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Arithmetic and algebra can profit from geometry, except, perhaps, for those
who like the juggling with numbers, with variables and constants, while
not in the least interested in the meaning of those numbers or something
with which they might correspond in (nonpropositional) reality.
For those who are not too fond of mathematical acrobatics which are
entirely divorced from reality --hard to learn and often
not beautiful to watch--
i* will offer here an
elementary method of presenting product integers as rectangles in a
geometric scheme.
Such a geometric scheme will not only give algebraic equations in this
field some real content, it will later help to better understand and,
perhaps, improve on them.
On the face of it, the term product integer should be clear enough:
it is an integer which is the product of two integers.
But the product of 1 and the number itself is also a product in the
mathematical sense, and allowing such a 'product' in the definition of
product integer would make it a useless term, because it would not
refer to any property which the one positive integer does and the
other does not have.
Therefore, we shall define it as the product of two other integers,
so that there is really something that is or can be divided.
Having said this, it seems that a product integer is, then, nothing else
than what is traditionally called "a composite number", a nonproduct
integer nothing else than what is traditionally called "a prime number",
with the possible exception of the number 1.
However, that conclusion would be jumped to a little bit too fast.
For a product may be the product of two integers indicating a surface area
in two-dimensional space, it can also be thought of as the product of
three integers indicating a volume in three-dimensional space, or a
product of four integers indicating an analogous four-dimensional
quality in four-dimensional space, and so on.
The number 8 is the product of a nonproduct integer (2) and a product
integer (4) in two-dimensional space, and to claim in that context that
8 = 2·2·2 is merely claiming that a surface of 4 can be
further subdivided into two surfaces of 2, because of a product on
a lower level.
In three-dimensional space, however, where products are the product of
three integers, treating 8 = 2·2·2 as a one-level product is
entirely relevant, just as relevant as 16 = 2·2·2·2
is in four-dimensional space.
A three-dimensional product integer such as 8 and a four-dimensional
product integer such as 16 may be 'composite' but, conversely, a 'composite
number' like 6=2·3 is not a three-dimensional product integer, and a
'composite number' like 30=2·3·5 not a four-dimensional one,
let alone one of an even higher dimension.
Interesting as they may be, these aspects we cannot deal with before we
have grappled the simpler cases, and here i shall indeed confine myself to
two-dimensional space.
There is one thing i shall definitely not confine myself to.
It is the use of the base-10 or 'denary' system in the discussion of
product integers.
Some may think that a difference in numeral system used will not play any
role in the reasoning about numbers whatsoever.
They sincerely believe in the simplicity or even 'perfect normality' of
their base-10 numbers.
In my paper
The irrelevance of denary arithmetic and the riddle
of pi's repetend i have already shed serious doubt on this assumption.
In this paper things will only get worse for the ardent devotees of
the ten digits.
It may be a consolation for them that, with the means i have, i will do my
calculations the denary way, but where the base of the numeral system has a
clear bearing on the two-dimensional conceptual framework i will use
the base-4 or quaternary notation of numbers instead, albeit in italics.
In the place-value or 'positional' notation of numbers a natural number s
is algebraically expressed as follows:
s = a0bn + a1bn-1+...+
an-1b1 + anb0
For the denary system the equation is:
s = a0·10n +
a1·10n-1+...+
an-3·1000 + an-2·100 +
an-1·10 + an
Every natural number in any numeral system can be represented by a figure
consisting of one or more units of 1 by 1, and by nothing else than such
units with a surface area 1.
But such a geometric representation of integers builds on sandy ground
exclusively: it provides us with no numerical structure whatsoever, and
therefore with no more insight into numbers and their interrelationships
then without the geometry.
For the first 99 positive integers the denary system suffers from precisely
that: under 100, we may arrange the units in rectangles of 2 by 5, because
2 · 5 happens to be the only product equal to 10, but how
are we to arrange the units in numbers over 999?
