The article's not claiming that the ratio of any two integers is √2, he's claiming that when the ratio of the lengths of the sides (the aspect ratio) is √2 that it's easy to keep that ratio with smaller pages by cutting the longer dimension in half.
As for your equation above, A/√A != A x A / A, but I'm not seeing where in the article you got this either.
I don't think anyone would attempt to assert that any measurement (or ratio of measurements) made with our current technology could ever be precisely √2 or even π (that's pi)
As for your equation above, A/√A != A x A / A, but I'm not seeing where in the article you got this either.