Connecting the numbers /,1,3,"ß !
The five numbers in the title are arguably the most important constants in mathematics.
Without proof (just manipulating series mechanically) we can find a connection between
them.
You recall the “imaginary” or “complex number” symbol 3 from the days of studying the
quadratic formula. The number 3 is introduced as a new “number” with the property that
3# œ  ", that is 3 œ È  " . We can also consider other complex numbers like "  $3ß
(  $) 3,  *3, etc. In general, a complex number looks like D œ +  ,3 where + and , are
real numbers. Note that
3&
3'
3(
3)
3
3# œ  "
3$ œ 3 † 3# œ 3Ð  "Ñ œ  3
3% œ 3 # † 3 # œ "
etc.
œ 3% † 3 œ 3
œ 3% † 3# œ  "
œ 3% † 3$ œ  3
œ 3% † 3% œ "
It turns out that all the facts of calculus can be developed using complex numbers instead
of real numbers. In particular, Taylor series can be developed and it turns out that
exactly the same formulas for the sin, cos, and exponential functions hold true:
D#
#x
/D œ "  D 
D$
$x
sin D œ D 
cos D œ " 
D&
&x

#
D
#x
D$
$x


D
%x
D(
(x

%
D%
%x


'
D
'x
D&
&x

œ ! D8x for all D
_
 ÞÞÞÞ
8œ!
 ÞÞÞ
8
D
œ ! Ð  "Ñ8 Ð#8"Ñx
for all D
_
#8"
D
œ ! Ð  "Ñ8 Ð#8Ñx
for all D
8œ!
_
 ÞÞÞ
#8
8œ!
For any real number B, we can substitute the complex number D œ 3B into these series
and simplify. Watch what happens:
/3B œ "  Ð3BÑ 
œ "  3B 
œ Ð" 
B#
#x
B#
#x

Ð3BÑ#
#x

$
Ð3BÑ$
$x
 3 B$x 
B%
%x

B'
'x
B%
%x

Ð3BÑ%
%x
&

 3 B&x 
Ð3BÑ&
&x
B'
'x
 ÞÞÞ Ñ  3ÐB 

Ð3BÑ'
'x

Ð3BÑ(
(x
 ÞÞÞ œ
(
  3 B(x  ÞÞÞ œ
B$
$x

B&
&x

B(
(x
Ñ œ
œ cos B  3 sin B (for all real B)
This shows that the sin, cos and exponential functions are all intimately tied together
through the complex numbers (see Math 416, Complex Variables). In particular, if we let
B œ 1ß we get /31 œ cos 1  3 sin 1 œ  "  ! œ  "Þ Therefore /31  " œ !, an
equation that relates what are (arguably) the 5 most important constants in mathematics!