of
these kinds of numbers.
Ramanujan was always looking for new ways to do things. He may not have
known of the traditional mathematical paradigm (of abundant, perfect, and
deficient numbers). Or if he did, he decided to explore on his own, and
discovered and classified "highly composite" numbers. As he said in his
now
somewhat famous letter to England in 1913, "I have not trodden through the
conventional regular course which is followed in a University course, but
am
striking out a new path for myself."
Let me give you some idea of the magnitude of his mathematical genius.
With
the help of a computer that calculated all night, I had determined the
first
30 versatiles up to 110 880 (144 factors). Without the use of a
computer, Ramanujan had calculated all the versatiles up to 6 746 328
388 800 (10 080 factors)! I was stunned.
I quickly looked through his table to see if there were
any more dominant versatile numbers. No. There were none. In fact, by
analyzing all this information, Ramanujan was able to prove that the ratio
of
two consecutive versatile numbers tends to unity
(two successive versatiles are asymptotically equivalent).
This means that as the versatiles get larger,
there is less and less chance of there being another
dominant versatile number. In effect, there are no more.
He came to a similar conclusion to mine with regard to predicting
versatile
numbers: "I do not know of any method for determining consecutive highly
composite numbers except by trial."
He described another class of numbers, which he called by the awkward
sounding "superior highly composite numbers." Compared to versatile
numbers,
it's definition is just as awkward: A number n may be said to be a
superior
highly composite if there exists a positive number x, such that d(n)/(n^x)
>=
d(n')/(n'^x) for all values of n' less than n, and d(n)/(n^x) >
d(n')/(n'^x)
for all values of n' greater than n. The first few are:
2, 6, 12, 60, 360, 2520, 5040, 55440, 720 720, 1 441 440, 4 324 320, ...
As you can see, with the exception of "1", which arguably could be
included,
his first six "superior highly composite numbers" are identical to
"dominant
versatiles." Whether he noticed the classification of what I call
dominant
versatiles, I do not know.
Translated into English, this mathematical expression above would be
written
like this:
"A number may be said to be a superior highly composite number if there
exists some positive exponent, such that the ratio of the number of
divisors
of the number to the number raised to that exponent is greater than or
equal
to the ratio of the number of divisors of any smaller number to the
smaller
number raised to that exponent, and the ratio of the number of divisors of
the number to the number raised to that exponent is greater than the ratio
of
the number of divisors of any larger number to the larger number raised to
that exponent."
Still don't get it? Don't worry. It's not exactly obvious. But
Ramanujan's
incredibly pure mathematical mind managed to ferret out and symbolize the
above rather hidden relationship. In fact, a pure mathematician might
look
at this feat with the same awe that a musician might regard a Beethovan
composition.
I prefer to call "superior highly composite numbers" by the name "superior
versatile numbers." Dominant versatiles compose the first seven of these
if
we include the number 1.
In summary, here is a possible scale of versatility:
VERSATILITY (ANTIPRIME) SCALE
DOMINANT VERSATILE : a number with more factors than any number from 0 up
to
(not including) double itself (the first seven of the superior versatile
series).
VERSATILE : a number with more factors than any smaller number (also known
as
Ramanujan's highly composite).
RIGID: A number with only two factors, one and itself (our old friend, a
prime).
For simplicity, I left off the scale the general versatile
numbers. Of course, I left off the scale the "abundant, perfect, and
deficient" numbers as they are archaic. I also left off the scale
Ramanujan's "superior highly composite numbers" because they are so
difficult to define.
This does not mean the superior versatile series is not of interest. It
may
have some significant applicability. In fact, I have thought about using a
number base that keeps changing to the next superior versatile number.
However, it's possible that Ramanujan,
in an effort to impress his European counterparts, simply went too far
with
his "superior highly versatile numbers." Knowledge that is complex and
difficult to remember has questionable usefulness in my mind.
So I am primarily interested in defining numbers that the public,
legislators, and mathematics teachers can easily remember and use. In
other
words, I am interested in large scale social application. I believe all of
the definitions of the numbers on this scale could be understood and
remembered by an average 12-year-old.
