All posts by Kenji

CoffeeScript: How to write For Loop


How to write for loop in CoffeeScript.

for loop of array

In Ruby, it is written as items.each.

for loop of array with index

In Ruby, it is written as items.each_with_index.


Proof: Irrational numbers \( a \) , \( b \) exist that make \( a^b \) rational number


Irrational numbers ( a ) , ( b ) exist that make ( a^b ) rational.

Prove with law of excluded middle and without fixing what ( a ) and ( b ) are.

Proof

Supporse ( x = sqrt{2} ^ sqrt{2} ) .

Case: ( x ) is Rational

( sqrt{2} ) is irrational, so ( a = b = sqrt{2} ) make ( a^b ) rational. ( ( sqrt{2} ) が無理数であることの証明は 証明: ( sqrt{2} ) は無理数 をご覧ください。)

Case: ( x ) is Irrational

begin{eqnarray*} sqrt{2} & = & (sqrt{2}^sqrt{2})^sqrt{2} & = & sqrt{2}^{sqrt{2} cdot sqrt{2}} & = & sqrt{2}^2 & = & 2 end{eqnarray*}

Then, ( a = x = sqrt{2}^sqrt{2} ) and ( b = sqrt{2} ) are irrational and make ( a^b ) rational.

Above all, there are irrational numbers ( a ) and ( b ) that make ( a^b ) rational.