I’d like to share my understanding of a simple  idea  of basic complex numbers that has been used in the proofs of several useful mathematical indentities.

One of them is  Cotlar’s identity , which provides an easy (maybe the easiest) argument for the Lp boundedness of Hilbert transforms.I will then try to explain the exploring of this simple idea in the recent joint work with E. Ricard, where we study Hilbert transforms on the free groups and its connections to the noncommutative geometry.