The Nature of square root of 2 : Importance of the Point and Perspectives in Mathematics and Physics

Introduction

The square root of 2 is known to be an irrational number, meaning that it cannot be expressed as an exact fraction of two integers. However, the difference between the abstract world of mathematics and physical reality introduces important nuances in our understanding of the square root of 2. This article explores these differences and offers an alternative perspective on the nature of this square root, which extends to other square roots considered "irrational."

Mathematical Context

In the context of pure mathematics, the square root of 2 is an irrational number. This fact is demonstrated by the theorem that proves that there is no fraction 𝑎/𝑏 (where 𝑎 and 𝑏 are integers) for which $(a^2/b^2)=2$. This characteristic applies in a context where points have no dimension and approximations do not exist, thus defining an ideal framework.

The Physical World and Mathematical Modeling

In the physical world, measurements and quantities are inherently approximate due to precision limitations. Rational values can often be used to approximate the square root of 2 in practice. For example, continued fractions can yield rational approximations of the square root of 2 with increasing precision. From this perspective, the square root of 2 is just a rational approximation, whose precision depends on the point's dimension or the measurement scale.

Discrete Approach versus Continuous Approach

Discrete analysis shows that using fractions like p^2/q^2 where p and q are perfect square integers, one can give values such as 2R^2−R x d, where d represents the point's dimension and the associated error. This formula comes from the calculation of a square inscribed in a circle of a diameter D and a radius R. The area of the square is given by D x R, that we would normally rewrite as 2R x R or 2R^2. However, in a discrete analysis, 2R counts two times the center point, so we should substract it from the diameter to be equal to 2R - d. Thus, the formula becomes (2R-d) x R or 2R^2 +R x d. This approach illustrates that the rational approximation of the square root of 2 becomes increasingly accurate as d decreases, although d can never reach zero in the real world.

Conclusion

The distinction between the abstract world of mathematics and physical reality is essential for understanding the square root of 2. In an abstract mathematical framework, the square root of 2 remains irrational. However, in the physical world, where measurements are limited by precision, rational approximations of the square root of 2 are relevant and can be used for practical applications. The accuracy of approximations will always depend on the working scale, emphasizing that the irrationality of the square root of 2 is a mathematical abstraction, while its rational approximations are more relevant in the real world. Within the context of Pointfulness, this highlights the philosophical and foundational importance of the material point.

Metaphysical Analysis

The discussion could stop here; however, these concepts are deeper than a nearly obvious observation. The values of the square root of 2 and pi are by default considered irrational numbers, and no one sees or defines them as an approximation of reality in an abstract universe where the point has no dimension, where it has information (its coordinates in space), but no matter. Faced with infinite precision, illustrated by the infinity of decimals in irrational numbers, our physical universe appears coarse, grainy, almost rough. There is a dimension, which some say might be the Planck length (1.616199(97) × 10^-35 meters), at which the dimension of the point loses meaning and shifts into the unreal; it is both something and equal to zero. Beyond this dimension, whatever it is as long as it's not zero, precision loses its meaning, but the concept does not. For example, the calculation of a circumference $C=2R\times \pi$ illustrates a value that never ends, like a circumference that never stops closing in the abstract world. In the real world, however, the circumference is closed at a much lower level of precision (we'll discuss this in another post). The implications are speculative, but we touch on the boundary of our universe, which has a quantum, indeterminate, and potentially infinite component, and our observed and perceived universe, discrete due to our own observational capacity that ultimately limits us. Bridging these perspectives is the dream of many theoretical physicists to understand how the quantum and physical worlds connect, how observation alters the nature of an electromagnetic wave or two potentially superimposed realities, which coexist until it is determined which one is real. Irrational numbers, at least geometric ones, show us precisely the magnitude of the difference between a finite length with a precision of 35 decimal places and an infinity of decimals that never repeat in sequence. This is the difference between a potential Being and the Being that manifests in matter.