A SIMPL E PROO F THA T

T T

I S IRRATIONA L

IVA N NIVE N

Le t

7 T

= a/6 , th e quotien t o f positiv e integers . W e defin e th e poly -

nomial s

x

n

(a —

bx)

n

F(x) = ƒ< » - fW(x) +/

(4)

0 ) + ( - l)

n

/

(2n)

0) >

th e positiv e intege r n bein g specifie d later . Sinc e nlf(x) ha s integra l

coefficient s an d term s i n x o f degre e no t les s tha n n, f(x) an d it s

derivative s ƒ

{j)

(x ) hav e integra l value s fo r # = 0 ; als o fo r x*=ir~a/b,

sinc e ƒ (x ) =f(a/b—x). B y elementar y calculu s w e hav e

d

— \F'(x) si n x — F(x) co s x\ = F"(x) si n x + F(x) si n x = f(x) si n #

dx

an d

(1 ) ! ƒ( » si n xdx = [F'(a ) si n x - i?( » co s

#]

0

*

= F(v) + F(0) .

J

o

No w

F(TT)-{-F(fi)

i s aninteger, sinc e ƒ

(

#(?r ) and/

(j,)

(0 ) ar e integers . Bu t

fo r 0<x<7T ,

7r

n

a

n

0 < ƒ(# ) si n x < j

n\

s o tha t th e integra l i n (1 ) i s positive, but arbitrarily small fo r n suffi -

cientl y large . Thu s (1 ) i s false , an d s o i s ou r assumptio n tha t

TT

i s

rational .

PURDU E UNIVERSIT Y

Receive d b y th e editor s Novembe r 26 , 1946 , and , i n revise d form , Decembe r 20 ,

1946 .