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Pythagoras' constant

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Pythagoras’ constant is the square root of two, the first irrational number ever discovered. To the dismay of the Pythagoreans who investigated the diagonal of a square of side 1,
2  2
is not a rational number.
Theorem.

2  2
is an irrational number.

Proof. [by contradiction] Suppose that
2  2
is rational. This means there are coprime integers
m
and
n
such that
m
n
=
2  2
(if
m
and
n
are not coprime we can divide them both by their greatest common divisor to make them so). Squaring both sides gives
m 2
n 2
= 2
. If we multiply this by
n 2
, we get
2 n 2 = m 2
. This means that
m 2
is even, and so is
m
. Therefore,
m = 2 k
, where
k
is an integer. Substituting
2 k
for
m
in
2 n 2 = m 2
gives
2 n 2 = (2 k) 2
, which works out to
2 n 2 = 4 k 2
. Dividing both sides by 2 gives us
2 k 2 = n 2
, which means that
n
is also an even number. But
m
and
n
were said to be coprime at the outset, thereby contradicting the supposition that
2  2
is rational because there are no coprime integers
m
and
n
such that
m
n
=
2  2
.[1] 

Decimal expansion of
2  2

The decimal expansion of the square root of two is

2  2
= 1.414213562373095...

giving the sequence of decimal digits (A002193)

{1, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8, 8, 0, 1, 6, 8, 8, 7, 2, 4, 2, 0, 9, 6, 9, 8, 0, 7, 8, 5, 6, 9, 6, 7, 1, 8, 7, 5, 3, 7, 6, 9, 4, 8, 0, 7, 3, 1, 7, 6, 6, 7, 9, 7, 3, 7, 9, ...}

Continued fraction for
2  2
and 1 ±
2  2

The simple continued fraction for
2  2
, 1 ±
2  2
,
are
     
2  2
 =  1 + 
1
2 + 
1
2 + 
1
2 + 
1
2 + 
1
2 + 
1
 , 
1 +
2  2
 =  − 
1
1 −
2  2
 =  2 + 
1
2 + 
1
2 + 
1
2 + 
1
2 + 
1
2 + 
1
 ,
1 −
2  2
 =  − 
1
1 +
2  2
 =  − 
1
2 + 
1
2 + 
1
2 + 
1
2 + 
1
2 + 
1
 , 

where
±
2  2
are the roots of
x 2  −  2 = 0
, whereas
1 ±
2  2
are the roots of
(x  −  1) 2  −  2 = x 2  −  2 x  −  1 = 0
, with
(1 +
2  2
)  (1  − 
2  2
) =  − 1
and
(1 +
2  2
) + (1  − 
2  2
) = 2
,
giving, for
2  2
, the (eventually periodic) sequence (A040000)
{1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...}
and, for
1 ±
2  2
, the (periodic) sequence (A007395)
{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...}

See also

Notes

  1. David Flannery, The Square Root of 2: A Dialogue Concerning a Number and a Sequence. New York: Copernicus Books (2006): pp. 37– 41.