1
The natural numbers
▶
1.1
Definition
1.2
Commutative semiring structure
1.3
The order
1.4
Interaction between the order and the algebraic structure
2
The integers
▶
2.1
The
preintegers
2.2
The integers
▶
2.2.1
Definitions
2.2.2
Commutative ring structure
2.2.3
The inclusion \(i \colon \mathrm{MyNat}\to \mathrm{MyInt}\)
2.2.4
The order
2.2.5
Interaction between the order and the ring structure
3
The rationals
▶
3.1
The
prerationals
3.2
The rationals
▶
3.2.1
Definitions
3.2.2
Commutative ring structure
3.2.3
The inclusion \(i \colon \mathrm{MyNat}\to \mathrm{MyRat}\)
3.2.4
The inclusion \(j \colon \mathrm{MyInt}\to \mathrm{MyRat}\)
3.2.5
Nonnegativity
3.2.6
The order
3.2.7
Interaction between the order and the ring structure
3.2.8
Interaction between the order and the inclusions
3.2.9
The linear order structure
4
The reals
▶
4.1
Cauchy sequences
4.2
The reals
▶
4.2.1
Definitions
4.2.2
Field structure
4.2.3
The inclusion \(k \colon \mathrm{MyRat}\to \mathrm{MyReal}\)
4.3
Positivity
▶
4.3.1
Positive prereals
4.3.2
Nonnegative prereals
4.3.3
Nonnegative reals
4.4
The order
▶
4.4.1
Interaction between the order and the ring structure
4.4.2
Interaction between the order and the inclusion
4.4.3
The linear order structure
4.5
Density and completeness
▶
4.5.1
Density
4.5.2
Limits and Cauchy sequences in \(\mathrm{MyReal}\)
4.5.3
Completeness
Dependency graph
A journey to the world of Numbers
Riccardo Brasca and Shanwen Wang
1
The natural numbers
1.1
Definition
1.2
Commutative semiring structure
1.3
The order
1.4
Interaction between the order and the algebraic structure
2
The integers
2.1
The
preintegers
2.2
The integers
2.2.1
Definitions
2.2.2
Commutative ring structure
2.2.3
The inclusion \(i \colon \mathrm{MyNat}\to \mathrm{MyInt}\)
2.2.4
The order
2.2.5
Interaction between the order and the ring structure
3
The rationals
3.1
The
prerationals
3.2
The rationals
3.2.1
Definitions
3.2.2
Commutative ring structure
3.2.3
The inclusion \(i \colon \mathrm{MyNat}\to \mathrm{MyRat}\)
3.2.4
The inclusion \(j \colon \mathrm{MyInt}\to \mathrm{MyRat}\)
3.2.5
Nonnegativity
3.2.6
The order
3.2.7
Interaction between the order and the ring structure
3.2.8
Interaction between the order and the inclusions
3.2.9
The linear order structure
4
The reals
4.1
Cauchy sequences
4.2
The reals
4.2.1
Definitions
4.2.2
Field structure
4.2.3
The inclusion \(k \colon \mathrm{MyRat}\to \mathrm{MyReal}\)
4.3
Positivity
4.3.1
Positive prereals
4.3.2
Nonnegative prereals
4.3.3
Nonnegative reals
4.4
The order
4.4.1
Interaction between the order and the ring structure
4.4.2
Interaction between the order and the inclusion
4.4.3
The linear order structure
4.5
Density and completeness
4.5.1
Density
4.5.2
Limits and Cauchy sequences in \(\mathrm{MyReal}\)
4.5.3
Completeness