VECTOR ALGEBRAIC THEORY OF ARITHMETIC:
Part 2 (March, 2006)
C.L. Greeno
The American Institute for the Improvement
of MAthematics LEarning and Instruction
(a.k.a. The MALEI Mathematics Institute)
greeno@mathsense.org
ABSTRACT: Virtually
unrecognized by curricular educators is that the theory of Arabic/base-number
arithmetic relies on equivalence classes of whole-scaler vectors. Sooner or
later the recognition will catalyze a major re-formation of scholastic
curricular mathematics — and also of the mathematics education of all teachers
of mathematics. Part 1 of this survey appears in the 2005 Proceedings of the
Background
Part
1 opened with an alphabetic construction of the Peano line of simple Arabic
digit-strings — devoid of any reference to numbers — and extended that
construction to achieve an Arabic
numbers kind of whole-numbers system. Part 1 concluded by disclosing how the
whole-scalers, vector-algebraic structure which underlies the Arabic arithmetic
for whole numbers can serve as a mathematical basis for greatly improving the
instructional effectiveness of curricular education in the whole numbers.
That
development described how the geometric-vectors — along the alphabetic line of
simple Arabic digit-strings — yield whole-number additions/subtractions, and
scaler multiplications/divisions by repeated additions/subtractions. (The geometric-vector
definitions construct those operations simply by counting spaces along the scale of any Peano line.) Imposing those
operations onto that line suffices for the resulting structure to satisfy the
axiomatic definition for systems of whole numbers.
Thereby,
the line of simple Arabic digit-strings becomes its own system of whole numbers
(as well as being serviceable as a “numerals” vocabulary for any other
whole-numbers line). Accordingly, once that concept of Arabic numbers has been achieved, even the very young can freely
use such numbers as the scalers for finite dimensional vector spaces. Even
before knowing how to perform any of the arithmetic calculations, children can rely
on hand-held calculators to count the
spaces — thereby attaining the vector algebra on which Arabic arithmetic
depends, as a rational means of learning the Arabic calculations theorems.
The
most primitive whole-scalers vector spaces were labeled, “inventory spaces”. Those are finite-dimensional, and become
whole-scaler “measurement spaces”
after they are duly partitioned by “conversion
equations” among their unit vectors. In turn, examination of measurement spaces
reveals that the familiar arithmetic of Arabic numerals uses a far more complex
whole-scaler vector-space, and relies on much higher developmental levels of
mathematical sophistication within the learners. It means that early childhood,
calculator-assisted, thorough learning about the vector-algebra of commonplace
measurement systems should precede education in the far more sophisticated arithmetic
of Arabic numerals.
While
inventory and measurement spaces are finite-dimensional, the Arabic numerals
are (singly-) infinite-dimensional vectors — normally truncated into finite simple forms. Generalizing from
measurement-spaces, the Arabic numerals are but the proper vectors within equivalence classes of the base-ten partition of the
singly-infinite dimensional, whole-scalers vector space. Within those cells
occur the “carrying” and “borrowing” operations on which the Arabic arithmetic
rely.
Thus
Part 1 discloses that Arabic arithmetic is simply vector algebra, performed on
the base-ten partition of the space of whole-scaler, singly infinite vectors — combined
with intra-cell, vector-conversions to, and from their equivalent Arabic numerals.
Beyond the vector operations, as such, is the operation of polynomial
multiplication (with conversions) — whereby the arithmetic of the Arabic numerals
is seen as taking place within the base-ten
partition of the space of whole-scaler polynomials.
If
the constructions in Part 1 have previously been published, they are not easy
to find. Probably, that much of the vector-algebra of arithmetic is of strictly
academic interest only within the foundations of mathematics. Therein, the above
cited constructions have their places among the classical (cardinal and
ordinal) constructions of the wholes, the difference-pairs construction of the
integers, the planer constructions of complexes, the quotient-pairs
construction of the fractions, the planer constructions of the rationals, and
the n-al constructions of the reals. The same is true of what follows.
