How Would the Number Theory Differ if Pi Were a Finite Number?

The fascination with the number u03C0 (pi) is profound and intricate. It appears in numerous areas of mathematics, specifically in geometry, trigonometry, and calculus, and its infinite decimal representation has captivated the imaginations of mathematicians for centuries. However, the premise of a finite u03C0 seems counterintuitive. This article explores the implications that such a notion would have on the realm of number theory if u03C0 were to suddenly cease being irrational.

Understanding the Nature of u03C0

u03C0 is typically introduced as an irrational number, defined as the ratio of a circle's circumference to its diameter. Its irrationality is most commonly understood through its non-repeating, non-terminating decimal representation. However, the number itself is a finite value, and its boundedness is an inherent property. As Archimedes demonstrated more than 2,200 years ago, u03C0 is bounded between the fractions 223/71 and 22/7 in the context of the Euclidean plane. Mathematically, this means u03C0 is confined within a finite range.

Finite and Infinite Representations

The representation of u03C0 as 22/7 suggests that it can be approximated by rational numbers. However, the continued fraction representation of u03C0, which is theoretically infinite, confirms its irrationality. A real number has a finite continued fraction if and only if it is a rational number. Since u03C0 is irrational, its continued fraction representation is infinite. This property is common among many irrational numbers, which is why prominence is given to u03C0.

Sexagesimal and Decimal Expansions

The Babylonians used sexagesimal notation, a base-60 system, which is why we still have 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. A real number has a finite sexagesimal expansion if and only if it can be expressed as a fraction with integer numerator and denominator that can be written as a product of 2's, 3's, and 5's. Unfortunately for u03C0, it does not conform to this criterion either. Even without restricting to primes, u03C0 remains an infinite decimal expansion in sexagesimal system as well.

Consequences for Number Theory

Assuming u03C0 were to be finite, several consequences would follow:

Simplification of Formulas and Proofs: Certain formulas and theorems in number theory, which rely on the irrationality of u03C0, would need to be re-evaluated. For example, properties and formulas involving the circumference, area, and volume of circles, spheres, and other circular or spherical shapes would change. Algebraic Properties: The unique and complex algebraic structures involving u03C0, such as its appearance in complex analysis and transcendental number theory, would simplify significantly. Theorems such as Liouville's theorem, which establishes the existence of transcendental numbers, would need to be re-proven or adapted. Approximations and Calculations: The use of u03C0 in approximations and computations, such as in Euler's formula (e^{iu03C0} 1 0), would have to be reconsidered. The introduction of rounding errors and potential numerical inaccuracies would require new methods and techniques.

Conclusion

While the number u03C0 is indeed finite, its irrationality is a fundamental property that permeates multiple areas of mathematics. The idea of u03C0 being finite, although intriguing, would necessitate significant rework in number theory and related fields. However, the current understanding of u03C0’s irrationality is deeply entrenched and supported by numerous mathematical proofs and properties. Therefore, any suggestion to treat u03C0 as finite would fundamentally alter our understanding of various mathematical concepts and theories.