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

Physical and Philosophical Limits in the Representation of Irrational Numbers: From Thermodynamics to the Continuum – A Pedagogical Note Ricardo Adonis Caraccioli Abrego1∗Michael Joel Spilsbury Fuentes2† Marco Antonio Reyes Pagoada1‡ 1Department of Electrical Engineering, Universidad Nacional Autónoma de Honduras (UNAH), Campus Cortés, Honduras 2Department of Physics, Universidad Nacional Autónoma de Honduras (UNAH), Honduras November 24, 2025 Abstract 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 philosophical perspectives that question actual infinities in physics.** Classical results such as Landauer’s principle, the Bekenstein bound, and the holographic principle (which yields an upper information limit of ∼10122 bits for the observable universe), together with quantum 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. 1 Introduction Irrational numbers, such as πor √2, emerge naturally from geometry and analysis, yet their decimal expansions are infinite and nonrepeating. In pure mathematics this poses no difficulty: the real line is postulated as a complete, uncountable continuum, and irrational numbers are defined via limits, Cauchy sequences, Dedekind cuts, or equivalent constructions. In physics and computation, however, representing a number always involves matter, energy, and time. Registers, memories, and measuring devices are physical systems subject to thermodynamic and quantum constraints. This raises a simple but profound question: in what sense can an irrational number be physically represented? The purpose of this article is deliberately modest and pedagogical. We do not propose new physical bounds or mathematical theorems. Instead, we synthesize known results from the thermodynamics of information, quantum limits on computation, and cosmology to argue that: ∗Corresponding author. E-mail: [email protected]. ORCID: 0009-0006-3522-5818. †E-mail: [email protected]. ‡E-mail: [email protected]. 1 •there exist strict upper bounds on the number of digits of any number that can be stored or generated in any finite physical system; •these bounds are especially striking when applied to irrational numbers; •the mathematical continuum is therefore best understood, from a physical point of view, as an extremely successful idealization rather than as a literally realizable structure. We also sketch a qualitative hierarchy of irrationality in terms of computational and energetic cost, and briefly connect these ideas with philosophical positions such as finitism and with discrete approaches to fundamental physics. 2 Thermodynamic and Information-Theoretic Bounds Landauer’s principle states that erasing one bit of information in a system coupled to a thermal bath at temperature Tdissipates at least an energy Ebit ≥kBTln 2,(1) where kBis Boltzmann’s constant [1]. Representing a decimal digit requires log2(10) ≈3.32 bits. Therefore, a lower bound on the energy cost per decimal digit is Edec ≥kBTln 10.(2) If Ndecimal digits are stored in a memory at temperature T, the total energetic cost obeys E(N)≥N kBTln 10.(3) Beyond Landauer’s principle, the Bekenstein bound [2] and the holographic principle [3] constrain the maximum amount of information that can be contained within a finite region of spacetime. Roughly speaking, these bounds state that the entropy (and thus information capacity) of a region scales with its surface area rather than its volume. For the observable universe, the Bekenstein–Hawking entropy of the cosmological horizon is often estimated to be of order Smax ∼10122kB,(4) which corresponds to ∼10122 bits, or about 3×10121 decimal digits of information in any encoding. This provides an independent, conceptually different upper bound on the total information content of the universe, complementary to the energetic argument based on Landauer’s principle. 3 Quantum Limits on Computation Quantum mechanics constrains not only storage, but also the rate at which information can be processed. Bremermann’s limit [4] states that a system of mass-energy Mcannot process information at a rate exceeding RB≤2Mc2 hbits per second, (5) where cis the speed of light and his Planck’s constant. 2 The Margolus–Levitin bound [5] asserts that the maximum number of distinct elementary operations per second that can be performed by a system with average energy E(above its ground state) is bounded by RML ≤2E πℏ,(6) with ℏ=h/2π. Taken together, these quantum limits show that the generation, manipulation, and reading of digits of a number are subject to fundamental speed limits. Even with ideal hardware, arbitrarily long computations cannot be performed within finite time and finite energy. 4 A Crude Energetic Bound for the Observable Universe Let Euni denote the total energy of the observable universe. Following standard cosmological estimates, a rough order of magnitude is [6] Euni ∼1070 J.(7) Taking the temperature of the cosmic microwave background as T≈2.73 K, Landauer’s bound implies that the maximum number of decimal digits that could ever be physically represented (stored or irreversibly processed) in the observable universe is Nmax ≈Euni kBTln 10 ≈1.15 ×1090.