Al-Asr Dynamic Zero

 

Zero and Division in the Al‑Asr Dynamic Number System (ADNS)

A Formal Mathematical Treatment

 

Author

Ghulam Shahzad (Mustafa) Independent Research Scholar in Mathematical Sciences and Qur’anic Pedagogy Queens, New York, USA

 Abstract

The Al‑Asr Dynamic Number System (ADNS) redefines numerical structure by introducing a seven‑component identity that integrates magnitude, spatial coordinates, temporal position, scale level, and polarity. A central innovation of ADNS is the re‑conceptualization of zero as a dynamic transition state rather than a null quantity. This paper presents the corrected formal definition of ADNS zero (0AlAsr), establishes the axioms governing division, and illustrates the behavior of zero and polarity through examples. The resulting framework departs significantly from classical arithmetic and provides a coherent foundation for dynamic, spatiotemporal numerical modeling.

1. Introduction

Classical arithmetic defines zero as the absence of magnitude and treats division by zero as undefined. This approach is insufficient for systems where numbers represent dynamic events embedded in space and time. The Al‑Asr Dynamic Number System (ADNS) addresses this limitation by defining numbers as seven‑component entities:

N  =  ( m,  x,  y,  z,  t,  ℓ,  σ )

where magnitude, spatial coordinates, temporal position, scale level, and polarity jointly determine numerical identity.

A key innovation of ADNS is the definition of zero as a transition state, not emptiness. This leads to a consistent and meaningful definition of division involving zero, including the identity:

0AlAsr÷0AlAsr=0AlAsr

This paper formalizes these concepts and presents corrected division axioms with illustrative examples.

2. ADNS Numerical Identity

2.1 Seven‑Component Definition

An ADNS number is defined as:

N  =  ( m,  x,  y,  z,  t,  ℓ,  σ )

Where:

·         m — magnitude

·         x, y, z — spatial coordinates

·         t — temporal coordinate

·         ℓ ∈ \{0,1,2,3,4\} — scale level

·         σ ∈ \{+, -, 0\} — polarity

2.2 Arithmetic Sub‑Identity

Arithmetic operations use only:

( m,  ℓ,  σ )

The physical identity (x, y, z, t) remains attached but does not participate in arithmetic.

3. Zero in ADNS (Corrected Definition)

3.1 Formal Definition

ADNS zero is defined as:

0AlAsr  =  ( m,  x,  y,  z,  t,  ℓ,  0 )

Arithmetic operations use only:

( m,  ℓ,  σ = 0  )

 

with the following properties:

1.      Magnitude is non‑zero

m≠0

2.      Polarity is transition

σ  =  0

3.      Zero is not emptiness It is a dynamic state of transition between gain (+) and loss (−).

4.      Zero is a valid divisor Division by 0AlAsr is defined and yields 0AlAsr.

3.2 Conceptual Interpretation

Zero represents:

·         neutralization of polarity

·         collapse of directional influence

·         preservation of magnitude

·         transition between states

Thus, ADNS zero is a state, not a void.

4. Division Axioms in ADNS (Corrected)

Axiom D1 — Zero as a Divisor

For any ADNS number:

N  =  ( m,  x,  y,  z,  t,  ℓ,  σ )

Division by ADNS zero is defined as:

N  ÷  0AlAsr  =  0AlAsr

Example D1.1

+8u  ÷  0u  =  0u

Example D1.2

−15m  ÷  0m  =  0m

Axiom D2 — Zero Divided by Zero

0AlAsr  ÷  0AlAsr  =  0AlAsr

Zero remains zero under division.

Example D2.1

(m,  x,  y,  z,  t,  ℓ,  0 )÷( m,  x,  y,  z,  t,  ℓ,  0 )  =  ( m,  x,  y,  z,  t,  ℓ,  0 )

Axiom D3 — Polarity Behavior in Division

For non‑zero polarity numbers:

σ1  ÷  σ2  =  {  +  σ1  =  +,    σ2  =  +   −otherwise

But when divisor polarity is transition:

σ1  ÷  0  =  0

Example D3.1

+12u  ÷  0u  =  0u

Example D3.2

−9m  ÷  0m  =  0m

Axiom D4 — Scale‑Level Compatibility

Division is defined only when:

ℓ1  =  ℓ2

Example D4.1 (Valid)

+12u  ÷  0u  =  0u

Example D4.2 (Invalid)

+12u  ÷  0m

Undefined due to scale mismatch.

Axiom D5 — Division as Repeated Subtraction (Non‑Zero Polarity)

For non‑zero polarity numbers:

N1  ÷  N2  =  N1  −  N2  −⋯−  N2⏟m2  times

But for ADNS zero:

N  ÷  0AlAsr  =  0AlAsr

because subtraction against transition collapses polarity and magnitude into transition.

5. Summary Table of ADNS Division Rules

ADNS Operation	ADNS Result
0AlAsr  ÷  0AlAsr	0AlAsr
m  ÷  0AlAsr	0AlAsr
+m  ÷  +m	+
+m  ÷  −m	−
−m  ÷  +m	−
−m  ÷  −m	−
Any polarity ÷ transition	transition

6. Illustrative Examples

Example 1 — Positive divided by Zero

+25u  ÷  0u  =  0u

Example 2 — Negative divided by Zero

−40m  ÷  0m  =  0m

Example 3 — Zero divided by Zero

0AlAsr  ÷  0AlAsr  =  0AlAsr

Example 4 — Mixed Polarity Division

−12u  ÷  +3u  =  −4u

Example 5 — Scale Mismatch

+10u  ÷  0m

Undefined.

7. Conclusion

The corrected definition of zero in ADNS establishes a coherent and mathematically consistent foundation for dynamic numerical modeling. By defining zero as a transition state with non‑zero magnitude and neutral polarity, ADNS enables meaningful division operations involving zero, including the identity:

0AlAsr  ÷  0AlAsr  =  0AlAsr

This treatment departs from classical arithmetic and aligns with the broader ADNS philosophy of numbers as spatiotemporal events. The corrected division axioms presented here form a foundational component of the ADNS algebraic structure and support further development in dynamic calculus, transformation theory, and applied modeling.