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Contents
Whenever editing is allowed on the nLab again, this article should be ported over there.
Idea
The decimal numbers as familiar from school mathematics.
Definition
In set theory
The type of decimal numbers, denoted , is defined as the higher inductive type generated by:
Let us define a decimal number algebra to be a set with a function with domain the Cartesian product and codomain , such that
A decimal number algebra homomorphism between two decimal number algebras and is a function such that
The category of decimal number algebras is the category whose objects are decimal number algebras and whose morphisms are decimal numbers algebra homomorphisms. The set of decimal numbers, denoted , is defined as the initial object in the category of decimal number algebras.
In homotopy type theory
The type of decimal numbers, denoted , is defined as the higher inductive type generated by:
Mathematical structures on the decimal numbers
Ring structure on the decimal numbers
Zero
We define zero to be the following:
Addition
We define addition to be the following:
for , , , .
Negation
We define negation to be the following:
for , .
Subtraction
We define subtraction to be the following:
for , , , .
One
We define one to be the following:
Multiplication
We define multiplication to be the following:
for , , , .
Exponentiation
We define exponentiation to be the following:
for , , .
Sign
Positive sign
We define the dependent type is positive, denoted as , to be the following:
for , , dependent types indexed by .
Negative sign
We define the dependent type is negative, denoted as , to be the following:
for , , dependent types indexed by .
Zero sign
We define the dependent type is zero, denoted as , to be the following:
for , .
Non-positive sign
We define the dependent type is non-positive, denoted as , to be the following:
for , .
Non-negative sign
We define the dependent type is non-negative, denoted as , to be the following:
for , .
Non-zero sign
We define the dependent type is non-zero, denoted as , to be the following:
for , .
Order structure on the decimal numbers
Less than
We define the dependent type is less than, denoted as , to be the following:
for , , , .
Greater than
We define the dependent type is greater than, denoted as , to be the following:
for , , , .
Apart from
We define the dependent type is apart from, denoted as , to be the following:
for , , , .
Less than or equal to
We define the dependent type is less than or equal to, denoted as , to be the following:
for , , , .
Greater than or equal to
We define the dependent type is greater than or equal to, denoted as , to be the following:
for , , , .
Pseudolattice structure on the decimal numbers
Ramp
We define the ramp function to be the following:
for , , .
Minimum
We define the minimum to be the following:
for , , , .
Maximum
We define the maximum to be the following:
for , , , .
Absolute value
We define the absolute value to be the following:
for , .
Properties
TODO: Show that the decimal numbers are a ordered Heyting integral domain with decidable equality, decidable apartness, and decidable linear order, and that the decimal numbers are a Heyting integral subdomain of the rational numbers.
10 is a unit term.
Thus this is true:
this forms the basis of scientific notation.
See also