However,
originality is far less important than is significance for curriculum reform. Each
year, millions of humans are being educated in Arabic arithmetic — often in troublesome
ways that mathematically make no common sense — while the learners are quite
unaware that they actually are using a humanly natural version of vector
algebra. Not only are partitioned, whole-scalers vector spaces not currently being
used as mathematical foundations for prevailing school curricula — that context
is not presently seen even in the teacher-education textbooks in mathematics.
So, the mathematical basis for curricular instruction in arithmetic is quite
shabby, weak, and confused — resulting in woefully ineffective learning and
instruction, which undermines students’ mathematical growth during the years
that follow.
Moreover,
schools now are being subjected to growing pressures — and also to strong skepticism
— about pressing “grade 9” algebra into lower grades. So, this mathematical news
that Arabic arithmetic actually is based on a mathematically rigorous,
“kindergarten” version of vector algebra opens the way for a major algebraic re‑formation
of the early school curriculum through whole-number arithmetic. That pre‑destined
reformation surely will begin within a decade, if not sooner.
So
arise two questions: one mathematical, and the other, instructological. First, does that vector-algebraic development
of Arabic arithmetic extend, in an equally natural way, to include the usual
arithmetics for the fractions? Yes — and
that is what the rest of Part 2 is about.
Then,
what does such a mathematical extension say, if anything, about a beneficial,
vector-algebraic re-formation of the core-curriculum through fractions? The
pragmatic need is to make the arithmetic of fractions more common-sensible to
the learners. For sure, part of the vector theory can be advantageously
invoked. At the very least, the vector-algebra of fractions must become a
compulsory topic in the education of all teachers of mathematics. Of course,
there is always the danger that some over-zealous missionaries could worsen the
situation by pressing the formal theory beyond the limits of the learner’s own
level of mathematical maturity.
Introduction
Classically,
the term, “fractions” refers to all number-systems that meet the axiomatic
definition for systems of fractions. It makes no sense to call them “non‑negative
rationals” without first having access to the concepts of “negative” and of
“rational-ness”. The “new math” reformers of the 1960’s chose to use the term,
“fractions” for all quotient-formulas that are expressed in the “over” format.
But the purpose of this paper is to provide mathematical clarification of the
arithmetic for the number system — for which the label, “fractions” suffices.
For
the vector-algebraic theory of fractions, the mathematics, itself, can be
formally (and inconsequentially) presented in a simple and straightforward manner.
But as a conceptual re‑view of school mathematics, useful insights from
vector algebra flow through a more constructive development. Much of the following
discourse is aimed at facilitating the theory’s interfacing with a duly re‑formed
curriculum. So, investing a bit of patience while tediously re‑viewing
some kiddy notions will return some insights that, while seeming to be
mathematically minor, are conceptually critical to the vector-algebraic reform
of the core curriculum.
The
vector theory of fraction arithmetic is a quantitative
theory. It conceptually contrasts to
the traditionally confusing “denominator” theory — which is not likely to be
replaced in public use or in the curriculum. Rather, the vector theory can be
used to greatly increase the learners’ conceptual understanding of traditional
notions — and this is a mathematical sketch of how.
As
with the (finite dimensional) whole-measurement spaces and with the (infinite
dimensional) base-ten numerals, the fractions are here constructed by partitioning
a whole-scalers vector space into cells of equivalent vectors. But this is not
merely an extension of the classical 2-dimensional construction of fractions —
which views fractions as being (essentially slopes of …) the proportion-lines whose
binary-ratios are ordered pairs of whole-numbers, within a coordinate plane. Rather
— as already suggested by the decimal construction of the reals — the vector
space underlying the fractions is infinite-dimensional, and the equivalence classes
are far from being geometric lines in that space.
The
most comprehensive vector-construction of fractions uses a generalization of
the standard decimal-point vocabulary for the decimal fractions — and of every
other n-al vocabulary for the n-al fractions. The fraction vectors are the finitely non‑zero vectors of a
singly-infinite, whole-scalers vector space — clustered into equivalence
classes. The resulting vector-algebraic structure does provide all of the usual
fraction arithmetic. But absorbing all those ingredients into single
vector-algebra takes too much sophistication for it to clearly be applicable
for purposes of improving curricular instruction. A more pertinent re‑view
is through incrementally developing the fractions from measurement spaces.
