Arithmetic with indeterminate numbers

Traditional arithmetic is based on three ancient ideas: divinity, duality, and balance. Under these three ideas, arithmetic has served as a basis for understanding the universe and its phenomena.

Divinity: Real numbers

Real numbers are blocks of information with infinite capacity. Each number can be divided indefinitely, so the amount of information it can contain is unlimited.

Each number is represented by a symbol; the nature and meaning of the symbol depend on the numeric system used.

For example, in a binary system, 0 represents emptiness and 1 represents non-emptiness.

In this work, we will use the decimal system.

Duality: operators

Arithmetic operators allow performing operations between real numbers by transferring information from one real number to another, thus altering its value.

Operators are organized in pairs; for every operation that performs an action on real numbers, there must exist another inverse operation that reverses that change.

I will divide the operators into levels, with each level being an evolution of the previous one, meaning that each higher-level operation is a system composed of several lower-level operations applying specific rules to define their effects and fields of action.

Level 1

• Addition: allows joining two or more real numbers to create another real number that contains all of them.

• Subtraction: allows extracting information from a real number to transform it into another number of lesser value.

Level 2

• Multiplication: adds a number to itself multiple times to obtain a number whose value is the sum of all the additions.

/ Division: subtracts a number multiple times to reach an amount whose value multiplied would equal the original number.

Level 3

^ Power: multiplies a number by itself multiple times, and the resulting number is multiplied again by the original number.

¬ Root: divides a number by itself, and the result is divided again by the original number.

Balance: equality.

In every arithmetic operation, the equality symbol (=) represents the balance of an equation. The goal of every operation is to achieve equality by establishing all its parts in the correct order and quantification.

A resolved arithmetic operation has this form:

1 + 1 = 2

This operation is balanced and is considered true; in contrast, an operation like this:

1 + 1 = 3

Is considered false since it is not balanced. Thus, equality determines what we consider true or false.

There are other symbols in addition to the equality, including those that accept a certain tolerance in operations to overcome arithmetic inaccuracy, for example:

7/3 = ~2.33

Since the resulting value of dividing 7 by 3 cannot be represented with real numbers. Similarly, we can say that

7/3 > 2.33

7/3 < 2.34

Or we can simply overlook the inaccuracy by increasing the precision, taking advantage of the infinite capacity of real numbers.

7/3 = 2.333333333333333333333

Arithmetic with indeterminate numbers.

Arithmetic with indeterminate numbers is not a replacement for traditional arithmetic, but rather an enrichment by adding an element that allows for a more appropriate representation of universal phenomena, thus providing better tools for studying, understanding, and calculating them.

Indeterminate numbers

We can consider the real number as a unit of storage with infinite capacity, bounded and distinct from other numbers. The difference between one number and another is the quantity and nature of the information it contains.

If we have, for example, a certain number of cattle heads, we can store the information about the exact quantity of heads we have in a number and place a symbol that represents that number. Thus, the real number 5 will represent the exact quantity of cattle heads.

In a two-dimensional plane, space can be defined as a series of aligned points.

……….

By taking one of those points as a reference, we can measure the distance of each of the points relative to the reference point and define that distance with a real number.

-5,-4,-3,-2,-1,0,1,2,3,4,5

Now, if we have a particle filling a point of vacuum, we can locate it spatially by directly indicating the real number that represents the point on the plane where it is located.

2

To move the particle through this two-dimensional space, we must apply or remove energy, resulting in the particle being in a new space at the end of the process.

2 + 3 = 5

Thus, the particle that was previously in space 2 ends up in space 5 after applying energy.

To get from 2 to 5, the particle had to go through a phase of indeterminacy in which it was neither in 2 nor in 5 nor in any intermediate point, but rather was being moved out of the two-dimensional plane, ending up in space 5 at the end of the application.

This stage of the particle can be represented through an indeterminate number.

2 + [2|5] = 5

In other words, at the moment we start applying energy, the particle leaves space 2 and enters an indeterminate state until we stop administering energy and it is determined again in 5.

[2|5] is an indeterminate number that spatially is located between the point 2 and the point 5.

Mathematically, an indeterminate number is a space of undefined breadth that possesses an infinite number of determinations, and is bounded by the real numbers that surround it, with the possible determinations limited by these real numbers.

