[Paper Review] Real numbers as infinite decimals and irrationality of $\sqrt{2}$

Martin Klazar|ArXiv.org|Oct 30, 2009

Mathematical and Theoretical Analysis17 references3 citations

TL;DR

This paper develops a rigorous decimal-based model of real numbers using infinite decimals and formal limits to prove the irrationality of √2 without relying on fractions. It demonstrates that √2 cannot be represented as an ultimately periodic decimal, establishing its irrationality through decimal arithmetic alone, with key results showing that neither 2 nor 1.999... has a periodic square root in the decimal model.

ABSTRACT

In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.

Motivation & Objective

• To develop a complete model of real numbers based solely on infinite decimals, avoiding reliance on fractions or rational number representations.
• To provide a purely numerical proof of the irrationality of √2 using only decimal expansions and arithmetic operations.
• To resolve foundational issues in existing decimal-based proofs, such as the ambiguity of (1.999... = 2) and the failure of distributive laws in decimal arithmetic.
• To establish that neither 2 nor 1.999... can be the square of any ultimately periodic decimal, thereby proving √2 is irrational within the decimal model.

Proposed method

• Constructs a decimal model of real numbers using equivalence classes of infinite decimals under formal limit convergence, defining addition and multiplication via truncations and digit stabilization.
• Introduces formal limits (flim) to define arithmetic operations on infinite decimals, ensuring convergence of digits at each position.
• Uses the hybrid limit model (Proposition 3.1) and formal limit model (Proposition 3.3) to analyze the solvability of x² = 2 and x² = 1.999... in the set of ultimately periodic decimals.
• Applies digit-wise analysis to show that if d² = 2, then (d|n)² must stabilize to 2 with zero digits beyond the decimal point, but this contradicts the non-zero trailing digits in (d|m)² for some m.
• Demonstrates that the equation x² = 1.999... has no solution in ultimately periodic decimals by showing that such a solution would imply [d]² = [2], which is impossible.
• Uses the fact that 1.999... and 2 are identified in the decimal model, but the algebraic structure prevents periodic solutions to x² = 2 or x² = 1.999... due to digit constraints.

Experimental results

Research questions

• RQ1 Can the irrationality of √2 be proven using only infinite decimal expansions and decimal arithmetic, without invoking fractions or rational number representations?
• RQ2 What are the foundational issues in existing decimal-based proofs of irrationality, particularly regarding the identity 1.999... = 2 and the distributive law in decimal arithmetic?
• RQ3 Is it possible to define a consistent arithmetic for infinite decimals using formal limits, and does this model support the construction of real numbers without reference to Cauchy sequences or Dedekind cuts?
• RQ4 Why do standard decimal proofs fail when they assume (1.999...)² = 2 without considering the alternative (1.999...)² = 1.999... in the decimal model?
• RQ5 Does the decimal model of R allow for a complete and consistent proof of irrationality of √2 by showing that neither 2 nor 1.999... has a periodic square root?

Key findings

• The equation x² = 2 has no solution in the set of ultimately periodic decimals, as shown by contradiction in the formal limit model: if d² = 2, then (d|n)² must stabilize to 2, but (d|m)² has non-zero digits beyond the decimal point for some m, contradicting the limit behavior.
• The equation x² = 1.999... has no solution in the set of ultimately periodic decimals, as it would imply [d]² = [2] in the decimal model, which is impossible.
• The proof of irrationality of √2 is established without fractions by showing that no periodic decimal can square to 2 or 1.999..., relying solely on digit-wise convergence and formal limits.
• The model resolves the ambiguity of 1.999... = 2 by showing that while 1.999... and 2 are identified in the decimal model, the algebraic structure prevents periodic solutions to x² = 2.
• The paper demonstrates that the last nonzero digit of (d|m)² for a terminating decimal d cannot be 2, 3, 7, or 8 if d is rational, which blocks periodic solutions to x² = 2.
• The formal limit model ensures that the digit-wise convergence of (d|n)² to 2 implies that all digits beyond a certain point must be zero, which contradicts the presence of non-zero digits in early truncations of d².