The key to the
construction is that the phonics for
the classical “over” symbols for
fractions comes directly from the quantities
used in inventory-spaces —whereby “3/4” is often pronounced as “3 fourths”,
which could just as well be written as “3(4ths)” or as “3F”. The “3(j)” expression, introduced by the UICSM
curriculum designers, likewise speaks of quantities — and it can effectively
assist the transition from “3(4ths)” to “3/4”.
As
always, each quantity consists of a denomination
(for all things of its kind), with a (coefficient) numerator depicting an amount of that kind — nD, as with
“3F(rogs)”. The phonic vocabulary presents the fraction denominations essentially
as nths — which presently are not
included in the read/write language of the core curriculum. The vector theory
focuses directly on the nths denominations, rather than on only on their
identifying denominators. Its vernacular includes the harmless, but helpful
frivolity of occasionally speaking in terms of 1ths, 2ths, 3ths, and 5ths.
Empirically,
nths are things, and each numerator speaks of an amount of things of that kind.
When fractions are so viewed as being whole-scaler quantities, serial
combinations in various denominations constitute an infinite-dimensional
whole-scalers vector space. When those vectors are viewed simply as strings of
coefficients, the vector space is seen to be the same as the one that underlies
the base-n arithmetics for the whole numbers. The essential difference comes in
how that vector space is partitioned by the conversion equations among the
unit-vectors — which serve as denominations.
The
stage for the arithmetic of fractions is set by the fractional partition of the vector space — which is where the
conceptual difficulties arise in calculating with the over-numerals. That partition
can glibly be achieved through formally imposing the conversion equations — thereby
equating various infinite-dimensional vectors with each other. The vector
operations being class-consistent, they nicely abstract to provide encompassing
operations among the cells. But
mathematical elegance of that kind would be of little help for purposes of
curriculum reform.
The
troublesome mathematical complications arise in connection with the intra-class
conversions. The ultimate need is to linearly order the cells into lines of
fractions — in such a way that each cell can be located with reference to other
cells. But unlike with
whole-measurements spaces or base-n spaces, not all fraction cells can be
aligned through reference to a single 1-denomination scale.
Instead,
the alignment takes recourse to an in‑common
denomination for each finite set
of fraction denominations. But a conceptual grasp of how that mechanism forces
all fraction-cells into a dense line requires an inductive generalization from
how it works in successive finite-dimensional cases. So, the focus, herein, is
on how such finite-dimensional constructions are successively extended, ad
infinitum.
Along
the way, the role of in-common denominations cannot be substantively detached from
construction of a “multiplication” among the unit vectors — which is not always
done on vector spaces. (Interestingly,
fraction-multiplication of fraction units is seen to be surprisingly parallel
to polynomial multiplication of polynomial units.) Of course, it is easy to
formally and arbitrarily define the multiplication table for the fraction
units. But a more plausible construction must be invoked — at least, if the
common-sensibility of the vector operations is to persevere onto the
unit-multiplications.
So
arises the operation of fractioning
things, by (natural) cut numbers —
notions that are essential in the traditional curriculum, but traditionally are
so thoroughly buried that they presently are of almost no instructional value. For
mathematical nicety, what gets fractioned are the spaces of the Peano scales for each of the fraction denominations —
spaces in which might be placed some fractionible things of any kind. The case
of
In
what follows, the usual vector algebra, as such, will be re-viewed only where
an unusual perspective might provide some additional perspective on the
algebraic re‑formation of the core curriculum. Instead, the major focus is on progressive
construction of the fraction-cells of whole-scaler vectors.
Fractional Measurement
Spaces
Measurement
spaces are partitions of finite dimensional vector spaces — by use of specific
kinds of conversion equations among the unit vectors. Some such spaces have
several “whole” denominations — commonly called “5’s” or “10’s” or “2’s”, etc.
