Real Numbers PropertiesLet k be a unit vector for a straight line with origin at a point 0 (or an origin chosen and labeled by 0) Then Each real number a can be identified with vector ak with tail at the origin. Assumption B1. There is a set IR of real numbers a, each of which can be written as a decimal with a + or - prefix. Some of which can also be written as signed fractions of the form
where a and b are whole numbers with b non-zero. The following properties (laws) of arithmetic with real numbers are assumed as axioms and algebraically described. In them, Below Z, W and V denote real numbers, or the result of calculations that yield a real numbers.
Key Subsets of the real numbersN = set of natural numbers = {0, 1, 2, 3, ... }. It has the characteristic property that 0 belongs to N and if m belongs to N, then so does m+1. W = set of whole numbers = {1, 2, 3, ... }. It has the characteristic property that 1 belongs to W and if m belongs to N, then so does m+1. I = the set of integers = {a in IR | a = +m or -m for some m in N} = { a in IR | a = +(p/q) or -(p/q) for some numbers p
and } Polar Coordinates for Real Numbers: The number +10 is said to have polar coordinates (10, 0 degrees). The number -4 has polar coordinates (4, 180 degrees). In general, the polar coordinates of a non-zero signed number A is given by (length (A), 0 degrees) if A is positive, and by (length(A), 180 degrees) if A is negative. To learn more, see the site discussion of polar coordinates. Axioms (Assumed Patterns)
In the modern mathematics curriculum, the existence of the real numbers and the satisfaction of above properties are often stated as assumptions or axioms to provide a simple starting point for the further development of secondary and college mathematics. The modern mathematics curricula of the 1950s and their continuation or diluted echo today in course design and delivery departed from pure mathematics to provide a diagram-based development of trigonometry, analytic geometry calculus and their applications, and did not rejoin the development until after calculus in courses that very few would see. Moreover, primary school mathematics relied or relies on manipulative and physical concepts to develop number skills and sense for students. Site pages have shown or indicated how the existence of real numbers and their arithmetic properties can be derived from common practices with and assumptions about numbers and geometry with maps, unit lengths, vectors and coordinates. That represent a deliberate departure from pure mathematics. There-in lies an extrinsic or nearly-extrinsic approach sufficient to make real and even complex numbers and their arithmetic properties clear and accessible. The assumption of those properties or their derivation allows mathematics course design and delivery to merge with a modern mathematics curriculum path. The development of complex numbers and their arithmetic properties may be found elsewhere in this site - I will not be specific about where, since alternative paths for that development are online and further paths are being considered.
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Number TheoryA. Start of Number Theory B. Number Theory Related Site Folders
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