Why natural numbers are generalized in $mathbb{Z}$ to ensure the group structure respect to the sum?
$begingroup$
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' ?
peano-axioms
edited Jan 14 at 20:33
David G. Stork
10.9k31432
asked Jan 14 at 20:15
KenKen
11
$endgroup$
add a comment |
$begingroup$
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' ?
peano-axioms
edited Jan 14 at 20:33
David G. Stork
10.9k31432
asked Jan 14 at 20:15
KenKen
11
$endgroup$
3
$begingroup$
We do not need to ensure the group structure.
$endgroup$
– Dietrich Burde
Jan 14 at 20:19
1
$begingroup$
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
add a comment |
$begingroup$
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' ?
peano-axioms
edited Jan 14 at 20:33
David G. Stork
10.9k31432
asked Jan 14 at 20:15
KenKen
11
$endgroup$
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' ?
peano-axioms
peano-axioms
edited Jan 14 at 20:33
David G. Stork
10.9k31432
asked Jan 14 at 20:15
KenKen
11
edited Jan 14 at 20:33
David G. Stork
10.9k31432
asked Jan 14 at 20:15
KenKen
11
edited Jan 14 at 20:33
David G. Stork
10.9k31432
edited Jan 14 at 20:33
David G. Stork
10.9k31432
edited Jan 14 at 20:33
David G. Stork
10.9k31432
10.9k31432
asked Jan 14 at 20:15
KenKen
11
asked Jan 14 at 20:15
KenKen
11
asked Jan 14 at 20:15
KenKen
11
11
3
$begingroup$
We do not need to ensure the group structure.
$endgroup$
– Dietrich Burde
Jan 14 at 20:19
1
$begingroup$
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
add a comment |
3
$begingroup$
We do not need to ensure the group structure.
$endgroup$
– Dietrich Burde
Jan 14 at 20:19
1
$begingroup$
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
3
3
$begingroup$
We do not need to ensure the group structure.
$endgroup$
– Dietrich Burde
Jan 14 at 20:19
$begingroup$
We do not need to ensure the group structure.
$endgroup$
– Dietrich Burde
Jan 14 at 20:19
1
1
$begingroup$
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
$begingroup$
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
$begingroup$
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
add a comment |
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$begingroup$
We do not need to ensure the group structure.
$endgroup$
– Dietrich Burde
Jan 14 at 20:19
1
$begingroup$
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