— each of which is equated with natural multiple of one particular denomination. Other measurement spaces have several
“fraction” denominations — and still others have both.
Herein,
the discourse is greatly simplified by neglecting the possibility of multiple whole
denominations — since those were attended in Part 1. Accordingly, we begin with
a singly-infinite, whole-scaler vector space with one dimension, called “W”,
for “wholes”. Following the classical language of vector algebra, all
denominations are expressed either as units
that classically are indexed by whole or natural numbers, as with “xn”— or are expressed as
successive places, [ | | | …, into which the corresponding numerators
are posted. (The standard use of parentheses does not speak of places nearly as
well as do the data-boxes.) Those two vocabularies are interfaced by equating
each xn to a
numerators-string of 0’s and a 1 — with the 1 occurring in whatever place
is numbered, n.
The
set of all whole multiples of a unit vector is a dimension or axis of the
space. Each is a Peano line for that place — discretely ordered, with a min and
no max. Along each such axis, adjacent points are separated by 2-point
intervals — the (line‑) spaces
of that line. The alternating succession of points and spaces is a Peano scale — whereon the multiples of
the unit vector are the marks of the
scale. So, the family of all such axes is a singly-infinite dimensional Arabian counter — portions of which are
viable empirical laboratories for representing the vectors.
In
harmony with the western convention for measurements and for polynomials, the unit-vectors’
indices decrease from left to right. But for our abstract discourse, it will be
convenient to have the indices also reveal which nths are identified with what
places. The compromise is to number the places with negative integers, starting
with ‑1. The succession of unit vectors thus is x-1,
x-2, x-3, … — with each x-n being called, “nths”. To
be sure, so labeling the unit vectors cause the indices to serve as denominators. But as seen below, their formal
use for numbering the places does not imply their visible use within the vector
language.
Within
the singly-infinite, whole-scalers vector space, the vectors to be used as fraction vectors are the “finite” ones
— those having at most a finite number of non‑0 numerators. Those
constitute a singly-infinite dimensional subspace. But for convenience, as with
the decimal-point vocabulary, the fraction vectors normally are truncated after
the lowest denomination that has a non‑0 numerator.
That
entire subspace of fraction-vectors is partitioned by the conversion equations, x‑1 = n(x-n). Use of the phrase, “fraction vectors” implies that
the whole-scaler vectors are clustered into equivalence classes, through those
conversion equations. The arithmetic of
fractions is concerned not so much with the usual (and simplistic) vector
operations with the vectors, as such, as with determining which vectors are
equivalent to each other. That is what
motivates this approach of developing the arithmetic through the context of
measurement spaces.
Within
the so partitioned subspace of fraction vectors, the fractional measurement spaces use only finite-dimensional subspaces
— always including x-1 as
the largest denomination — and always, there also is another one that a
smallest denomination for that space. As
with the whole-measurements spaces, each denomination is a natural-number
multiple of that space’s smallest denomination.
For the novice, the latter condition greatly aids progressive
construction of the number system — by gradually easing toward fluency with in‑common
denominations.
As
before, if the selected dimensions constitute a regular system, each non-smallest unit is a scaler multiple of the
next smaller one. If a regular system also is consistent, all of those multipliers are the same — as with the
commonplace “yardstick”, whose denominations make it actually an “inchstick”. Of
course, the total family of all successive fraction denominations is not even
regular — and that is the mathematical source of the curricular dilemma with
fractions.
Simple Fraction
Scales
The simplest fractional measurement
spaces have only two denominations — x‑1 and,
for some natural n>1, x-n. Our development begins with partitioning
those 2‑dimensional vector spaces.
When the largest (x‑1) denomination is called W (for wholes), the x‑2 denomination may be called H (for halves). That puts the
vector-algebra language within easy reach even of those too young to spell the
words. Notice that along each axis, the additions/subtractions of
quantities, and their scaler multiplications/divisions are fully natural — as
with 5H+4H , and with 3·7H. For sure, that much of the theory is quite
helpful to childhood students — because using that “phonic” vector-algebraic
quantity-format initially circumvents the confusions that are induced by the
appearance of numerical denominators.
