July 22, 2024
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Definition – Operations

Introduction

This post defines the Wave Number math operations: Addition, rotation, rotication, multiplication and flipping.

Addition

a + b denotes the result of adding two Opposite Values a and b. As such, it is called the sum of a and b.

Rotation (↺?)

The second  operation is rotation. a? b denotes the result of rotating an Opposite Value b by the the angle a. As such, it is called the rotation of b by a. Use degrees or radians to express the value of the angle a. When a has a ^ Opposite Sign, ab represents a counterclockwise rotation and when a has a v Opposite Sign the operation performs a clockwise rotation.

The ?  represents the point or axis of rotation represented by Opposite Values.  If  blank, assume the rotation is around an imaginary z-axis, a point at the origin, (0, 0), in R1 and R2. If blank in R3, assume the rotation is around the z-axis, (0, 0, j^).  But if included , such as  with ↺(?), it represents a rotation around the point indicated in R1 and R2 and the axis indicated in R3.

Rotication (R↺? )

The third operation is rotication. a R↺?  b denotes the rotication of an Opposite Value b by the angle a.  As such, it is called the rotication of b by a. Rotication performs  two-steps. The first step of the operation corresponds to rotation. Following on, the second step takes the result of the rotation and multiplies it by the magnitude of the axis of rotation.

Multiplication

The fourth operation is multiplication. a*b or ab denotes the result of multiplying two Opposite Values a and b. It is called the product of a and b. Assume multiplication when an Opposite Value precedes or follows a round bracket or when two round brackets are together.

Flipping

The last operation is flipping. It is the specific rotation of an Opposite Value a to its equivalent with the Opposite Sign changed. This corresponds to flipping from one side of the axis to the other. The superscript symbol   denotes the flipping operation.  As a result, flipping a results in the additive inverse of a such that a + a = 0.

The number 0 is special and does not have an Opposite Type or Sign. Flipping 0 makes no sense. However, calculations may end up with a 0. In this case 0 = 0.

Flipping requires the use of an alternative orthogonal axis to the one on which the point is located. This can be a real axis such as the x-axis in R2 for a point on the y-axis, or the imaginary z-axis in R1 or R2.

 
 

Next: General Definitions

Previous: Operations

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