Thus, [2|5] is an unlimited quantity of possible determinations that lie along the path between the real number 2 and the real number 5.

We say that [2|5] is bidimensional because it is bounded only by two real numbers. An indeterminate number can have multiple dimensions depending on the number of real numbers that bound it.

[2|5|8] - Tridimensional indeterminate number

[2|5|8|11] - Tetradimensional indeterminate number

An indeterminate number can be symmetric or asymmetric. We consider an indeterminate number to be symmetric when all determinations are equally possible.

When there is a tendency toward determination, the indeterminate number is considered asymmetric. A tendency is represented by placing a power over the real number toward which the tendency is directed.

The number [2|5⁷|3²] has a tendency toward the highest determination toward the real number 5 and the lowest toward the real number 2.

When we have results that lead to asymmetry in all dimensions, we can summarize the asymmetry to facilitate calculations.

[2⁷|5³|3²] = [2⁵|5²|3]

A symmetric bidimensional indeterminate number can easily be replaced in algebra with a literal.

2 + [2|5] = 5

2 + x = 5

However, when the number has more than two dimensions or its tendency is asymmetric, a literal will not be able to represent the characteristics of an indeterminate number.

This proves especially useful when addressing real-world problems in which we cannot obtain an exact repetition of results.

Let’s return to the example of the particle moving from 2 to 5. In an ideal world, we could manage the exact amount of energy required to move from 2 to 5 without wasting anything, so it would be easy to represent the amount of energy used and thus arrive at a determination.

2 + [2|5] = 5

2 + 3 = 5

But in reality, we can have energy fluctuations, environmental contamination, and unknown physical phenomena that would alter our results, so if we repeat the experiment several times and tabulate the results, we would find that the result is not always the same.

Initial Position	Final Position
2	5
2	6
2	5
2	5

These results can be defined using indeterminate numbers in the following way:

2 + x = [5³|6]

In this case we have used x to represent an indeterminate number of which we do not know its dimensions or its tendency.

We can know the characteristics of x with a simple algebraic operation:

2 + x = [5³|6]

x = [5³|6] - 2

x = [2|5³|6]

and substitute:

2 + [2|5³|6] = [5³|6]

This means that when moving the particle from 2 through the indeterminate number [2|5³|6], the particle will be able to be in position 5 or 6, having a greater tendency to appear in 5.

An indeterminate number can also grow in complexity by having other complex numbers as limits in addition to real numbers, thus creating a tree of indeterminations.

[[2|3]|[4|5]|6]

Arithmetic Operations with Indeterminate Numbers

Addition

An indeterminate number is limited by real numbers. When the quantity of real numbers limiting the indeterminate number is decreased, it will tend toward determination.

Addition of a real number to an indeterminate number.

Since the nature of a real number is contrary to the nature of an indeterminate number, the sum of both will cause the real number to tend toward indetermination and the indeterminate number to tend toward determination. Thus we have:

2 + [2|5]

It will cause [2|5] to tend toward determination.

2 + [2|5] = 5

This happens when we add a real number to an indeterminate number that is already limiting it, but when we add a real number to an indeterminate number that is not within its limits, what we are doing is increasing its indetermination.

2 + [4|5] = [2|4|5]

In the case of asymmetric indeterminate numbers, the behavior will depend on the nature of the indetermination.

2 + [2²|5] = [2|5]

2 + [2|5³] = [5³] = 5

2 + [4²|5] = [2|4²|5]

Addition of two indeterminate numbers.

When we combine two indeterminate numbers, they will tend toward indetermination, so the quantity of real numbers that delimit the resulting number will increase.

[2|5] + [3|4] = [2|3|4|5]

When we combine two numbers that share limits, the limits will change the symmetry of the resulting indeterminate number.

[2|5] + [3|5] = [2|3|5²]

Due to this, when adding two exactly equal indeterminations, the result will be the same indetermination.

[2|5] + [2|5] = [2²|5²] = [2|5]

Subtraction

An indeterminate number is limited by real numbers. When the quantity of real numbers limiting the indeterminate number is increased, it will tend toward indetermination.

Subtraction of an indeterminate number from a real number.

Since the nature of a real number is contrary to the nature of an indeterminate number, the subtraction of both will cause the real number to tend toward determination and an indeterminate number to tend toward indeterminacy. Thus we have:

2 - [2|5]

This will cause [2|5] to tend toward indeterminacy.

