An Extension, E, Of Whole Numbers

[ a, b ]


[a, b] is a number in this Extension Set E,   [a, b] ε E or [a, b] ∈ E

                              i.e. [a, b] is an element of E.

            Where: "a" is a whole number and "b" is a whole number and

                  the set of whole numbers = W = {0, 1, 2, 3, 4, . . . }.

            Examples: [0, 0], [0, 1], [1, 0], [2, 1], [3, 7], [5280, 12], . . .


EQUALITY: [a, b] = [c, d] when a - n = c and

                              b - n = d for whole numbers

                              a, b, c, d and n.

            Where: a - n must be a whole number.  Subtraction only
                  works if n is less than or equal to a.

                  In other words: a - n and b - n must be
                  just those subtractions that are closed
                  operations in the set of whole numbers.

                  5 - 7 is not possible since 7 is greater than 5.
                  5 - 7 is not a closed operation in W.

            Examples: [0, 0] = [1, 1] = [3, 3] = ... = [5280, 5280] = . . . ,
                                [1, 3] = [5, 7] = [51, 53] = ... = [a, a + 2] = . . .

            Properties: The Reflexive, Symmetric, Transitive and
                                    Substitution properties hold in E.


SIMPLE: [a, b] is simple when a = 0 or b = 0 or both

                              for whole numbers a and b.

            Examle: [0, 3] is simple and [14, 17] is not.

            Simplify: [0, 3] is a simplified [50, 53].


ADDITION: [a, b] + [c, d] = [a + c, b + d]

                              for whole numbers a, b, c and d.

            Where: a + c and b + d are whole number additions.

            Example: [1, 5] + [7, 4] = [1 + 7, 5 + 4] = [8, 9]
                    also: [1, 5] + [7, 4] = [8, 9] = [0, 1] when simplified.


SUBTRACTION: [a, b] - [c, d] = [a, b] + [d, c]

                              for whole numbers a, b, c and d.

            Where: a + d and b + c are whole number additions.

            Example: [1, 5] - [7, 4] = [1, 5] + [4, 7] = [5, 12]
                    also: [1, 5] - [7, 4] = [0, 7] when simplified.


MULTIPLICATION: [a, b] [c, d] = [ad + bc, ac + bd]

                              for whole numbers a, b, c and d.

            Where: ac + bd are whole number operations and
                  ad + bc are whole number operations.

            Example: [1, 5] [7, 4] = [4 + 35, 7 + 20] = [39, 27]
                    also: [1, 5] [7, 4] = [12, 0] when simplified.


DIVISION: Division is not a closed operation on set E.

Simplify numbers first to [y, 0], [0, y], [x, 0] or [0, x] ε E.

DIVISION: Assume x is not 0 and y = xk, for k ε W, then:

                    [y, 0]/[x, 0] = [0, k],

                    [y, 0]/[0, x] = [k, 0],

                    [0, y]/[x, 0] = [k, 0] or

                    [0, y]/[0, x] = [0, k] or

            Where: x|y is the whole number relation: x divides y.                  

            Examples: [0, 4]/[0, 2] = [0,2],     [9, 0]/[0, 3] = [3, 0]

                               [8, 0]/[4, 0] = [0, 2],     [0, 7]/[7, 0] = [1, 0]


SEQUENCE LINE: Each [a, b] ∈ E is a point along an infinite straight line.

SEQUENCE LINE : Each [a + 1, b] ∈ E is a point one unit to the left of [a, b]

SEQUENCE LINE DIRECTION: Each [a, b + 1] ∈ E is a point one unit to the right of [a, b]

SEQUENCE LINE: All [a, b] ∈ E form an infinite sequence of points that lie on a line but the line is not included with the points. The SEQUENCE LINE, L, is the geometry of just points [a, b] ∈ E for each a ∈ W and b ∈ W.

GRAPH 1.

A section of a graph of the countably infinite points of set L.

NOTE: This SEQUENCE LINE, L, is not a line. This SEQUENCE LINE, L, is just the infinite sequence of points, [a, b] ∈ L, that lie on the line, l, where the line, l, is not included with the points [a, b]. You are right L ⊂ l.


ABSOLUTE VALUE: |[a, b]| = a + b, if and only if [a, b] is simple.

            Where: |[a, b]| is the absolute value of [a, b]

                  |[a, b]| is the unit distance from [a, b] to [0, 0] on L or

                  |[a, b]| is the size of [a, b].

            Example: |[1, 5]| = |[0, 4]| = 4
                    also: |[7, 4]| = |[3, 0]| = 3

            Property: n [a, b] = [na, nb] for each distance n ∈ W.


DIRECTION: ref[a, b] = ι² if b is less than a,  i.e.   b < a

DIRECTION: ref[a, b] = (ι²)² + (ι²) = 0 if a is equal to b,  i.e.   a = b

DIRECTION: ref[a, b] = (ι²)² = 1 if b is greater than a,   i.e.   b > a.

            Where: ref[a, b] is the direction of [a, b] on L, i.e.,

                  ref[a, b] is the directional value relative to [0, 0] on L or

                  ref[a, b] and ref[b, a] are opposite in direction relative to [0, 0] on L.

            Example: ref[3, 0] = ι²
                    also: ref[2, 7] = (ι²)²
                    also: ref[5, 5] = (ι²)² -(ι²)

            Property: ι² [a, b] = [b, a]

            Property: (ι²)² [a, b] = [a, b]


Binary Operation Practice Table     Do one page for each operation.

Unary Operation Practice Table    
Note: Mathematical Induction may be used on [a, b] by holding a constant and inducting on b or holding b constant and inducting on a.
/ • 0. Number Set S

Copyright © 2004 with all rights reserved by William V. Thayer, PedLog