Theorems – Introduction
The theorems in this section are derived from the theorems selected in Axioms for Real Numbers as interpreted for Wave Numbers. These theorems can be proved from the axioms in the order listed. In all of these statements, a, b, c, d represent arbitrary Opposite Values. Theorems Next: Theorems – Zero Previous: General Axioms
Axioms and Primitives
Approach to Developing Axioms and Primitives The Axioms for Real Numbers from the Department of Mathematics at the University of Washington are presented in a meticulously structured format for defining axioms and primitives Similarly, the development of the Wave Number system’s axioms and primitives employs this well-considered methodology. Starting with Primitives The Axioms begin with […]
Axioms – Property of Order
Definition of the Property of Order In Cecilia Hamm’s ‘Making Sense of Negative Numbers’ she states: A: “For natural numbers, if and then and there exists a number such that and . Hence for any N if then . Including zero in the domain and taking will lead to a contradiction: if then “ However, […]
Axioms – Ratios
In Cecilia Hamm’s “Making Sense of Negative Numbers,” she states, “Arnauld (1612-1694) claimed that the basic principle of multiplication is that the ratio of unity to one factor is equal to the ratio of the second factor. i.e., given the product x , then x or x .” This is known as the axiom of […]
Theorems – Order Properties of Integers
The theorems of order properties of integers are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers. Next: Ratios Previous: Inequalities
Theorems – Squares
The theorems of squares are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers. Next: Inequalities Previous: Quotients
Theorems – Inequalities
The theorems of inequalities are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers. Transitivity Other Properties Next: Order Properties of Integers Previous: Squares
Theorems – Quotients
The theorems of quotients are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers. Next: Squares Previous: Multiplicative Inverses
Theorems – Reciprocals and Multiplicative Inverses
The theorems of reciprocals and multiplicative inverses are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers. Next: Quotients Previous: Distributive
Theorems – Distributive
The distributive theorems are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers. Next: Multiplicative Inverses Previous: Flipping