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

. □

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

• Golden ratio

  ϕ± = 1 ± 2√  5 2