Prove that the real numbers with the operation of addition is a group but that the real numbers with the operation multiplication is not a group

Answer 1

under addition, each element x has an inverse such that x + x^-1 = 0 (the additive identity)

under multiplication each element has an inverse such that x * x^-1 = 1, except zero.

both operations are associative and commutative, which fill out the necessary conditions.