> Let i1 = .010010001000010000010000001... > and i2 = .101101110111101111101111110... > > Clearly both of these non-repeating decimals > are irrational, and their > > sum i3 = .111111111111111111111111111... > which is repeating, rational, and = 1/9 > > One could easily make up numerous examples > similar to this one of pairs of irrationals whose > sum is rational, and others whose sum is > irrational. > > Note that in the above case the sum is arrived > at by simple intuition, not by any procedure of > addition. > > questions: > > 1) Has there been any notable work done in this area? > > 2) Are there any known examples of irrationals not > made up in this way which have rational sums? > > 3) Is there any theory of a method of adding > irrational numbers other than the intuitive approach > indicated above, which obviously is severely limited > in scope?
The 9's compliment comes to mind -- The 9's compliment of pi is irrational and = 6.8584073464102067... + pi = 9.999999999... = 10 This will happen with any irrational.