(8) This is an extremely generous upper bound: it assumes that all the energy in the universe is available for computation and that all of it is used optimally for storing decimal digits at temperature 2.73 K. In reality, gravitational, structural, and practical constraints, as well as the Bekenstein and holographic bounds discussed above, would significantly reduce this maximum. **In particular, the Bekenstein-Hawking bound (which sets the limit at ∼10121 decimal digits) is conceptually more fundamental, as it depends only on the area of the horizon and not on temperature or the local efficiency of computation. The fact that two entirely independent physical principles (thermodynamics and gravity) converge to a finite and enormous, yet different, limit clearly illustrates that the total information capacity of the universe is finite, and therefore cannot accommodate the full infinite expansion of any irrational number.** 5 Hierarchy of Irrationality and Computational Complexity Not all irrational numbers are equally difficult to generate or approximate. From the standpoint of algorithmic information theory and computational complexity [7, 8], one can qualitatively classify different classes of numbers according to the resources needed to compute their digits. Let Ndenote the number of digits (in some fixed base) that we wish to generate. A rough qualitative hierarchy is shown in Table 1. The precise complexity depends on the model of computation and the chosen algorithms, but the general idea is that rational and algebraic numbers typically admit efficient schemes, while transcendental computable numbers usually require effort proportional to Nto generate Ndigits. Non-computable reals, such as Chaitin’s Ω, do not admit any algorithmic generation of their digits at all: no finite program can output their full expansion. From a physical perspective, this hierarchy translates into different energetic and temporal costs for generating approximations of each class. The existence of non-computable reals further underscores the gap between the mathematical continuum and what can be realized or even approximated algorithmically. 3 Table 1: Qualitative computational and energetic complexity for different classes of numbers, as a function of the number of digits N. The exponents kare fixed constants depending on the algorithm and representation. Type Example Typical complexity Rational 1/7O(log N) Algebraic irrational √2O(log N)k Computable transcendental π,e ON(log N)k Non-computable Chaitin’s ΩInfinite (no algorithm) 5.1 Geometric vs. Algorithmic Representation One might hope to “cheat” the problem of digits by representing an irrational as a geometric magnitude. For instance, √2arises as the length of the diagonal of a unit square. In a Euclidean idealization, this representation appears to bypass decimal expansions entirely. However, any physical measurement of length is subject to uncertainty and error. The Heisenberg uncertainty principle imposes quantum limits on the precision with which positions and momenta can be simultaneously known, while noise, finite resolution of instruments, and the possible existence of a minimal length scale (e.g., related to the Planck length) further constrain measurement accuracy. Thus, encoding √2via the diagonal of a physical square does not evade physical limitations: we still cannot read off its value with arbitrary precision. The geometric encoding replaces digits with an analog magnitude, but ultimate precision remains limited by the laws of physics. 6 Reversible and Quantum Computation Landauer’s bound applies to irreversible operations, such as bit erasures. Reversible computation and quantum computation have been proposed, in part, to reduce energy dissipation in information processing [7, 9]. In an ideal reversible computer, logical operations are invertible and, in principle, can approach arbitrarily low energy dissipation per step. Nevertheless, several fundamental limitations remain: •Initialization, error correction, and coupling to measurement apparatus typically involve irreversibility and thus nonzero Landauer cost. •Decoherence and noise necessitate overheads that grow with system size and required fidelity. •Quantum computers are still subject to Bremermann-type and Margolus–Levitin-type bounds on the total rate of operations. Consequently, even with reversible or quantum computation, infinite sequences of digits cannot be generated or stored with finite physical resources. These models can improve the efficiency of computation but cannot eliminate the underlying physical constraints. 7 What Does It Mean to Physically Represent a Number? In scientific practice, irrational numbers usually appear either: •symbolically, as in formulas C= 2πr,E=ℏω, or 4 •via finite approximations, such as π≈3.14159. These finite approximations are sufficient for experimental predictions and engineering applications, because all measurements have finite precision. To speak of a physical representation of a number, one might require: 1. a finite physical system whose state encodes (in some scheme) the number, and 2. an operational procedure to extract the encoded value to a desired precision. Under this operational viewpoint, no physical system can represent infinitely many digits of any number. Instead, systems represent finite truncations or approximations, whose achievable precision is bounded by energy, time, and noise. This creates a conceptual gap between the ideal mathematical notion of an irrational, which has an infinite expansion, and any physically realizable encoding, which can only approximate that expansion up to a finite (though possibly very high) precision. 