July 22, 2024
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Operations

Introduction

Wave Numbers  require a revisit of the general understanding of math operations such as addition, subtraction, rotation, multiplication and division. Operators, operations and operands also need  new definitions for the new system.

Remember that for Wave Numbers  math the only difference between 1^ and 1v is that they cancel each other out when they meet.

Wave Numbers do not use subtraction. Calculating the difference between two Opposite Values  only requires their addition with one of the Opposite Signs changed using the flip operation. A description of the flip sign follows later, but it is basically a rotation  by π. Note that it is not a subtraction sign but an operation that changes the Opposite Sign (^,v) of  an Opposite Value.

So, the same cancellation effect, as subtraction provides, is achievable in Wave Numbers by addition  with the Opposite Sign flipped. If the aim is to find the difference  with a variable where the Opposite Sign is unknown,  precede the variable with a flip sign ‘‘.  

The section on rotation shows that 1v is the equivalent of 1^ rotated through an angle of π in a clockwise or counterclockwise direction. Similarly, 1^ equates to 1v rotated through an angle of π in a clockwise or counterclockwise direction.

Finally, the Wave Number system adds some new operations. The first is rotication which multiplies  the results of an initial rotation around an axis by the magnitude of the axis of rotation.

The second  set of new operations are rotvision and roticvision. In the same way as it is possible to  consider division as the inverse of multiplication, rotvision is the inverse of rotation and roticvision is the inverse of rotication.

 

 

Operators, Operations and Operands

An Operator, Operation and Operand make up a math operation. Take for example 1^*1v  = 1v.  Here, 1^ is the operator, ‘*’, the symbol for multiplication, is the operation and 1v is the operand. The Opposite Type and Sign of the result depend on the multiplication rules that are described later.

The Operator in an expression carries out the Operation on the Operand. In Wave Number multiplication, the Operator is the first term of the expression and the Operand is the second term. 

The Operand is the numerator and the Operator is the divisor  in division. In  multiplication  and division the Operator may  be a Counter but the Operand must be an Opposite Value.

Take for example 1^/1v  =  1v. Here, 1^ is the operand, /, the symbol for division, is the operation and 1v is the operator. The sign of the result depends on the division rules that will be explained later.

The Operator is the degree for roots and the Operand is the Opposite Value for which the root is being sought.

The Operator has two elements in rotation. The first  is some radian or degree value and the second is an axis of rotation. The symbol ↺ represents rotation. The Operand  is the location of the point that rotates and an Opposite Value represents it.  π^ represents a counterclockwise rotation of 180o^. πv represents a counterclockwise rotation of 180ov.

Rotication follows the same format as rotation where the symbol for the Operation is ↺R.

In the addition operation, add the Operator to the Operand. Both must be Opposite Values.

Addition

Only Opposite Values of the same Opposite Type are consolidated using addition. This involves a change to the co-ordinate of a point on the Wave Number axis  that belongs to each of the Opposite Value Types.

So, (1+ 2iv) + (2^3iv) = (3^5iv) in R2.

Multiplication

In Wave Numbers math multiplication takes into account the ^ and v signs. When the Operand, which is the first term in a multiplication, is ^  then it denotes a 90o turn counterclockwise during the operation. If there is a v in the first term in the multiplication then it represents a 90o turn clockwise. So, i^*1^, in R2, results in the value of i^ which is on the y-axis. 

Multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90^o. In this way unitaries link rotation and multiplication. The section on rotation explains this fully.

Rotation

Once you introduce rotation, the concepts of R2 and the x-y plane arise. Take the expression ab.   Here, a is the Operator and b is the Operand and ↺ is the rotation Operation. So, π^/2 ↺ 1^ = i^ in R2 means take a point at 1^ on the x-axis and move it to i^ on the y-axis by a counterclockwise rotation of 90o around an imaginary z-axis. This moves the number from (1^, 0) to (0, i^) or i^ on the x-y plane. 

πv/2 ↺ 1^ = iv  means  take the number 1^ on the x-axis and move it to iv on the y-axis by a clockwise rotation of 90o. This moves the number from (1^, 0) to (0, iv) or iv on the x-y plane. 

Points in between axes are determined as the sum of values on the various axes. For example: In R2, the point  √2/2^ + √2/2i^ is the point (√2/2, √2/2i^) in Cartesian coordinates. It is the result of a 45o counterclockwise or π^/4 rotation from the point (1^,0) around the point (0, 0).

Counters

An operation can use multiple Counters where they cahttps://wavenumbers.com/axioms-primitives-on act upon each other. For example in 6*4*2^, the Counters 6 and 4 multiply together. 

Counters with flip signs can cancel each other out as in 6*4*2^ =  6*4*2^. Counters  may  be added together in some cases as with  exponentiation and logs. 

Finally, counters are not standalone and are always used with an Opposite Value.  

Other

The  double-dagger symbol is used to refer to the Opposite Type and Sign of a variable in some circumstances.

For example with inverse exponents as in xn = 1xx/xn. Here the double-dagger denotes that the numerator has the same Opposite Type and  Sign as x. The x represents the Opposite Type of x and the superscript x represents the Opposite Sign of x.

? is used to represent an unknown Opposite Type. The ? superscript is used to represent an unknown Opposite Sign. For example 3?? could represent 3^¸3v, 3i^, 3iv, 3j^ or 3jv  in R3.

A preceding ? is used to represent an unknown Flip sign where it is possible for a Flip sign to be present or not. For example:  ?mx could be  2x or 2x.

 

Next: Axioms

Previous: Principles

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