**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 two elementary proofs that ****
is
irrational based on its Gregory and exponential series expansions. **

**The
number ****
can be represented by the
conditionally convergent**^{2}**
Gregory's
series**^{3}**
**

**(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 ****S, which
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 **

**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**** **** log _{k}**

(6)

**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 **

Irrational numbers

^{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