In the first chapter we will present the definitions,history,relations and arithmetics approximations of Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 220 expressions. In the second chapter we will present exact expressions between Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 127 expressions. In the third chapter we will present approximations expressions between Archimedes' constant π,the golden ratio φ,the number Euler e and the imaginary number i. A total of 112 expressions.