At least 100=10·10, so that some nonarbitrary structure can be
created in the numbers from 100 to 999, all of them having at least one
square of 10 by 10.
However, just as there is no square with the surface 10 and integer sides,
so there is no square with the surface 1000 and integer sides.
Do the denary thousands form rectangles of 2 by 500, 5 by 200 or rectangles
with some other arbitrary width-to-length ratio?
The answer is not really worth considering or waiting for.
This is what a number such as 27 may look like on the basis of its denary
notation:
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denary
notational
representation
of 27
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Since 27 is such a small number, we know by heart that it is a product
integer, and that the two-dimensional product is 3 · 9.
But how are we going to represent this geometrically in the denary
system?
Perhaps thus:
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denary
rectangular
representation
of 27=3·9
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In the denary system bn can only be represented by a square if
n is even, in the quaternary system bn can be represented by a
square for any n, and that makes all the difference.
The notational equation for b=4 is:
s = a0·4n +
a1·4n-1+...+
an-3·64 + an-2·16 +
an-1·4 + an
Here we can make every 4n represent a square with a surface
area of 4n and sides √4n.
In this way every natural number can be represented by a figure consisting
of squares only, including 41-surface squares of 2 by 2 and
40-surface squares of 1 by 1.
By carefully following the equation, in which each coefficient stands for
the number of squares of the order concerned, the surface area of all
squares together will be equal to the number.
The quaternary number 123 (denary 27) will now look like:
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quaternary
notational
representation
of 123 (27)
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To geometrically ascertain that such a number is a product integer
we must eventually gain the insight to make it look like:
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quaternary
rectangular
representation
of 123
(27=3·9)
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The notational representation of a number does not guarantee a
rectangular shape for the total number whatsoever, if only because it
includes two-dimensional nonproduct integers (those 'primes' that
distinguish themselves by lacking a property, the property of
divisibility, that is, the kind of divisibility which not all integers
have).
But even the notational representation of product integers is often not
rectangular (as the number 123 above testifies), if only because
this representation starts out from (multiples of) the square with the
largest possible surface area (a 42-surface square in
123).
By rearranging squares so that a rectangle is created with a surface area
equal to the natural number, if necessary after splitting a square up in
four squares of a lower order, we prove geometrically that the number is a
product number.
For in the rectangular representation of an integer the width of the
rectangle corresponds with its smaller factor, the length with its larger
factor.
This can easily be corroborated for 123:
27 = 1·42 + 2·41 +
3·40 =
0·42 + 4·41 +
2·41 + 3·40 =
4·41 + 0·41 +
8·40 + 3·40 =
4·41 + 11·40 =
16 + 2 + 8 + 1 = (2+1)·(8+1) =
(1·√4+1·√1) ·
(4·√4+1·√1) = 3 · 9.
Of course, by playing with numbers in this way, i have already some kind of
universal formula in mind, a formula in which the product of the width of
the rectangle and its length is going to play a major role.
I have been using the word rectangle as a generic term, a hyponym of
square too.
Not only is this much more convenient than having to use the cumbersome
rectangle or square all the time, there is also a good substantive
reason for doing so.
After all, a square is a limit case of vertical rectangles on the one hand
and horizontal rectangles on the other, while having only right angles
itself too.
(There is nothing in the etymology of rectangle which precludes its
length from being equal to its width.)
As it does not make any difference for the theory of numbers what direction
a nonquadrate rectangle has, we shall only work with nonvertical
rectangles here, that is, squares or horizontal rectangles.
In order to prepare for the algebraic gymnastics later on, we should first
do some more practical exercises with examples of the geometrical
representation of two-dimensional product integers.
Unless we want to pay special attention to the relationship the number has
with certain other numbers, we will always start from the notational
representation of the quaternary integer given, and then attempt to build a
rectangle from the set of one, two or three largest quaternary squares in
the notation by extending it to the right and/or downward with smaller
quaternary squares.
