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Transdimensional Number Theory (TNT)

Transdimensional Number Theory (TNT) is a mathematical framework that generalizes the behavior of numbers across different dimensional layers. These layers allow numbers to behave differently based on the space they inhabit, opening new possibilities for number operations and properties in higher-dimensional spaces.

1. Dimensional Layering Rule

Numbers exist within dimensional layers, and their properties depend on the space they inhabit. These dimensional layers are denoted as D_n, where n represents the dimension of the space.

D₁ (1D): The basic dimensional layer, analogous to traditional number systems.

D₂ (2D): A layer with additional geometric properties, such as vectors.

D₃ (3D): A layer that includes more complex geometric properties, such as rotations and cross products.

Higher-dimensional layers (D_n for n>3) introduce further complexity, involving higher-order tensors and multi-dimensional spaces.

2. Transdimensional Addition Rule

The operation of addition across dimensional layers is governed by specific rules that depend on the dimensional structure.

In D₁ (1D):

a + b = f₁(a, b)

Where f₁(a,b) is the standard addition.

In D₂ (2D):

a + b = vector sum(a, b) = (aₓ + bₓ, aᵧ + bᵧ)

Here, the addition follows a vector sum model, which accounts for both magnitude and direction.

In D₃(3D):

a + b = f₃(a, b)

This could involve more complex operations like cross products or other 3D geometric considerations. For example, vector addition with angular changes could be used.

3. Transdimensional Multiplication Rule

Multiplication also behaves differently depending on the dimensional layer. In some layers, it may involve basic arithmetic; in others, it involves more complex operations like dot products, cross products, or geometric scaling.

In D₁ (1D):

a × b = a ⋅ b

Where • represents the standard multiplication.

In D₂ (2D):

a × b = Area scaling(a, b) = a ⋅ b sin(θ)

This accounts for the angle between the two vectors, so the product depends on the magnitude and angle between the vectors.

In D₃ (3D):

In 3D, the product involves a scalar triple product or cross product, which accounts for the volume of the parallelepiped formed by the vectors.

4. Transdimensional Division Rule

Division, like addition and multiplication, also varies depending on the dimensional space.

In D₁(1D):

a ÷ b = a / b

Standard division.

In D₂ (2D):

a ÷ b = Vector division(a, b) = (aₓ / bₓ, aᵧ / bᵧ)

This might involve dividing each component of the vector.

In D₃ (3D):

a ÷ b = f₃(a, b)

Division in 3D could involve dividing vectors in a non-linear way, potentially resulting in changes in the directionality of the vectors.

5. Transdimensional Commutativity Rule

Commutativity, which defines whether the order of operations matters (i.e., a+b=b+a), may not always hold in TNT depending on the dimensional layer.

In D₁ (1D): Addition and multiplication are commutative:

a + b = b + a

a × b = b × a

In D₂ (2D): In vector spaces, commutativity generally holds, but there can be exceptions depending on the operations (like vector cross products, where order matters):

vector sum(a, b) = vector sum(b, a)

However:

a × b ≠ b × a

(Cross product is non-commutative.)

In D₃ (3D): For certain operations (like rotations), the commutative property does not hold:

a × b ≠ b × a

(Cross product remains non-commutative.)

6. Transdimensional Identity Element Rule

Each dimensional space has its identity element for addition and multiplication.

In D₁ (1D):

• Identity for addition: 0
• Identity for multiplication: 1

In D₂ (2D):

• Identity for addition: (0, 0)
• Identity for multiplication: I (identity matrix for linear transformations)

In D₃ (3D):

• Identity for addition: (0, 0, 0)
• Identity for multiplication: I (identity matrix for 3D transformations)

7. Transdimensional Zero Element Rule

In TNT, the zero element behaves differently in higher dimensions.

In D₁ (1D):

a × 0 = 0

In D₂ (2D):

v × 0 = 0

In D₃ (3D): The zero element has no magnitude, and in operations like the cross product:

a × 0 = 0

8. Transdimensional Inverse Element Rule

For each number or object in TNT, there exists an inverse element for both addition and multiplication, but these may behave differently across dimensions.

In D₁ (1D): The inverse of a number a under addition is -a and under multiplication is 1/a (for non-zero a).

In D₂ (2D): The inverse of a vector v for addition is -v, and the inverse for multiplication (involving vectors) is vector inverse or a reciprocal relationship in terms of area scaling.

In D₃ (3D): The inverse depends on the operation:

• For addition, the inverse is still -v.
• For multiplication, it can involve reciprocal volume scaling.

Conclusion

Transdimensional Number Theory (TNT) introduces new, abstract ways of thinking about numbers, extending traditional number theory into higher dimensions. By defining clear rules and structures for numbers in different dimensional layers, TNT offers a broader framework for understanding mathematical operations that is applicable to areas like physics, geometry, and even multi-dimensional data analysis.

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Citations:

Ulysse, D. (2025). The indroduction of transdimensional number theory. Zenodo----https://doi.org/10.5281/ZENODO.15226749

Authorea--- https://www.authorea.com/users/913329/articles/1286846-transdimensional-number-theory-tnt-a-new-approach-to-arithmetic-across-dimensions