Extrinsic Numbers Theory
Added June 20, 2008
I. What are numbers
Whole numbers in the first instance may appear as multipliers to
say or describe how many units of measure or unit vectors are
present. These whole numbers may be given on as marks on a tally
stick or as decimal notation on paper.
Decimal Representation of Whole Numbers: The assumption
that regrouping of elements in a set does not change the number
provides a justification for the decimal representation of
numbers in which elements are counted in terms of units, then
groups of ten, then further powers of ten with coefficient of
each group being digits in the range 0 to 9.
Proper and Improper fractions too may be regarded as multipliers to
describe how many and how much of a unit of measure is
required. Which multipliers are applicable depends on
the divisibility of the unit - is the unit divisible? Can it be
divide into 2, 3, 5 or an unlimited number of parts of equal
value? Finite decimals are examples of fractions - fractions
that have denominators equal to a power of ten or have the same
value as a fraction with a denominator that is a power of
ten. Finite decimal expansions can be identified with an
improper fraction or with a whole number plus a proper fraction
where the denominator in the latter is a power of ten.
Completion: Infinite decimal expansions with tails,
periodic and non-periodic, via a limiting process provide further
multipliers to describe how many or much. The non-periodic
expansions extend the concept of multiplier from whole and
fractional to irrational.
The foregoing multipliers in the first instance do have signs
(prefixes to provide a sense of sign and direction). Unsigned
numbers may be used to measure, order and compare amounts.
Unsigned whole numbers may also be used to indicate position in
queue: first, second, third and on on.
The addition and subtraction of collinear vectors (displacements)
is easily defined and illustrated along with the additive inverse
of each. Next multiplication of vectors by whole numbers,
fractions and in the decimal limit (convergence assumed) irrational
numbers.
Signed Numbers: Let plus and negative
signs are employed as prefixes in superscript position.
Then signed (superscript, prefixed) numbers (signed multipliers)
+N and -N can be used as a
coordinates along a number line. They can also be used as
unit vector multipliers. Here the multiple N or
+N of a vector corresponds to a sum in which
the vector occurs as the only addend, N-times while
-N times a vector is given by the additive
inverse of N-times the vector.
II: Operations on Multipliers - Extrinsically Implied
In the following, a unit refers to a unit amount or a unit
vector. The size of the unit is arbitrary. Changes of unit are
possible.
The introduction and use of numbers as multipliers (coefficients)
turns them into coefficients. For example 5 meters
points to the multiple 5 of meters. In general, we speak of N
units, where N is a multiplier.
Arithmetic operations (+,- *, /) with whole numbers and fractions,
and the decimal representation of whole numbers alone or in
denominators and numerators, stem from physical operations on
multipliers of unit amounts or vectors.
The concept or definition of addition of multipliers M and N stem
from the physical addition of M units with N units and the question
of how describing the result as K units where K is expressed in
terms of the multipliers M and N.
The concept or definition of subtraction of multipliers M and N,
when M < n, stem from the physical subtraction of M units with N
units and the question of how describing the result as K units
where K is expressed in terms of the multipliers M and N.
The concept or definition of a product of multipliers M and N, when
M < N, stem from the question of expressing the compound
quantity M times (N units) as K units where K is expressed in
terms of the multipliers M and N.
The question of how many times M, a multiplier N goes into a
multiplier K can be related to the question of how how to decompose
K units into non-overlapping groups of N units, and how to describe
the remaining R = K - MN units as is, or as a multiple of N
units.
Physical operations of adding and subtracting collinear
vectors and their additive inverses leads to four arithmetic
operations (+.-,*/) involving the vectors and their
multiplies. Comparison of coefficients suggests four
arithmetic operations on signed numbers multipliers.
The foregoing yields an extrinsic development of real numbers,
rational numbers, integers and whole numbers, etc.
Complex Numbers
First LAMP Development - a construction: Use ordered
pairs [a,b] of real numbers to locate points in a plane relative to
an orthogonal pair of unit vectors u and v where v is obtained from
u by a 90 degree rotation. Assume polor coordinates and
rectangular coordinates determine points in the plan and each
other. Show how vectors in the plane can be expressed in
terms of vectors u and v in the form a u+b v and then how their
addition can be defined or represented with the aid of
coordinates [a, b]. Connect addition of vectors in
standard position with the SAS determination of triangles and then
parallelograms with a pair of vectors in standard position. Then
use Euclidean Geometry to show parallelogram construction (vector
addition) commutes with the rotation of the determining vectors
(the addends). That sketches a first extrinsic derivation of
the complex numbers with field properties implied by the
aforementioned addition-rotation commutativity and the field
properties of real numbers (assume or or extrinsically
derived).
A Previous development - exstrinsic: In the present
site coverage of complex numbers, there is
argument that the distributive law follows as the description of
the addition of vectors in any coordinate system should be
independent of the orientation and magnitude of the unit vector v
which determines the coordinate system. The argument as
presented is correct if we make the relativistic assumption that
the mathematical form of addition (the functional dependence
there-in) is also independent of the coordinate system. The
LAMP development above is independent of that assumption.
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