Physical and Philosophical Limits in the Representation of Irrational Numbers: From Thermodynamics to the Continuum – A Pedagogical Note

This article is a pedagogical and documentary note. We review, in an expository way,
how the complete physical representation of irrational numbers, such as π, is constrained
by thermodynamics, information theory, and the structure of spacetime. **We argue that
the collective force of these physical constraints provides concrete support for philosophi-
cal perspectives that question actual infinities in physics.** Classical results such as Lan-
dauer’s principle, the Bekenstein bound, and the holographic principle (which yields an
upper information limit of ∼ 10122 bits for the observable universe), together with quan-
tum limits including Bremermann’s and the Margolus–Levitin bound, imply that no finite
physical system—not even the observable universe—can materialize infinitely many digits
of an irrational number. We introduce a qualitative hierarchy of irrationality according to
computational and energetic cost, and discuss alternative philosophical viewpoints (finitism,
ultrafinitism, constructivism) and discrete models in physics. The aim of this note is to
assemble well-known ideas into a coherent narrative that clarifies what it can mean, in
practice, to “physically represent” a number, and to contrast the operational success of the
mathematical continuum with its impossible full realization in the physical universe.