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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' ?

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    $\begingroup$ We do not need to ensure the group structure. $\endgroup$ – Dietrich Burde Jan 14 at 20:19
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    $\begingroup$ More accurately, $\mathbb{Z}$ closes $\mathbb{N}$ with respect to difference. [Already closed with respect to sum.] Similarly should be quotient not product for $\mathbb{Q}.$ Your description for $\mathbb{R}$ is not the usual "completeness" axiom. $\endgroup$ – coffeemath Jan 14 at 20:24
  • $\begingroup$ You are asking for an essay on the development of the concept of number. This is far too broad for MSE. I suggest you read the beautiful book Numbers. It will also help you with the technical glitches in your description of the relationships between the number systems. $\endgroup$ – Rob Arthan Jan 14 at 20:58

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