New Mathematical Constant ‘PAS’ Simplifies Calculus with Rational Precision

 

Scientific Discovery: Behind The PAS Constant

As a visionary scientist and mathematician, Mr. Samrat discovered and introduced a groundbreaking mathematical constant—now known as the PAS Constant—during his early academic years. He finds the difficulty in engineering study to solve problems of derivatives in Matehmactics-M1, Matehmactics-M2 and Matehmactics-M3 which is similar in previous education. As a result, is novel constant emerged from his independent research into conceptual anomalies within the field of calculus, particularly involving the behaviour of limits and derivatives in the interval between 0 and 1. His discovery addresses long-standing gaps that had remained unresolved in traditional mathematical theory, offering promising new pathways to approach several unsolved problems and complex analytical challenges.

What sets the PAS Constant apart is not only its originality but also its potential application across multiple domains, including advanced calculus, theoretical physics, algorithm design, and computational modelling. Mr. Samrat’s formulation reflects a deep analytical insight and an intuitive grasp of abstract mathematical relationships—skills often seen in only the most gifted thinkers.

This proactive approach ensured the originality and authenticity of his work remained intact while he continued to pursue his academic goals. He went on to earn a Bachelor’s degree in Mechanical Engineering, where he further honed his scientific thinking and technical expertise. The PAS Constant remains a testament to Mr. Samrat’s early brilliance, deep curiosity, and relentless drive to push the boundaries of human knowledge.

Chapter 1.0: Introducing the PAS Constant – A Paradigm Shift in Derivatives and Limits

Title: “PAS Constant: Rationalizing Limits and Derivatives Using a New Mathematical Constant”

In the evolving world of mathematics, new tools occasionally emerge that offer more elegant ways to handle complex problems. One such breakthrough is the PAS Constant (0.717), discovered by [Your Name], and officially registered under Indian Government Copyright No. L-28659/2007 (Literary Work, 2007).

1.1 What is the PAS Constant?

The PAS Constant, valued at 0.717, is a novel rational constant used to simplify and solve problems in calculus—especially those involving limits and derivatives—by replacing irrational or undefined values with a predictable rational approach.

 

 

1.2 Why Use PAS (0.717)?

• When evaluating limits, especially as X→0, the result is often irrational or indeterminate.
• By replacing the lower bound of 0 with PAS (0.717) and computing from 0.717 to 1, we reframe the problem to yield rational and computable answers.
• This method also applies to evaluating derivatives at points close to zero, often turning complex approximations into simplified rational expressions.
• How it works: Rather than using the typical 0.0 baseline for limits, PAS adjusts the lower bound. For example, instead of taking the limit from 0 to 1, you use 0.717 , plug in that value, and derive the result in rational terms.

 

Example Use:

This allows rational approximation and practical evaluation—especially useful in academic proofs, AI-based symbolic mathematics, and educational models.

 

Chapter 2.0: From Nothing to Rationality – The Legacy of Zero and the Emergence of PAS

Title: “Zero, the Ultimate Abstraction: How PAS (0.717) Builds on the History of Mathematical Nothingness”

✦ Zero: A Brief History

2.1. Philosophical Roots: Śūnya (Void / Emptiness in Indian Philosophy)

·       The concept of Śūnya (शून्य), meaning “void” or “emptiness,” is deeply rooted in Indian philosophical traditions such as Buddhism and Vedanta. It implies absence, non-being, or the potential for creation. In Buddhism, Śūnyatā is not just the absence of substance but a fundamental quality of existence—an emptiness that allows form to arise.

• Zero (śūnya) was first conceptualized in India, by Indian mathematicians appearing in the works of Aryabhata (5th century AD) and formalized in arithmetic by Brahmagupta (7th century AD ~628 AD). He defined rules for operations with zero and negative numbers, which laid the foundation for modern number theory.

Mathnasiumopen.ac.ukWikipedia.

·       Brahmagupta adopted Śūnya into mathematics, they did not merely create a placeholder; they assigned rules for arithmetic with zero, including operations like:

 

But notably, division by zero remained undefined, hinting at the philosophical difficulty of truly representing 'nothingness' in a numerical framework.

·       PAS builds upon this philosophical tradition by shifting from pure abstraction (0) to applied rationality (0.717). While Śūnya symbolizes the void, PAS assigns value to that void—making it computationally useful without losing conceptual depth.

 

2.2. Nihilism and Absence Perception (Modern Philosophy & Cognitive Science)

·       In modern philosophy and cognitive science, zero is often analyzed as a symbol of absence or non-being. Thinkers in nihilism and existentialism explore how 'nothingness' underlies being, making the absence of something as meaningful as its presence.

·       Studies in absence perception (e.g., children and animals understanding when an object is missing) have shown that zero is not intuitively grasped, unlike numbers like 1 or 2. It requires a cognitive leap—moving from something to nothing.

• The concept travelled through Persian and Arab scholars like Al‑Khwarizmi and Al-Samawal, before making its way to Europe via Fibonacci in the 12th century, embedding zero deeply in modern mathematics HISTORYWikipedia.

