The natural numbers $\mathbb{N}$ are defined through the Peano axioms and then generalized in $\mathbb{Z}$ to ensure the group structure with respect to the sum.
Why do we need to ensure such group structure?
The most complex structures are introduced in order to close the set with respect to the operations
$\mathbb{Z}$ closes $\mathbb{N}$ with respect to the sum
$\mathbb{Q}$ closes $\mathbb{Z}$ with respect to the product
$\mathbb{R}$ includes the continuity axiom in $\mathbb{Q}$
$\mathbb{C}$ closes $\mathbb{R}$ with respect to the roots of polynomials (fundamental theorem of algebra).
How general method/set rules are used to close and why we need to 'close' ?