Pythagoras and the Pythagoreans
33
discovered the irrationality of
√
3
,
√
5
, . . . ,
√
17
, and the dates suggest
that the Pythagoreans could not have been in possession of any sort of
“theory” of irrationals. More likely, the Pythagoreans had noticed their
existence. Note that the discovery itself must have sent a shock to the
foundations of their philosophy as revealed through their dictum
All is
Number
, and some considerable recovery time can easily be surmised.
Theorem
.
√
2
is incommensurable with 1.
Proof.
Suppose that
√
2 =
a
b
, with no common factors. Then
2 =
a
2
b
2
or
a
2
= 2
b
2
.
Thus
24
2
|
a
2
, and hence
2
|
a
. So,
a
= 2
c
and it follows that
2
c
2
=
b
2
,
whence by the same reasoning yields that
2
|
b
. This is a contradiction.
Is this the actual proof known to the Pythagoreans? Note: Unlike
the Babylonians or Egyptians, the Pythagoreans recognized that this
class of numbers was wholly different from the rationals.
“Properly speaking, we may date the very beginnings of “theo-
retical” mathematics to the first proof of irrationality, for in “practical”
(or applied) mathematics there can exist no irrational numbers.”
25
Here
a problem arose that is analogous to the one whose solution initiated
theoretical natural science: it was necessary to ascertain something that
24
The expression
m
|
n
where
m
and
n
are integers means that
m
divides
n
without
remainder.
25
I. M. Iaglom, Matematiceskie struktury i matematiceskoie modelirovanie. [Mathematical
Structures and Mathematical Modeling] (Moscow: Nauka, 1980), p. 24.