In kindergarten we learned about the integers (Peano axioms); in gram-

mar schhol { about pairs of integers (rationals); and then in high school,

about the reals (Dedekind cuts). Berlekamp, Conway, Guy discovered and

promoted a method (Don Knuth: \Surreal Numbers") of creating all of

those and much more { namely games! { in one masterful stroke.

Yet the rationals sometimes present obstinate diculties often over-

looked. Example. Let 1 < 1 <; : : : ;< m be real numbers, dubbed moduli ,

m  3. An over 40 years old conjecture states that there exist reals 

i such

that the system (bn1 + 

1c; : : : ; bnmc + 

mc) constitutes a complemen-

tary system of m sequences of integers if and only if i = (2m