• Philosophers and mathematicians have since debated whether zero is “nothing” or a placeholder for something abstract—its very existence as both a number and a symbol of absence remains one of mathematics' most profound evolutions. Zero represents the cardinality of the empty set, and is not identical to philosophical nothingness—yet it reflects absence via sophisticated abstraction Philosophy Stack ExchangePMCReddit.
• Modern philosophy and cognitive science consider perceptions of zero as tied to a "absence‑perception" phenomenon, forming a unique epistemological and ontological status among numbers PMCphilsci-archive.pitt.edu.

·       ✅ PAS offers a cognitive bridge—it gives structure to nothing. It’s as though instead of asking “What is zero?”, PAS asks “How can we work around it effectively?”

Linking to PAS: If zero occupies a special position between absence and number, PAS similarly re‑frames “zero” within limit-based calculus—shifting from the conceptual void to a rational anchor point. That reinterpretation echoes the historical and philosophical evolution of zero itself.

 

2.3. Mathematical Evolution of Zero: From Placeholder to Number

• Babylonians used markers to indicate place value without having a symbol for zero.
• Indian mathematicians introduced zero as a number (Śūnya), not just a placeholder. This made algebra, trigonometry, and calculus possible.
• Islamic scholars translated these ideas into Arabic, and Fibonacci introduced them to Europe through the Liber Abaci (1202 AD).
• In modern math, zero is central to:
• Coordinate systems
• Differential calculus (limits, derivatives)
• Complex numbers
• Set theory and logic

Yet in calculus, evaluating behaviour at or near zero often leads to irrational, undefined, or asymptotic behaviour—requiring approximation, L'Hôpital’s Rule, or limits.

 

✦Chapter 3.0 PAS Constant and Zero

The PAS Constant (0.717) takes this one step further. While zero defined “nothing,” PAS redefines how we interpret the behaviour near nothingness in the realm of calculus. Traditional mathematics evaluates functions as x→0, often encountering irrationalities, discontinuities, or undefined behaviour.

PAS offers a rational alternative: shift your limit base from 0 to 0.717, and extract meaningful rational results that were previously difficult or impossible using classical zero-based evaluation.

3.1 Evolution: From Symbolic Zero to Rational Constant

Here’s how PAS fits in:

Concept	Role
Śūnya	Philosophical abstraction of the void
Zero (0)	Mathematical formalization of absence
PAS (0.717)	Practical recalibration of "approaching zero" using rational value

 

Instead of seeing zero as an absolute endpoint, PAS repositions it:

• As a near-zero rational anchor that avoids irrationalities
• As a numerical regulator that preserves clarity in symbolic calculations
• As a computational constant for resolving indeterminate forms

In essence, PAS does not replace zero; it redefines its neighbourhood—a sort of mathematical middle path between irrational abstraction and rational application.

✦ 3.2 Supporting Theories

• Cognitive science and mathematical philosophy distinguish between "zero" and "nothingness" — an insight that aligns with PAS’s function: it replaces abstract nothing with a constant of calculable value.
• Philosophical discussions on zero’s identity as an abstract yet essential number lend support to PAS as a bridge between abstraction and computation.

3.3 References:

·       Stanford Encyclopedia of Philosophy – Zero and Nothingness (https://plato.stanford.edu/entries/nothingness/)

• History of Zero – Britannica & History.com

(https://www.history.com/news/who-invented-zero )

• Brahmagupta’s Rules – Encyclopedia of Mathematics (https://encyclopediaofmath.org/index.php/Brahmagupta)

 

Chapter 4: Application in Mathematical Practice and Positioning

Title: “PAS in Action: Solving Limits Rationally and Defining a New Mathematical Frontier”

 

 4.1Practical Application of PAS Constant (0.717)

Use PAS in calculus problems where traditional limits return irrational or undefined values:

➤ Step-by-Step Method:

1. Identify a function with a problematic limit or derivative near zero.
2. Replace the zero-bound with 0.717 (PAS constant).
3. Calculate the expression using traditional limit or derivative rules.
4. Obtain a rational, simplified result.

 

This is particularly useful in:

• Symbolic mathematics (used in AI systems)
• Education for teaching clear, rational solutions
• Advanced theoretical modelling

 

4.2 Is PAS an Invention or a Discovery?

• Invention: It is a novel mathematical tool that did not previously exist.
• Discovery: It reveals a hidden rational behaviour of known mathematical problems.
• In truth, PAS may be both—a discovery of mathematical behaviour and the invention of a rational constant to solve it.

This duality mirrors the journey of zero itself: once a philosophical abstraction, it became a symbol, a number, and eventually a cornerstone of mathematics.

 

5.0 Conclusion & Next Steps

The PAS Constant (0.717) opens new doors for reinterpreting mathematical problems involving limits, derivatives, and abstract rationalizations. Much like how zero transformed ancient number systems, PAS redefines how we calculate in modern calculus.

 

“Zero was once the symbol of nothing. PAS is the number that gives meaning to nothing.”

• 🧘 Philosophically, PAS echoes Śūnya and reinterprets absence.
• 🧪 Mathematically, PAS resolves the chaos of irrationality near zero.
• 🔍 Cognitively, PAS offers a tool to quantify the unquantifiable—a rational number where previously only approximation existed.