Since there are only seven dominant versatile numbers, we have a loose
analogy to the limited number of Platonic solids: the
tetrahedron (four-corners), the octahedron (six-corners), the cube
(eight-corners), the icosahedron (twelve-corners), and the dodecahedron
(twenty-corners). Euclid proved, at the end of his book (The Elements,
one
of the most popular books of all time) that there can be only five of
these
regular, solid
figures.
Notice that the number of corners of these figures are all general
versatile
numbers.
It was partly due to my study of these figures that I discovered
versatile numbers. In addition, so much of geometry and mathematics is
potentially infinite in its process, I was fascinated when I discovered
that
there were only five of these figures. This fact may have led me to
wonder
if there were only seven dominant versatiles.
Although Ramanujan made a breakthough in our knowledge concerning
numbers with relatively large numbers of factors, or versatiles, in 1915,
has
this information found itself into the mainstream of
society? Have the public and legislators used this information to bring
about ease of computation, packing and distribution?
The answer must be a resounding "no," and then some.
G .W. Hardy, the brilliant British mathematician who brought Ramanujan to
England, in 1917 (see Collected Papers of G.W. Hardy, Vol. VII), called
Ramanujan's paper "... the largest and perhaps the most important
connected
piece of work which he has done since his arrival in England."
Ramanujan died in 1920, at a young age, and Hardy went on to live many
more
years, until 1947. I wondered why Hardy did not try to apply this work to
society. However, Hardy himself answers this question: "If
asked to explain how, and why, the solution of the problems which occupy
the
best energies of my life is of importance to the general life of the
community, I must decline the unequal contest." (in the book, The Man
Who Knew Infinity, by Kanigel)
This raises many questions concerning the relationship between
mathematicians
(and scientists) and the rest of society. For example,
whose responsibility is it to take the products of mathematical
intelligence
and apply them to society? Unfortunately, these questions are beyond the
scope of
this article.
I have been told by some, upon reading my theories, that "so and so
already
knows all about that." "So and so" is usually some currently fashionable
mathematician with some popular books on the market. However, when I look
through "so and so's" books I find nothing of the sort. Perhaps hidden
away
in section of one of "so and so's" books is a curious interest in numbers
with
relatively large numbers of factors, but no analysis, and no grasp of the
importance of these
numbers for civilization. For if "so and so" knows all about these
numbers
and their importance, why has the public remained ignorant of them? Why
has
the world continued to slip toward base ten and the "scientific" metric
system?
For example, in 1971, the British switched from half-pennies, pennies,
threepence,
sixpence, shillings, half-crowns, pounds, and guineas (a 1/2, 1, 3, 6, 12,
30, 240, 252, system which uses all versatile
numbers except for 1/2 and 252--but if you double them all,
then they are all versatile), to a decimal, semi-rigid, monetary
system. The versatile numbers were in use when London was the foremost
financial center of the world. The English system of measurement with 12
inches to the foot was in use when the U.S. put a man on the moon. Now
the
U.S. is trying to go to a non-versatile metric system.
It's a noble goal to align all your measuring systems with your number
system, and those who have tried to do so should be thanked for their
efforts. However, measurements should be derived from a dominant
versatile
number.
Why haven't versatile numbers, in a sense the shelters and the great
cathedrals of all our
numbers, been more intensively studied and taught? I believe we've been
surrounded by versatile numbers for so long (months, seconds, minutes,
hours,
dozens, grosses, feet, six-packs, twelve-packs, twelve notes, twelve
pence,
etc.) that we have forgotten about them. And even if one does discover
them,
they are an embarassment to a society that uses "ten," a semi-rigid
number,
as the core of its number system.
I believe that school children should be able
to define and list versatile numbers and dominant versatile numbers as
well
as they do prime (rigid) numbers. They should also be taught to use
versatile numbers in real situations as mentioned at the beginning of this
article. Businessmen, politicians, military leaders, in fact, all
citizens
could benefit from knowing these numbers.
I teach these numbers NOW. I don't wait for approval from some d____
curricular committee. It'll be a snowy day in Miami before they finally
put
it on the curriculum. JUST DO IT. By the time the adminstrators figure
out
that its not an approved curricular item, everyone will already be
teaching
versatile numbers.
