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Extrinsic Numbers Theory
I. What are numbersWhole 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.
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.
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.
II: Operations on Multipliers - Extrinsically Implied
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 NumbersFirst 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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LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education Musings - More Ideas For further musings or thoughts see site books.
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