That [W | H ] space consists of binomial vectors — as with 7W+5H —
among which the vector operations are naturally inherited from inventory
spaces. But the H-dimension truly becomes “halves” only after the conversion
equation is imposed: 2H = 1W. It forces the equivalence of each
W-quantity with exactly one H-quantity — whereby, the family of all aW+bH
vectors is partitioned into equivalence classes.
Each cell has exactly one “H-vector”, 0W+bH. So, the Peano
ordering of the Halves-axis induces a corresponding order of the
equivalence-classes. Each W-cell has exactly one aW+0H vector — but between
successive W-cells is exactly one non-W cell. In effect, that line of equivalence classes overlays the H-axis, onto the W-axis. That process inserts 1H into the [0W, 1W]
space on the W-scale (actually, by inserting 0W+1H in between 0W+0H and 1W+0H —
which has been equated with 0W+2H). That same construct likewise injects
exactly one H-quantity into every W-space — thereby fracturing each whole-space in(to) 2 (spaces).
For that passage from the W-scale, to the W&H‑scale, the conversion factor, 2, is the cut number for that construction. Although essential for converting fraction quantities,
cut numbers are not overtly attended in the traditional curriculum — much to
the conceptual loss of the students.
The resulting Peano line of cells — the W&H fraction‑scale — has its own line-spaces, and thus
its own scale of alternating marks and spaces. Along that scale, each cell is a
mark — and each comprises a single proper
vector which locates that cell as
being a W‑cell, or as being between successive W‑cells. Thereby,
each of the W&H‑scale’s spaces
is numbered by the proper vector for its lower end-point.
Each cell also has a unique reduced
vector whose numerators-sum is
the minimum among all numerator-sums for that cell — which establishes the
intra-cellular conversions of “carrying” and “borrowing”, as with 7W+5H
“carrying” to 9W+1H. Notice that the reduction is strictly about converting to
smaller coefficients — not about the “location” of a vector, or of a cell. But
for 2-place systems of this kind, each cell’s reduced form is also its proper
form.
That construction of the W&H-scale illustrates the general
nature of “fractioning” any Peano scale, by any natural number, done by
overlaying another scale onto the original. But the simple-fractions scales are those in which the W scale has been
fractioned by the nths scale — resulting in a discrete line of
equivalence-class cells of proper or improper “mixed” binomials. For more scales of the simple kind. BROY G BIV offers a colorful rainbow of 8‑dimensions,
for generating seven simple scales. For even more learning-space, English offers
26 alphabetic dimensions — notably including the (daily) 24ths-scale.Such
families of 2-dimensional spaces give ample room for
developing the needed vector operations and conversions — even for clarifying the
denominators-vocabulary for fractions. Of course, the cross-combinations of non-W
denominations do not yet surface.
The
immediate instructional significance of the construction of the family of simple
fraction scales is that — even before introducing numerical denominators — much
headway can be made into the arithmetic of fractions, by exploring the vector
algebra of simple scales. Their constituent binomial
vectors already are widely used in the core-curriculum. But at present, the
visual display of numerical denominators obscures the equivalence classes of
binomial vectors — and conceals the fact that “doing fractions” is an arena of
“doing algebra”. Evidently, premature curricular injection of numerical
denominators (in the standard “over”‑vocabulary for fractions) has long
hidden the underlying, more conceptual vector algebra of fractions.
Fractioning Fractions
As a portal into the general
theory, the simple-scales provide the conceptually crucial notion of fractioning any Peano scales and its spaces
— by any natural cut number. In the simple cases, each W&nths scale is got by overlaying the nths-scale, onto the W‑scale
— done by partitioning the [W|nths] space, relative to the conversion
equation. But complications can arise when trying to extend such constructs by overlaying,
onto the W&nths scale, a third scale — to achieve a W&nths&mths‑scale
of cells of equivalent trinomials.
So, from the entry through simple fraction scales, the development
must somehow explore the arena of whatever nths‑scales can be overlaid
onto which others.
