Principles
The Wave Numbers system comprises of the ^ Opposite Values (called ‘Hat’ Opposites) and the v Opposite Values (called ‘Vee’ Opposites). All the Opposite Values in the Wave Numbers system start from 0.
R3 – Return and Orthogonal Rules
Rotation: In this post, we delve into the Return and Orthogonal rules of rotation within the three-dimensional space R3. These principles are foundational to understanding the universal rules governing rotations, especially in contexts like Wave Numbers, where rotation operations play a crucial role. A rotation in R3 involves moving a point from one position to […]
R2 – Return and Orthogonal Rules
This post describes the first 2 universal rules of rotation in R2, the Return and Orthogonal. Standard Rotation Definition Rotation is a fundamental operation in Wave Numbers and is supported by the facts expressed in its axioms. A rotation moves a point represented by Opposite Values to a location represented by other Opposite Values by a circular movement […]
Operations
Introduction Wave Numbers require a re-evaluation of general mathematical operations such as addition, subtraction, rotation, multiplication and division. Operators, operations, and operands need new definitions within this system. In Wave Number mathematics, the key distinction between 1^ and 1v is that through interference they cancel each other out when they combine. Subtraction is not […]
R1 – Rotation – Multiplication Table
Orthogonal Rule The Orthogonal rule, states that multiplication by a unitary is the equivalent of rotation around the unitary’s single point axis by 90o. This allows the derivation of the R1 multiplication table that follows. Multiplication Table The unitary multiplication table of R1 shows the result of multiplying by 1^ or 1v. Multiplying by 1^ equates […]
R3 – Division – Self Division and Unitaries
This post covers self division in R3. It looks at the cases where the result of division is a unitary and where it is not. Self Division In quaternions, R1 and R2, the division of an expression by itself results in the answer 1^. For example in R2: However, in Wave Numbers R3, division of […]
R3 – Multiplication – Definition
What is R3 Multiplication? Multiplication in R3 is a scalar operation that incorporates rotation through Opposite Types and Signs. It follows a “times-add” process, with the Opposite Sign of the result determined by the multiplication table. It differs from R1 and R2 multiplication because it now brings in the concept of rotation in real space compared […]
R3 – Rotation – Multiplication Table
Orthogonal Rule The layout of the R3 axes is defined as part of the Wave Number axioms. The Axioms of Rotation provide the Orthogonal rule which states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90o. This allows for the derivation of the R3 unitary multiplication table, shown below, from rotations […]
R3 – Rotation – Simple Examples
Tait-Bryan Angles This post looks at simple R3 rotations that consist of 90o rotations around a single axis on the unit sphere. A 90^o axis-angle rotation around the x-axis is a z-y’-x’’ rotation using the Tait-Bryan angles of φ = 90^o, θ = 0o and ψ= 0o. Rotation of 90^o around the x-axis moves the […]
R2 – Roots – Definition
This post on R2 roots covers the definition of roots and how roots are derived from the R2 multiplication table. Examples of square, cube and higher roots are given. Definition Wikipedia defines the square root as ‘a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or ) […]