The extension to the right we will call "a horizontal extension",
because it will make the total rectangle longer in the horizontal direction.
The extension downward we will call "a vertical extension",
because it will make the total rectangle wider in the vertical direction.
There are also combined, bidirectional extensions:
largest quaternary square
(1, 2 or 3) |
largest quaternary square
(1, 2 or 3) |
largest quaternary square
(1, 2 or 3) |
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I: horizontal extension
II: vertical extension
I+II+III: bidirectional
extension
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Note that the 'smaller quaternary squares in the notation' also include
40-surface squares, in other words, squares of 1
by 1 or units.
The scheme above shows the largest number possible of largest notational
squares --if there are more than 3 largest squares, then the
largest notational square has been split up in four squares of a lower
order-- but sometimes one (largest) quaternary square
suffices, as in the representation of 1000 (64), simply because
1000 = 43 =
√43 · √43 =
23 · 23 = 8 · 8, or
√1000 = 20, the side of the square.
(Mind you, under denary lordship √1000 = 20 may cost you
your head!)
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notational
and rectangular
representation
of
1000 (64=8·8)
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The quaternary number 22 (10) has two largest notational squares,
which are both retained.
Whereas the units would be put to the right in one row in the notational
representation, in the rectangular representation they are put to the right
in one column, so that the two 41-surface squares
are horizontally extended:
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rectangular
representation
of 22 (10=2·5)
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In the number 33 (15) two of the three largest notational squares
are retained, while the third one is split up in four new units.
With the three notational units this makes seven units together which are
arranged in a bidirectional extension of the rectangle formed by the two
remaining largest notational squares:
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rectangular
representation
of 33 (15=3·5)
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Somehow, the 42-surface square of 102 (18) has to go,
for the sides of a 42-surface square are 41 and
these are too long to accommodate the two notational units in a horizontal
or vertical extension.
But
1·42 + 0·41 + 2·40 =
0·42 + 4·41 + 2·40,
and this can be represented as:
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rectangular
representation of 102 (18=2·9)
with a horizontal extension
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One of the 41-surface squares obtained from the higher-order
notational square can in turn be split up into four
40-surface squares, so that we have not two but six units in
total:
0·42 + 4·41 + 2·40 =
3·41 + 1·41 + 2·40 =
3·41 + 4·40 + 2·40 =
3·41 + 6·40.
This is represented by:
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rectangular
representation of 102 (18=3·6)
with a vertical extension
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To prove geometrically that 18 is a product integer both rectangles will do
equally well, yet the one with the horizontal extension is to be preferred
in a sense, since it makes use of a smaller number of units.
A subdivision into units besides the ones given in the notation should only
be done if no other way of accomplishing a rectangular shape is possible.
Again, units do not add any structure to the rectangle or to the procedure
for getting this geometric form.
We may be glad that in the quaternary system the only positive integers to
be represented by units and no larger square at all are 1, 2 and 3.
Nonetheless, when we are not merely interested in proving that 18 is a
product integer, but that it follows from 18's being a product integer that
other natural numbers must be product integers as well, the representation
of 18 with the greater number of units is definitely as important as the
one with the smaller number of units.
This matter, however, is not our present concern.
Clearly, our conscious choice of using the quaternary system in
two-dimensional mathematics keeps us away from the quicksands of other
numeral systems, yet even the base-4 framework cannot prevent us from
having to cut some of the smallest bricks into even smaller pieces of
heat-hardened clay now and then.
We saw it happening with the number 102 (18), but there the number
of different orders of squares in the rectangular representation remained
the same as the number of powers of 4 in the notation, namely 2.
A number like 130 (28), however, will not show the same picture.
Its standard notational representation does not reveal a need for unit
squares:
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notational
representation
of 130 (28)
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Since the number of 41-surface squares is more than 1, we could
also try:
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intermediate
representation
of 130 (28)
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On the basis of such a shape we would have to decide that 28 is not a
product integer, that is, if really no rectangle could be formed.
