# OrderedCompletion RΒΆ

- R: SetCategory

Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989

+: (%, %) -> % if R has AbelianMonoid

-: % -> % if R has AbelianGroup

- <: (%, %) -> Boolean if R has OrderedSet
- from PartialOrder
- <=: (%, %) -> Boolean if R has OrderedSet
- from PartialOrder
- =: (%, %) -> Boolean
- from BasicType
- >: (%, %) -> Boolean if R has OrderedSet
- from PartialOrder
- >=: (%, %) -> Boolean if R has OrderedSet
- from PartialOrder
- ~=: (%, %) -> Boolean
- from BasicType
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer
- from RetractableTo Fraction Integer
- coerce: Integer -> % if R has RetractableTo Integer
- from RetractableTo Integer
- coerce: R -> %
- from RetractableTo R

- finite?: % -> Boolean
`finite?(x)`

tests if`x`

is finite.- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- infinite?: % -> Boolean
`infinite?(x)`

tests if`x`

is +infinity or -infinity,- latex: % -> String
- from SetCategory
- max: (%, %) -> % if R has OrderedSet
- from OrderedSet
- min: (%, %) -> % if R has OrderedSet
- from OrderedSet

- minusInfinity: () -> %
`minusInfinity()`

returns -infinity.

- plusInfinity: () -> %
`plusInfinity()`

returns +infinity.

- rational: % -> Fraction Integer if R has IntegerNumberSystem
`rational(x)`

returns`x`

as a finite rational number. Error: if`x`

cannot be so converted.

- rational?: % -> Boolean if R has IntegerNumberSystem
`rational?(x)`

tests if`x`

is a finite rational number.

- rationalIfCan: % -> Union(Fraction Integer, failed) if R has IntegerNumberSystem
`rationalIfCan(x)`

returns`x`

as a finite rational number if it is one and “failed” otherwise.- retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
- from RetractableTo Fraction Integer
- retract: % -> Integer if R has RetractableTo Integer
- from RetractableTo Integer
- retract: % -> R
- from RetractableTo R
- retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
- from RetractableTo Fraction Integer
- retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
- from RetractableTo Integer
- retractIfCan: % -> Union(R, failed)
- from RetractableTo R
- smaller?: (%, %) -> Boolean if R has OrderedSet
- from Comparable

- whatInfinity: % -> SingleInteger
`whatInfinity(x)`

returns 0 if`x`

is finite, 1 if`x`

is +infinity, and`-1`

if`x`

is -infinity.

Comparable if R has OrderedSet

OrderedSet if R has OrderedSet

PartialOrder if R has OrderedSet

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer