# 22.5 Infinity Rings

Module: `sage.rings.infinity`

Infinity Rings

The unsigned infinity ``ring'' is the set of two elements

```        * infinity
* A number less than infinity
```

The rules for arithmetic are that the unsigned infinity ring does not canonically coerce to any other ring, and all other rings canonically coerce to the unsigned infinity ring, sending all elements to the single element ``a number less than infinity'' of the unsigned infinity ring. Arithmetic and comparisons then takes place in the unsigned infinity ring, where all arithmetic operations that are well defined are defined.

The infinity ``ring'' is the set of five elements

```        * plus infinity
* a positive finite element
* zero
* a negative finite element
* negative infinity
```

The infinity ring coerces to the unsigned infinity ring, sending the infinite elements to infinity and the non-infinite elements to ``a number less than infinity.'' Any ordered ring coerces to the infinity ring in the obvious way.

We fetch the unsigned infinity ring and create some elements:

```sage: P = UnsignedInfinityRing; P
The Unsigned Infinity Ring
sage: P(5)
A number less than infinity
sage: P.ngens()
1
sage: oo = P.0; oo
Infinity
```

We compare finite numbers with infinity.

```sage: 5 < oo
True
sage: 5 > oo
False
sage: oo < 5
False
sage: oo > 5
True
```

We do arithmetic.

```sage: oo + 5
Infinity
```

Note that many operations are not defined, since the result is not well defined.

```sage: oo/0
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '/': 'The Unsigned Infinity
Ring' and 'Integer Ring'
```

What happened above is that 0 is canonically coerced to "a number less than infinity" in the unsigned infinity ring, and the quotient is then not well defined.

```sage: 0/oo
A number less than infinity
sage: oo * 0
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'The Unsigned Infinity
Ring' and 'Integer Ring'
sage: oo/oo
Traceback (most recent call last):
...
TypeError: infinity 'ring' has no fraction field
```

In the infinity ring, we can negate infinity, multiply positive numbers by infinity, etc.

```sage: P = InfinityRing; P
The Infinity Ring
sage: P(5)
A positive finite number
sage: oo = P.0; oo
+Infinity
```

We compare finite and infinite elements

```sage: 5 < oo
True
sage: P(-5) < P(5)
True
sage: P(2) < P(3)
False
sage: -oo < oo
True
```

We can do more arithmetic than in the unsigned infinity ring.

```sage: 2 * oo
+Infinity
sage: -2 * oo
-Infinity
sage: 1 - oo
-Infinity
sage: 1 / oo
Zero
sage: -1 / oo
Zero
```

If we try to subtract infinities or multiply infinity by zero we still get an error.

```sage: oo - oo
Traceback (most recent call last):
...
SignError: cannot add infinity to minus infinity
sage: 0 * oo
Traceback (most recent call last):
...
SignError: cannot multiply infinity by zero
sage: P(2) + P(-3)
Traceback (most recent call last):
...
SignError: cannot add positive finite value to negative finite value
```

TESTS:

```sage: P = InfinityRing
True
```

```sage: P(2) == loads(dumps(P(2)))
True
```

The following is assumed in a lot of code (i.e., "is" is used for testing whether something is infinity), so make sure it is satisfied:

```sage: loads(dumps(infinity)) is infinity
True
```

Module-level Functions

 is_Infinite( x)

Class: `FiniteNumber`

class FiniteNumber
 FiniteNumber( self, parent, x)

Functions: sqrt, square_root

Special Functions: __abs__, __cmp__, __invert__, _add_, _div_, _latex_, _mul_, _neg_, _repr_, _sub_

Class: `InfinityRing_class`

class InfinityRing_class
 InfinityRing_class( self)

Functions: fraction_field, gen, gens, ngens

Special Functions: __call__, __cmp__, _coerce_impl, _repr_

Class: `LessThanInfinity`

class LessThanInfinity
 LessThanInfinity( self, [parent=The Unsigned Infinity Ring])

Special Functions: __cmp__, _add_, _div_, _latex_, _mul_, _repr_, _sub_

Class: `MinusInfinity`

class MinusInfinity
 MinusInfinity( self)

Functions: lcm, sqrt, square_root

 lcm( self, x)
Return the least common multiple of -oo and x, which is by definition oo unless x is 0.

```sage: moo = InfinityRing.gen(1)
sage: moo.lcm(0)
0
sage: moo.lcm(oo)
+Infinity
sage: moo.lcm(10)
+Infinity
```

Special Functions: __abs__, __cmp__, __invert__, _add_, _div_, _latex_, _maxima_init_, _mul_, _neg_, _repr_, _sub_

 _maxima_init_( self)

```sage: maxima(-oo)
minf
```

Class: `PlusInfinity`

class PlusInfinity
 PlusInfinity( self)

Functions: lcm, sqrt, square_root

 lcm( self, x)
Return the least common multiple of oo and x, which is by definition oo unless x is 0.

```sage: oo = InfinityRing.gen(0)
sage: oo.lcm(0)
0
sage: oo.lcm(oo)
+Infinity
sage: oo.lcm(10)
+Infinity
```

Special Functions: __abs__, __cmp__, __invert__, __repr__, _add_, _div_, _latex_, _maxima_init_, _mul_, _neg_, _sub_

 _maxima_init_( self)

```sage: maxima(oo)
inf
```

Class: `SignError`

class SignError

Class: `UnsignedInfinity`

class UnsignedInfinity
 UnsignedInfinity( self)

Functions: lcm

 lcm( self, x)
Return the least common multiple of oo and x, which is by definition oo unless x is 0.

```sage: oo = UnsignedInfinityRing.gen(0)
sage: oo.lcm(0)
0
sage: oo.lcm(oo)
Infinity
sage: oo.lcm(10)
Infinity
```

Special Functions: __cmp__, _add_, _latex_, _maxima_init_, _mul_, _repr_, _sub_

Class: `UnsignedInfinityRing_class`

class UnsignedInfinityRing_class
 UnsignedInfinityRing_class( self)

Functions: fraction_field, gen, gens, less_than_infinity, ngens

Special Functions: __call__, __cmp__, _coerce_impl, _repr_