In the long term, our numbering system should have a versatile number at
its
core rather than a non-versatile number. In the booklet NATURE'S NUMBERS
and
in the video NUMBERS OF THE FUTURE, I discuss this further, and also
propose
what I believe are more efficient, easily learned numbering systems (not
using 0-9 but entirely different symbols of my own invention) derived from
versatile numbers. I have personally taught these numbers to over 2800
students as of this date; that doesn't include those that have seen my
video.
Again I don't wait for curriculum commitees to vote their approval.
How do we reconcile our human anatomy of five fingers per hand to a
versatile
number base? Well, how many fingers DO you have on each hand? Most
people
will immediately say five. However, I can just as easily say that they
are
wrong, and that we have four fingers and a thumb on each hand. The
five-fingers-per-hand is a MODEL, and like all models is inherently
imperfect.
I say that we lie to our children a little when we tell them we have
five-fingers-per-hand. It's not a big lie. It's no worse than using
newtonian physics or any other imperfect model.
However, the question I want to raise is: Is this the best model? I can
think of two other models that I believe are better.
First, one can maximally extend all the fingers and the thumb of the hand
as
far apart from each other as possible. Outline all of these with a
pencil,
including the wrist. You will notice an approximate circle shape at the
tips.
Draw this circle with a compass if possible. Outline all the forms inside
the circle, and you will see SIX extensions
radiating from a center (like spokes of a wheel), because the wrist is now
included in the circle as
an extension. (Try this.) Now we have six-per-hand, and this model is
just
as valid as the five-per-hand.
In addition, this model more closely corresponds to the spherical nature
of
the earth (and most of the rest of the visible mass in the universe).
It's
my believe that the five-per-hand model helps to fosters the still
existing
subconscious belief in a flat earth (i.e. using the words "up" and "down"
when they have no longer any meaning on a round earth). In fact, this
model
I chose as the cover of the first edition of my book NATURE'S
NUMBERS. (However, since too many people thought the book had something to
do
with palm reading, I have a different cover now.)
Second, one can use the thumb as a counter, and count on the bony segments
of
each finger. Start with the little finger and work out. One, two, three.
Next finger: four, five, six. Next finger: seven, eight, nine. Last
finger:
ten, eleven, twelve. (Try this, too.) A dozen-per-hand. This model, as
some
of you may know, is not my invention, but exists in many parts of the
world
today, despite the current predominance of ten grouping. It exists in
many
parts of the Middle East (see Ifrah's book, From One to Zero), and I found
out recently the the native Indians of Paraguay also count like this.
Both of these models are preferable for the simple reason that twelve is a
versatile number (see my article on antiprimes or versatile numbers), but
ten
is not. Because we are USED TO the ten-grouping system, we tend to think
it
is easier and better.
If we had been taught a twelve-grouping system when we were five orbits
old,
if we had a simple symbology to carry it, we would be better off. For
that
matter, almost any dominant versatile number as a core (base), would be
better than using the non-versatile number ten.
Currently almost 6 billion people think in terms of tens, a non-versatile
number. What would be the synergetic effect if all these people minds
were
in tune with a versatile number system?
(c) July 1995 revised Aug 1995 and Sep 1995 by Bill Lauritzen
(constructive comments and suggestions welcomed.)
WLauritzen
re your comments:
1) thankyou.
2) I assume your joking and thanks again.
3)Have I ever taught real live human beings? Well, actually I was sent
here from the planet Celta to teach Earth how the rest of the Galaxy
counts. This article was just the beginning of my campaign. : )
Scott Senft
Three (3) thoughts on your article:
1. Your math history was fascinating (until, towards the end, you started
frothing at the mouth).
2. Could you write your thoughts up in a łthesis format?˛ Išll buy it
from you and then submit it as my thesis for my masters degree in
mathematics at California State University at Northridge.
3. My God, man! Have you ever taught real-live human beings?
My three thoughts above notwithstanding, I really enjoyed your article!
Scott Senft
Mathematics Student/Teacher
Cal State Northridge
Scott Senft
(WLauritzen) wrote:
I thought my original post was semi-intelligent. What the hell is the above?