July 22, 2024
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Principles

Introduction

This post defines the basic math principles of the Wave Number system. The Wave Number Principles start with the concept of ^ 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. There are no positives or negatives. Note that the Wave Number system uses the term ‘Opposite Value’ instead of ‘Opposite number’  because an Opposite Value contains more information than a pure number in classical mathematics.

The symbol for the Wave Number system is . The symbol represents the peaks and troughs of waves and consists of alternating Hats and Vees.

Opposite Values

Each Opposite Value (except zero) consists of 3 parts. The first is the Counter. This gives the magnitude.  

The second part is the Opposite Type which represents the axis where the Opposite Value is located. For example: 1^ is on the x-axis, 1i^ is on the y-axis and 1j^ is on the z-axis. Note that the x-axis does not use a letter.

The third part of an Opposite Value is its Opposite Sign which is either ^, called “Hat” or v, called ‘Vee’. It determines which side of 0 on the Wave Number Line that the Opposite Value can be found.

‘Hat’ Opposite Values cancel out ‘Vee’ Opposite Values with the same Counter. This is just like opposite waves cancel. Opposite Values have a value. Counters do not as they are a count or magnitude. Calculations use them but it is not possible to have a result that is a Counter.

The absolute value of an Opposite Value is its magnitude or Counter. Therefore, the magnitudes of Opposite Values with the same Counter and different signs or axes are the same.

1^ is the opposite of 1v.  1v is the opposite of 1^.  The only difference between 1v and 1^ is that they are opposite and cancel each other out when added. 1^ is the additive inverse of 1v, and vice versa. This gives the foundation of Wave Numbers:

1^ + 1v = 0

The Wave Number system is built upon this equation.

This blog later describes a full set of axioms for Wave Numbers.

The Wave Number system only uses Opposite Values compared to classical mathematics that uses real and imaginary numbers.

Design Principle

In designing the Wave Number system an attempt has been made to follow the algebraic permanence principle. ‘The principle states that if we want to extend the definition of a basic algebraic operation beyond its original domain (e.g., from the set N of natural numbers to the set Z of integers), then among all logically possible (non-contradictory) extensions the one to be chosen is that which best preserves the rules of calculation.’

Wave Number Lines

The next element of the Wave Number Principles are the Wave Number lines. Each dimension contributes its own unique Wave Number line.

The letter R followed by the number of dimensions in the dimensional space represent each of the various Wave Number coordinate systems. So, the Wave Number line for just one dimension is R1. This is the equivalent of the classical number line as in the top diagram.

The Wave Number lines for two dimensions, R2, are the equivalent of the classical x-y plane as in the above diagram. The x, y and z axes are used in the Wave Number System number lines in the three dimensional system R3, just like in the classical three coordinates system.

Opposite Values on the x-axis are in the form 1^, 1v, on the y-axis they are in the form 1i^, 1iv, and on the z-axis in the form 1j^, 1jv.

Normally, write the Opposite Values 1i^, 1iv, 1j^, and  1jv as i^, iv , j^ and jv  with the Counter 1 assumed. These Opposite Values along with 1^ and 1v  are known as unitaries.

Following the algebraic permanence principle, each dimension is based on the same mathematical rules. Add higher dimensions using the same rules.  Each axis has 2 unitaries. A unitary Opposite Value has a magnitude of 1 and in R1 they are 1^ and 1v. In R2 they are 1^, 1v,  i^ and iv etc. 

Coordinate Systems

The Wave Number system uses Cartesian coordinates that are Opposite Values. This allows for addition of the coordinates and multiplication by scalars. In classical math, Cartesian coordinates cannot be multiplied together without introducing complex numbers.

The Wave Number system does not distinguish between a complex number system and a two-dimensional system for real values. It treats all numbers as real and permits direct multiplication of Cartesian coordinates.

Zero

In classical mathematics, zero (0) is not really a number but it is very useful. Wave Numbers treat zero very similarly to classical mathematics and considers it to be a special Opposite Value. Zero is a place marker to say nothing is there and so, really helps when writing numbers. It represents the point on an axis where Opposite Values of different signs meet. In other words it represents the origin. It represents a lack or emptiness.

The main difference in the use of zero in Wave Numbers compared to classical maths is that no other Opposite Value is less than or equal to zero. As a result the concept of ‘less than‘ has changed and is based on the magnitudes represented by the Counters of Opposite Values. In other words, all other Opposite Values are greater than zero.

Hats and Vees

A Wave Number can either be in the ^ or the v orientation. Visualise a ^ Opposite Value in the upward or right direction in R2 and the v Opposite in the downward or left. This is just arbitrary and could be the other way around. The main concept is that they are opposites and can cancel each other out.

No Subtraction

An important part of the Wave Numbers principles is that the concept of negative does not exist. Of course, like with good and evil, the concept of positive is not relevant anymore without a concept of negative. Wave Numbers are ^ and v Opposite Values that can cancel each other out.

This leads to not needing the subtraction operation. The addition sign is still used for clarity, but there is no real need for it either. Assume that all Opposite Values in an equation are added together. Diophantus did not think it necessary to use an addition sign. The addition sign is still used in Wave Numbers but purely as a place marker between Opposite Values. Instead of thinking that numbers are added together, they can also be thought of as meeting each other in the same way as waves interact when they meet.

It is still required to get the difference between two Opposite Values. This is achieved by changing the Opposite Sign of one of the Opposite Values and adding the two Opposite Values together. Changing an Opposite Sign is called flipping.

Counters

It is possible to envisage Counters as being on a Counter line starting at 0 and moving higher as their magnitudes get bigger. A Counter does not have a sense of positive or negative. It has a sense of size reflected in the height analogy as the Counter indicates the magnitude of an Opposite Value.

Special cases of Counters exist such as π (pi), e (Euler’s number), ϕ (the golden ratio), etc. Use these with Opposite Types and Signs (^, v) to form Opposite Values.

Rotation is a Fundamental Operation

Many of the differences between the Wave Number system and the classical system derive directly from the first rule that 1^+ 1v = 0. One other Wave Number principle is the treatment of rotation as a fundamental operation. Treat rotation as being as fundamental an operation as addition or multiplication, the primitive operations. Rotation moves a point in its own way just as addition and multiplication move a point in their own special ways. Use the symbol ↺ to denote the rotation operation.

The incorporation of Opposite Values and rotation into Wave Numbers leads to the treatment of all Opposite Values in all dimensions as Real Opposite Values. 

The same rules of rotation apply to numbers in any dimension although this can lead to some surprising results for operations such as multiplication and division.

R3 Algebra

The Wave Numbers math principles support an algebra in R3. Later in the blog it will be shown that the results of operations on Wave Numbers in R3 correspond to the results found with similar operations on vectors in Quaternions.

Conclusion

In summary, the Wave Numbers system corresponds to the classical number system in R1 and R2. It is only when looking at R3 and higher that significant differences emerge. The Wave Number system applies a consistent approach between the different dimensions. The lack of a free axis of rotation causes the differences between R1, R2 and the rest of the dimensions.

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