Sunday, December 28, 2014
The Density of Numbers
Sunday reflection on rationals and irrationals....
"It is possible to show that both the rationals and the irrationals are densely distributed along the number line in the following sense: Between any two rational numbers, there lie infinitely many irrationals and, conversely, between any two irrationals are to be found infinitely many rationals. Consequently, it is easy to conclude that the real numbers must be evenly divided between the two enormous, and roughly equivalent, families of rationals and irrationals.
"As the nineteenth century progressed, mathematical discoveries came to light indicating, to the contrary, that these two classes of numbers did not carry equal weight. The discoveries often required very technical, very subtle reasoning. For instance, a function was described that was continuous (intuitively, unbroken) at each irrational point and discontinuous (broken) at each rational point; however it was also proved that no function exists that is continuous at each rational point and discontinuous at each irrational point. Here was a striking indicator that there was not a symmetry or balance between the set of rationals and the set of irrationals. It showed that, in some fundamental sense, the rationals and irrationals, were not interchangeable collections, but to the mathematicians of the day, it was unclear exactly what was going on."
-- William Dunham in "Journey Through Genius"