But the width of the rectangle which is formed by the 42-surface
square and the first two 41-surface squares is
√41 + √41 = 4, and this is exactly the
number of units which can be made out of the protruding
41-surface square.
Hence, there is an all-encompassing rectangle for 130.
It is:
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rectangular
representation
of 130 (28=4·7)
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Algebraically, we have just done the following:
s = 1·42 + 3·41 +
0·40 =
1·42 + (3-1)·41 +
1·41 + 0·40 =
1·42 + 2·41 +
4·40.
And algebraically, this may seem too trivial to write down, but it is only
the algebra of this particular case, which later on will help us to come up
with and to better understand the much more complicated universal
formulas.
In the examples given so far the rectangular representation of 130
is the first one with quaternary squares of three different orders.
The next product integer, 132 (30), even has squares of three
different orders in both its notational and its rectangular
representation:
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notational
representation
of 132 (30)
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rectangular
representation
of 132 (30=5·6)
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This rectangular representation is the first one without an extension of
lowest-level squares to the right but with a necessary extension of
lowest-level squares downward.
(102 or 18 could be represented by a rectangle with either a
vertical or a horizontal extension.)
It is important to notice that in the rectangle with the total area of
the product integer all squares of order n and higher form in their turn
subsidiary rectangles.
It is only squares of the lowest order which do not necessarily form a
rectangle, or merely a rectangle of which the width is 1 (in the case of
units).
The rectangular representation of the number 210 (36) contains an
example of a bidirectional extension with squares of the lowest order which
are not as small as unit squares:
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rectangular
representation
of 210 (36=6·6)
created with one
bidirectional extension
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But even this rectangle can be interpreted as a bidirectional extension
with units of the rectangle for 121 (25) which is itself a
bidrectional extension with units of the rectangle for 100 (16):
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rectangular
representation
of 210 (36=6·6)
created with two
bidirectional extensions
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The difference between the above two rectangular representations of
210 corresponds with a difference in procedures.
We end up with the first rectangle by starting from the notational
representation and splitting one of the two 42-surface squares
up into four 41-surface squares, as s =
2·42 + 1·41 + 0·40 =
1·42 + 4·41 + 1·41 =
1·42 + 5·41 =
1·42 +
(√42/√41+√42/√41+1)·41.
We end up with the second rectangle by starting from the rectangle for a
smaller number, first with the rectangle for 100 (16), of which the
length of 4 units is extended to the right by 1 unit and the width of 4
units downward by 1 unit, yielding a new square, now of 5 by 5.
(The result when starting from the notational representation would be the
same: 42 + √42 +
√42 + 1 = 1·42 + 2·41
+ 1·40 = 121 or 25).
Subsequently, we extend this rectangle bidirectionally again, that is, to
the right by 1 unit and downward by 1 unit, yielding the final square of
6 by 6 for the number 36.
Even when we confine ourselves to two-dimensional space, product integers
may sometimes be represented in different ways.
There are numbers such as 18 which may be represented by two different
rectangles with different widths and lengths (corresponding with
18=2·9 and 18=3·6).
And there are numbers such as 36 for which the width and the length of the
quadrate rectangle is fixed, but of which the rectangles differ internally
on the basis of their formation.
Such variations only enhance the usefulness, if not fruitfulness, of the
geometric approach to numbers.
It stands to reason that with the geometric representations of numbers in
mind, and the conditions which govern these representations, and the
procedures which lead to them, the arithmetic and algebra of
two-dimensional product and nonproduct integers will become considerably
easier and clearer than without.
If the corollary is that it will also put 'prime numbers' in their proper
place (and their traditional name in the annals of history), so much
the better.
67.MNW
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The first-person singular pronoun is
spelled with a small i, as i do not consider myself a Supreme
Being or anything else of that Ilk. |
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