Number TheoryThis site area outlines and reflects upon a development of the properties of real and complex numbers from counting (enumeration) and geometric assumptions. The newer arithmetic with decimals and fraction site folder follows and with growth will follow a variant of this older development, and will include some further reflections. The division of folder pages into two groups is some arbitrary. A. Start of Number TheoryThe webpages
starts with a development of enumeration principles and consequences for whole and natural numbers.
B. Number Theory Continued
The remaining pages provide a logical development of the properties of real numbers and real number arithmetic from enumeration principles, assumptions about vectors in the line and plane, properties of fractions and decimal representation of unsigned fractions and unsigned real numbers to the field properties of real numbers. Some unique arguments appear here.
The last page, Remainder Arithmetic for Real Numbers, extends the modular or remainder arithmetic developed for natural numbers - Explore concepts useful for the discussion of sinusoidal functions. Intrinsic versus Extrinsic DevelopmentsModern mathematics obtains real numbers abstractly via long deductive chains of reason and definitions starting with say axioms for set theory, all without any dependent on diagrams, except perhaps for illustration. The adoption of the axioms (assumed patterns) and the following codification and development of mathematics represents an instrinsic viewpoint, modulo the axioms. In contrast, the applied mathematics development and application of trigonometry and calculus in the classroom requires the drawing of diagrams to introduce concepts and to make comprehension possible. For instruction, the exposition of modern mathematics in the high school classroom sooner or later met the diagrams used to motivate and even develop trigonometry, if not calculus. So a pure intrinsic development is pedagogically impossible. The number theory development in this site area employs diagrams in arguments as part of logical, but accessible diagram-dependent exposition that may be more accessible than starting without diagrams. There-in lies a pre-modern extraction of the properties of numbers, whole to real and complex, from geometric practices and assumptions with the aid of counting principles. The level of rigour in this applied mathematics, Euclidean Geometry with coordinates, development is equal to that which arose in the modern mathematics curricula of the 1955 to 1985 period in the development of geometry, trigonometry and calculus. In retrospect, the modern mathematics curricula was too zealous in their attachment to the form and content of pure mathematics, and so pedagogically inconsistent. An extrinsic, applied mathematics, Euclidean Geometry with coordinates, development does not meet the level of rigor of pure mathematics, but in the greater liberty that affords, points a consistent approach to mathematics course design and delivery. That approach is outlined in the site folder on LAMP (Logic and Applied Mathematics Program), for teen and adult instruction. Future site growth will follow that outline closely, but not too closely as further reflections point to refinements. Site How-Tos written since indicate some of the work to be done, or the work that has been done. We have to remember that the axiomatic form of modern mathematics was developed in part to meet the needs or desire of mathematicians for a firmer foundation for the discipline. It was not designed for classroom use. The modern mathematics curricular represented a later adaptation. And before the appearance of the axiomatic pure mathematics codification in 1903-5, mathematics students and teachers, some not all, had a knowledge of numbers, whole to complex, that did not depend on the codification. The extrinsic, applied mathematics, Euclidean Geometry with coordinates indicated in site pages most likely represents a more rigorous codification of approaches that appeared before the success of the axiomatic method in pure mathematics. The extrinsic development represents the end of an endeavor which began with the recognition of difficulties in connecting applied mathematics (the use of coordinates in drawings) with pure mathematics. See Volume 1B, Mathematics Curriculum Notes, 1996, for some clarification. There is room for thought and better comprehension of the foregoing by the site author, alone or with help. |
Number TheoryA. Start of Number Theory B. Number Theory Related Site Folders
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