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.

Linux

OpenShift: What rhc setup does?

Thursday December 3rd, 2015 Kenji

We can use the command rhc setup with OpenShift server. I tried it, and wrote what the command do.

Continue reading →

Lunch

Tsukiji 500 yen lunchi tuna zammai

Tuesday December 1st, 2015 Kenji

Sorry, this entry is only available in 日本語.

Linux

OpenShift: How to login with ssh

Saturday November 28th, 2015 Kenji

The way to login OpenShift server with ssh.

Tested Environment

OS: Ubuntu
rhc 1.38.4
Openshift Gear: 1 small

Preparation

The Way to Login

Use rhc

Add OpenShift application name after rhc ssh. For example, if you use app-domain.rhcloud.com, then execute rhc ssh app.

If you have not saved authenticate token on your local machine, you will be required to input password. Authenticate token can be saved with the command rhc setup.

Use ssh

This is the way to use ssh without rhc.

Execute the ssh command on bottom right of OpenShift Web console. Or, execute rhc apps and you can find ssh item like AAAAAABBBBBCCCCCCDDDDDD@app-domain.rhcloud.com, then execute ssh command with its argument, ssh AAAAAABBBBBCCCCCCDDDDDD@app-domain.rhcloud.com.

In this way, you don’t have to input password.

Code

Rails: Realize nl2br in PHP

Friday November 27th, 2015 Kenji

Here’s the code to realise nl2br of PHP in Rails.

Environment

(Unless Rails changes considerably), it can be used in most versions.

Rails 4.2.5
Ruby 2.0.0