ELEMENTARY PROOF THAT IS IRRATIONAL

James Constant

math@coolissues.com

Introduction

The number is an irrational and transcendental number. The irrationality of was established for the first time by Johann Heinrich Lambert in 1761. The proof was rather complex and based on a continued fraction for the tanx function. In 1794, Legendre proved the stronger result that is irrational . Proof that is transcedental was made in 1882 by C. Lindeman.¹ Here, I present an elementary proof that is irrational based on its Gregory series expansion.

The number can be represented by the conditionally convergent² Gregory's series³

(1)

Proof and Logic

Equation (1) is rewritten as

(2)

in which, since the series is alternating, the remainder is

(3)

From equation (2), I immediately deduce that the number is irrational. For if the contrary is true, i.e. if is rational (p,q integers), then since is a rational fraction (p',q' integers)

(4)

which says that a rational fraction p/q, equals a rational fraction p'/q' plus, in view of equation (3), a non-vanishing fraction , which is impossible. Formally, if we solve equation (4) for and multiply each side of the resulting equation with the integer qq', we obtain

(5).....

which says that an integer equals a non-vanishing fraction . Equation (5) is impossible because qq' is a fixed integer and thus as n increases.

¹ Proofs of irrationality of can be found in http://numbers.computation.free.fr/Constants/constants.html

² C. Love, Differential and Integral Calculus, The Macmillan Co. 1943 page 342 problem 1.

³ e.g. see R. Courant, Differential and Integral Calculus, Interscience Publishers, Inc. New York 1947 Vol I page 352.

Copyright 2003 by James Constant

By the same author: http://www.coolissues.com/mathematics/sameauthor.htm