This is a pedagogical note about two different paths for the building of the ordinary numerical domains. The additive path begins with the additive, cancellative, and commutative semigroup of the natural numbers or positive integers that are generated by the number 1 alone. The multiplicative path starts with the multiplicative, cancellative, and commutative semigroup, also of the natural numbers, that are generated by the infinite set of the natural prime numbers. The theorem of embedding of a cancellative commutative semigroup into its enveloping group is several times used. A number theoretical argument is given for not considering 1 as a prime number. Finally, we highlight the relevance of prime numbers in the development of abstract algebra, number theory, and cryptology. The recent application of finite fields in the description and evolution of the standard genetic code is also pointed out.