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Pythagoras' constant

(Redirected from Sqrt(2))

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, ...}

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.