Occasionally it is convenient to work with infinite quantities (for example, when working with valuations or cardinalities). {Magma} provides two such objects, the positive and negative "integer" infinities. This section describes the {Magma} facilities for dealing with such objects.

Subsections

• Creation
• Arithmetic
• Comparison
• Miscellaneous

Certain system intrinsics such as Valuation which normally return an integer may return an infinite object for appropriate exceptional cases. Two special intrinsics are also provided to create infinite objects.

Infinity() : -> Infty

The positive infinity object.

The negative infinity object.

Only basic arithmetic operations are provided for infinite objects. Here the infinite objects are viewed as belonging to the integers in that either of the arguments to the operation may be an integer. Certain forms (such as infinity - infinity or infinity * 0) are not well defined and will trigger a runtime error.

- x : Infty -> Infty

x + y : Infty, Infty -> Infty

x + y : Infty, RngIntElt -> Infty

x + y : RngIntElt, Infty -> Infty

x - y : Infty, Infty -> Infty

x - y : Infty, RngIntElt -> Infty

x - y : RngIntElt, Infty -> Infty

x * y : Infty, Infty -> Infty

x * y : Infty, RngIntElt -> Infty

x * y : RngIntElt, Infty -> Infty

x / y : RngIntElt, Infty -> RngIntElt

x / y : Infty, RngIntElt -> Infty

x ^ n : Infty, Infty -> Infty

x ^ n : Infty, RngIntElt -> Infty

Infinite objects may be compared with themselves and integers.

x eq y : Infty, Infty -> BoolElt

x eq y : Infty, RngIntElt -> BoolElt

x eq y : RngIntElt, Infty -> BoolElt

x ne y : Infty, Infty -> BoolElt

x ne y : Infty, RngIntElt -> BoolElt

x ne y : RngIntElt, Infty -> BoolElt

x lt y : Infty, Infty -> BoolElt

x lt y : Infty, RngIntElt -> BoolElt

x lt y : RngIntElt, Infty -> BoolElt

x le y : Infty, Infty -> BoolElt

x le y : Infty, RngIntElt -> BoolElt

x le y : RngIntElt, Infty -> BoolElt

x gt y : Infty, Infty -> BoolElt

x gt y : Infty, RngIntElt -> BoolElt

x gt y : RngIntElt, Infty -> BoolElt

x ge y : Infty, Infty -> BoolElt

x ge y : Infty, RngIntElt -> BoolElt

x ge y : RngIntElt, Infty -> BoolElt

Maximum(x, y) : Infty, Infty -> .

Maximum(x, y) : Infty, RngIntElt -> .

Maximum(x, y) : RngIntElt, Infty -> .

Minimum(x, y) : Infty, Infty -> .

Minimum(x, y) : Infty, RngIntElt -> .

Minimum(x, y) : RngIntElt, Infty -> .

Returns 1 if x is the positive infinite object, -1 if x is the negative infinite object.

IsFinite(x) : RngIntElt -> BoolElt

True if x is finite, otherwise false. This is more convenient than checking the type of x.