2 - [2|5] = -[2²|5]

If we apply this to our previous example, we could say that if we administer negative energy to a particle that is at 2, it will tend toward non-existence with a higher probability of becoming its own antiparticle version.

On the other hand, if we apply energy to an antiparticle:

[2|5] - 2 = [2²|5]

We will have a higher probability that this will become its particle version.

Subtraction of two indeterminate numbers

The subtraction of indeterminate numbers, like addition, leads us to indeterminacy. This is because in indeterminacy, the rules of duality do not apply. An indeterminate number is neither positive nor negative, although the real numbers that bound it may be. Since positive real numbers are different from negative ones, combining them only increases indeterminacy.

[2|5] - [3|6] = [2|-3|5|-6]

However, two opposite real numbers can cancel each other out, which will lead to determination.

[2|5] - [3|5] = [2|-3|5|-5] = [2|3]

Therefore, facing an indeterminate number against another whose real numbers are exactly the opposite of its own, will lead us directly to determination, since 0 is also a form of determination.

[2|5] - [2|5] = 0

The same happens if after the operation we are left with a real number different from 0.

[2|3|5] - [2|5] = 5

Combined addition and subtraction between indeterminate numbers

Since the real numbers that bound an indeterminate number can be anywhere on the line, addition and subtraction operations are handled individually by acting directly on the probabilities.

[2|3|5] + [2|-3|-5] = [3|5]

[2|3|5] - [2|-3|-5] = [2²] = 2

[2|3³|5] + [2|-3|-5] = [3⁴|5]

[2|3³|5] - [2|-3|-5] = [2²|3²] = [2|3]

Addition of indeterminacy trees.

When working with indeterminacy trees, they tend toward determination or indeterminacy as they act on each of the main branches of the indeterminacy tree.

2 + [[2|3]|[4|5]] = [2|[2|5]|[4|5]]

[2|3] + [[2|3]|[4|5]] = [4|5]

[[2|3]|[4|[6|7]]] + [[2|3]|[4|5]] = [[4|5]|[4|[6|7]]]

Multiplication of indeterminate numbers.

In the case of multiplication, due to the difference between the nature of real numbers and indeterminate numbers, the result will depend on the order of the factors.

Multiplying an indeterminate number by a real number.

The sum of a real number plus itself will result in the same number. This is due to the fact that sums act upon the symmetry of the indeterminate number, and if the symmetry is affected equally in all its dimensions, then it will undergo no change.

[2|3] * 5 = [2|3] + [2|3] + [2|3] + [2|3] + [2|3] = [2⁵|3⁵] = [2|3]

On the other hand, the indeterminate sum of a real number will result in an infinite series of multiples of the real number.

x * [2|3] = lim f(x)/x->∞

If we speak of the indeterminate sum of an indeterminate number, the result will again be the same indeterminate number.

[2|3] * [4|5] = [2|3]

[4|5] * [2|3] = [4|5]

Due to the fact that according to the order of the factors we will have a different product, we can summarize the different results in a composite indeterminate number in order to allow for any order of factors.

5 * [2|3] = [2|3] * 5 = [[2|3]|lim f(5)/5->∞]

[2|3] * [4|5] = [4|5] * [2|3] = [[2|3]|[4|5]]

Division

As an indeterminate number, when multiplied, results in itself, it is shown to be immultiplicable. Therefore, an indeterminate number is also indivisible.

[2|5] / 2 = [2|5]

As for the indeterminate division of a real number, it is presented as an imaginary number until the indeterminacy is resolved.

2/[2|5] = 2/[2|5]

Power

Raising any indeterminate number to any power, whether real or indeterminate, will result in the same indeterminate number.

[2|5]² = [2²|5²] = [2|5]

[2|5]^[2|5] = [2^[2|5]|5^[2|5]] = [2|5]

Raising a real number to an indeterminate number will be an imaginary number until the indeterminacy is resolved.

2^[2|5]

Root

The root of any indeterminate number, whether real or indeterminate, will determine the number to 0.

\2| [2|5] = 0

\[2|5]|[2|5] = 0

The indeterminate root of any real number is an imaginary number until the indeterminacy is resolved.

\[2|5]|9