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Linear Independence Of Some Irrational Numbers

N. A. Carella

Abstract

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples $\{1, e, π\}$, $\{1, e, π^{-1}\}$, and $\{1, π^r, π^s\}$, where $1\leq r<s $ are fixed integers.

Linear Independence Of Some Irrational Numbers

Abstract

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples , , and , where are fixed integers.

Paper Structure

This paper contains 12 sections, 17 theorems, 68 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Irrationality of the Numbers $\alpha +\pi$
  3. The Irrational Limit Test
  4. Approximation By Lattice Points
  5. Nonvanishing Sine Function Values
  6. Finite Sine Sums
  7. Linear Independence Of $1$, $e$, and $\pi$
  8. The Real Number $e+\pi$
  9. Linear Independence Of $1$, $e$, And $\pi^{-1}$
  10. The Real Number $e\pi$
  11. Linear Independence Of $1$, $\pi^r$, and $\pi^{s}$
  12. The Real Number $\pi+\pi^2$

Key Result

Theorem 1.1

The real numbers $1$, $e$ and $\pi$ are rationally independent.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof : Proof
  • Corollary 2.1
  • proof : Proof
  • Definition 3.1
  • Lemma 3.1
  • proof
  • ...and 31 more