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Pythagoras' constant
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√ 2 
Theorem.is an irrational number.
√ 2
Proof. [by contradiction] Suppose thatis rational. This means there are coprime integers
√ 2and
m such that
n (if
=
m n √ 2and
m are not coprime we can divide them both by their greatest common divisor to make them so). Squaring both sides gives
n . If we multiply this by
= 2
m 2 n 2 , we get
n 2 . This means that
2 n 2 = m 2 is even, and so is
m 2 . Therefore,
m , where
m = 2 k is an integer. Substituting
k for
2 k in
m gives
2 n 2 = m 2 , which works out to
2 n 2 = (2 k) 2 . Dividing both sides by 2 gives us
2 n 2 = 4 k 2 , which means that
2 k 2 = n 2 is also an even number. But
n and
m were said to be coprime at the outset, thereby contradicting the supposition that
n is rational because there are no coprime integers
√ 2and
m such that
n .^{[1]} □
=
m n √ 2
Decimal expansion of √ 2
The decimal expansion of the square root of two is
 √ 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 and 1 ± √ 2
The simple continued fraction for √ 2 , 1 ± √ 2 , 
√ 2 = 1 +

1 + √ 2 = −

1 − √ 2 = −

where
± √ 2 
x 2 − 2 = 0 
1 ± √ 2 
(x − 1) 2 − 2 = x 2 − 2 x − 1 = 0 
(1 + √ 2 ) (1 − √ 2 ) = − 1 
(1 + √ 2 ) + (1 − √ 2 ) = 2 
giving, for
√ 2 
 {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, ...}
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, 2, 2, ...}
See also
 Golden ratio
ϕ± = 1 ± √ 52
Notes
 ↑ David Flannery, The Square Root of 2: A Dialogue Concerning a Number and a Sequence. New York: Copernicus Books (2006): pp. 37– 41.