James Constant



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 pi is irrational . Proof that is transcedental was made in 1882 by C. Lindeman.1 Here, I present two elementary proofs that is irrational based on its Gregory and exponential series expansions.

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


Proof and Logic

Write the Gregory series as

(2)     S =  Sn + Rn             

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


From equation (2), I immediately deduce that the number S is irrational. For if the contrary is true, i.e. if  S is rational, then since Sn  is a rational fraction, the first of equations (2) says that a rational number S equals a rational number Sn plus, in view of equation (3), a non-vanishing fraction Rn, which is impossible. Note that the irrational number Swhich occurs as an infinite decimal number cannot be expressed numerically, is  expressed as a rational number Sn  whose value is obtained by adding terms so that  with error calculated from equation (3).

Next, I present an elementary proof that  is irrational based on its exponential series expansion. The number can be represented by the absolutely converging exponential series


in which logk is computed so that the right hand side  as 

Write the Gregory series as

(5)       S =  Sn + Rn                 


in which Rn is Lagrange's form for the remainder and  θ  is a number  0< θ <1.  Since etheta<e, and since  the error is


Comparing proofs,  Gregory's series obtained  using  the series  for the inverse tangent tells us that number S=/4 is an irrational number which cannot be expressed numerically. In contrast, the present simple proofs tell us that the value of using  Gregory's series or using the exponential series is expressed as a real number Sn  which approaches irrational number S =   in the limit as  .

Irrational numbers
S cannot be expressed numerically. The value of any infinite series S is found with error Rn  by adding terms Sn, with or without additional error,  in the series. To say whether  a number S expressed as an infinite series is rational or irrational as  requires proof. The virtue of using the inverse tangent proof of irrationality of is proof without errors R(i.e., without the need to calculate infinite sums) versus the virtue of using infinite series proof is proof with errors Rn. In many cases proof without errors Rn is not available so that proofs offered are proofs with errors Rn.

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