In order to achieve that, the overlaid mths scale must fraction
the nths scale by some cut number — meaning that m must be a natural multiple
of k. Accordingly, the natural
refinements of the nths‑scale are those having denominations (2n)ths,
(3n)ths, etc. Such refinements are tacitly
used when cutting 3(4ths) into 6(8ths), 9(12ths), etc.
A serial overlayment of
several scales is most easily done when each newly appended scale is a natural
refinement of the preceding one — and therefore of all that occur earlier in
that series. Of course, the existence of infinite series of that kind
guarantees the existence of dense lines of fractions.
For the finite-dimensional series, such a refinement chain of invoked dimensions constitute a regular system fractional-measurement. Those
constitute a nice next-step in the development, because they are structurally
identical to the regular whole-measurements spaces — both of which are naturally convertible measurement
spaces. The only conceptual difference lies in which denomination is to be
called “W” — with all larger denomination being “whole” denominations, and with
smaller denominations being fraction-denominations.
It means that, through vector algebra, there is a natural
transition from regular whole-measurement spaces to regular fractional-measurement
spaces. In the latter, every vector equates to a quantity in the smallest
denomination. Using those, the regular measurement spaces accommodate much
learning about ordering, addition, and subtraction with multiple-denominations.
The regular spaces also are natural arenas for generating the
“multiplication” of denominations by successive refinements — as mths of nths give (mn)ths, by using m
as a cut number for fractioning the nths-scale. In that context, the
refinements can be used for constructing the regular system and for converting
the vectors — without yet broaching the search for in‑common
denominations. Of course, the special case is the consistent systems, including the finite decimals:
aW+b(10ths)+c(100ths)+….
Greater complications arise when trying to join two nths-scales when neither index is a natural multiple of
the other. The search arises when trying to construct non-regular systems of fractional measurement. The [W|2ths|3ths] space cannot be
partitioned into a measurement space, since the smallest denomination is not a
natural multiple of all others. The minimal encompassing space is [W|2ths|3ths|6ths]
— which does yield a scale of equivalence classes. The [W|3ths|4ths]
space is completed by appending
12ths, or 24ths, etc. But the minimal full
system of measurements from [W|3ths|4ths] includes not only the join
of the proposed scales, but each factor of that join: [W|2ths|3ths|4ths|6ths|12ths].
The [W|2ths|3ths|4ths|6ths|12ths] full-system completion of
the [W|3ths|4ths] subspace enjoys a realism that cannot be matched by
merely calculating the LCM. Plus, the process tacitly requires conceptual identification
also of the cut numbers that are needed for the various conversions. So, the
vector algebra enables even the most complicated aspects of fraction arithmetic
to be resolved within the limited confines of finite-dimensional measurement
spaces.
[This development further discloses that the search for in-common
denominations is not strictly a fraction-theoretic problem. Consider the
non-regular, [50’s|20’s|10’s|5’s|W] subspace of the
Of course, the multiplication of fraction-vectors with each other entails
the crossing of each quantity of one, with all quantities of the other — by
using denominations of the first to fraction denominations of the second. But access to in-common denominations allows
that to be more simply done by simply crossing two single quantities. Likewise, the division of one fraction vector
by another could be unduly complicated. Instead, converting both to an
in-common denomination reveals that the quotient is a scaler — the quotient of
the numerators — and that the division of fractions actually amounts to finding
a per-unit ratio.
Ad Infinitum
While all of the above can be achieved through the vector-based,
finite-dimensional measurement spaces, no one space of that kind can
accommodate all of the fraction-vectors. Finite-dimensionality must be outgrown
— and with it, the presence of a smallest denomination that is in-common for
all others.
However, even in the infinite-dimensional space of all “finite”
fraction vectors, operations on any finite set of them can be done in a
finite-dimensional subspace. Moreover, each
finite family of successive fraction denominations — W, 2ths, 3ths,
et al — has its own join, which is a refinement of all of those scales. Since
every fraction denomination eventually occurs within one such
finite-dimensional space — and in all subsequent spaces, the scale-alignment is
preserved, thereafter. It means that the full line of fraction-vector cells is
dense.