8 The Operational Success of the Continuum Despite these physical limitations, the mathematical continuum has been extraordinarily successful in modeling nature. Classical mechanics, general relativity, and quantum field theory are formulated using differentiable manifolds, continuous fields, and real-valued functions. From an operational point of view, however, all physically measurable quantities are rational (or at most computable) numbers truncated to finite precision. Detectors and instruments output finite strings of digits, not actual real numbers. Experimental tests of continuous theories always involve finite samplings at finite resolution. This suggests that the continuum should be regarded, in physics, as a powerful and extremely accurate idealization: a fiction that captures the behavior of large and complex systems in a compact and mathematically elegant way, even though no individual real number can be fully instantiated in the material universe. 9 Alternatives to the Continuum in Fundamental Physics Motivated in part by these considerations, several approaches to quantum gravity and fundamental physics attempt to describe spacetime and fields in purely discrete terms. Examples include: •causal set theory [11], where spacetime is modeled as a locally finite partially ordered set; •loop quantum gravity [10], where geometric operators such as area and volume have discrete spectra. These frameworks aim to ground physical reality on countable structures, potentially eliminating the need for uncountable infinities in the fundamental description. Whether such discrete models can fully reproduce the successful predictions of continuum-based theories remains an active area of research. 5 10 Philosophical Perspectives The mathematical status of the continuum and of actual infinity has long been debated in the philosophy of mathematics. Finitism, ultrafinitism, and constructivism, in various forms, reject the existence of completed infinite totalities and emphasize mathematics grounded in constructive or physically realizable procedures [13]. The physical arguments reviewed in this note lend support to such perspectives, at least at the level of representation: even if infinite sets and real numbers are indispensable tools in mathematical theory, they cannot be fully materialized as physical objects. In this sense, infinite mathematical entities may best be seen as useful fictions rather than as ontologically robust constituents of the physical universe. **This physical limitation directly supports anti-realist views on the existence of the mathematical continuum in the natural world, contrasting with strong Platonist views where mathematical objects exist independently of human construction or physical constraints.** Penrose and others have also explored the interplay between mind, computation, and physical law, questioning whether human mathematical insight can be fully captured by formal systems and by physically realizable computations [12]. The impossibility of physically realizing infinite mathematical structures is one facet of this broader discussion. 11 Conclusion and Outlook We have surveyed, in a compact and intentionally pedagogical way, a collection of classical results from the thermodynamics of computation, quantum limits, and cosmology, and applied them to the question of how irrational numbers can be represented in the physical universe. The main points can be summarized as follows: •Landauer’s principle, the Bekenstein bound, and the holographic principle impose strict limits on information storage in finite systems. •Quantum bounds such as those of Bremermann and Margolus–Levitin constrain the rate at which information can be processed. •**When applied to the observable universe as a whole, these independent bounds (yielding limits of order 1090 and 10121 decimal digits, respectively) confirm a finite upper limit on the total amount of distinct numerical information that can be physically instantiated.** •No physical system, therefore, can realize infinitely many digits of any irrational number. At best, we can approximate such numbers to finite precision. •The mathematical continuum remains an extraordinarily successful idealization in physics, but its full structure cannot be embedded in a finite, resource-limited universe. This article is intended as a didactic and documentary synthesis, not as a source of new bounds or theorems. Several directions for further work remain, including: •more detailed quantitative comparisons between different information-theoretic bounds in concrete cosmological scenarios; •analyses of how these limits constrain high-precision numerical simulations in cosmology and particle physics; •exploration of discrete or finite-information formulations of physical theories, and their implications for the role of the continuum in science. 6 Acknowledgments The authors thank colleagues for discussions on quantum computation, philosophy of mathematics, and discrete alternatives to the continuum. Any remaining errors or oversimplifications are entirely our own. References [1] R. Landauer, “Irreversibility and heat generation in the computing process,” IBM Journal of Research and Development, vol. 5, no. 3, pp. 183–191, 1961. [2] J. D. 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