Signed and Unsigned Infinities

The unsigned infinity “ring” is the set of two elements

1. infinity

2. 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 take 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

1. plus infinity

2. a positive finite element

3. zero

4. a negative finite element

5. 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.

Note

The shorthand oo is predefined in Sage to be the same as +Infinity in the infinity ring. It is considered equal to, but not the same as Infinity in the UnsignedInfinityRing.

EXAMPLES:

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: unsigned_oo = P.0; unsigned_oo
Infinity

We compare finite numbers with infinity:

sage: 5 < unsigned_oo
True
sage: 5 > unsigned_oo
False
sage: unsigned_oo < 5
False
sage: unsigned_oo > 5
True

Demonstrating the shorthand oo versus Infinity:

sage: oo
+Infinity
sage: oo is InfinityRing.0
True
sage: oo is UnsignedInfinityRing.0
False
sage: oo == UnsignedInfinityRing.0
True

We do arithmetic:

sage: unsigned_oo + 5
Infinity

We make 1 / unsigned_oo return the integer 0 so that arithmetic of the following type works:

sage: (1/unsigned_oo) + 2
2
sage: 32/5 - (2.439/unsigned_oo)
32/5

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

sage: unsigned_oo/0
Traceback (most recent call last):
...
ValueError: quotient of number < oo by number < oo not defined

What happened above is that 0 is canonically coerced to “A number less than infinity” in the unsigned infinity ring. Next, Sage tries to divide by multiplying with its inverse. Finally, this inverse is not well-defined.

sage: 0/unsigned_oo
0
sage: unsigned_oo * 0
Traceback (most recent call last):
...
ValueError: unsigned oo times smaller number not defined
sage: unsigned_oo/unsigned_oo
Traceback (most recent call last):
...
ValueError: unsigned oo times smaller number not defined

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

The symbol oo is predefined as a shorthand for +Infinity:

sage: 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
0
sage: -1 / oo
0

We make 1 / oo and 1 / -oo return the integer 0 instead of the infinity ring Zero so that arithmetic of the following type works:

sage: (1/oo) + 2
2
sage: 32/5 - (2.439/-oo)
32/5

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

Signed infinity can also be represented by RR / RDF elements. But unsigned infinity cannot:

sage: oo in RR, oo in RDF
(True, True)
sage: unsigned_infinity in RR, unsigned_infinity in RDF
(False, False)

class sage.rings.infinity.AnInfinity

Bases: object

lcm(x)

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

EXAMPLES:

sage: oo.lcm(0)
0
sage: oo.lcm(oo)
+Infinity
sage: oo.lcm(-oo)
+Infinity
sage: oo.lcm(10)
+Infinity
sage: (-oo).lcm(10)
+Infinity

class sage.rings.infinity.FiniteNumber(parent, x)

Bases: RingElement

Initialize self.

sign()

Return the sign of self.

EXAMPLES:

sage: sign(InfinityRing(2))
1
sage: sign(InfinityRing(0))
0
sage: sign(InfinityRing(-2))
-1

sqrt()

EXAMPLES:

sage: InfinityRing(7).sqrt()
A positive finite number
sage: InfinityRing(0).sqrt()
Zero
sage: InfinityRing(-.001).sqrt()
Traceback (most recent call last):
...
SignError: cannot take square root of a negative number

class sage.rings.infinity.InfinityRing_class

Bases: Singleton, CommutativeRing

Initialize self.

fraction_field()

This isn’t really a ring, let alone an integral domain.

gen(n=0)

The two generators are plus and minus infinity.

EXAMPLES:

sage: InfinityRing.gen(0)
+Infinity
sage: InfinityRing.gen(1)
-Infinity
sage: InfinityRing.gen(2)
Traceback (most recent call last):
...
IndexError: n must be 0 or 1

gens()

The two generators are plus and minus infinity.

EXAMPLES:

sage: InfinityRing.gens()
(+Infinity, -Infinity)

is_commutative()

The Infinity Ring is commutative

EXAMPLES:

sage: InfinityRing.is_commutative()
True

is_zero()

The Infinity Ring is not zero

EXAMPLES:

sage: InfinityRing.is_zero()
False

ngens()

The two generators are plus and minus infinity.

EXAMPLES:

sage: InfinityRing.ngens()
2
sage: len(InfinityRing.gens())
2

class sage.rings.infinity.LessThanInfinity(*args)

Bases: _uniq, RingElement

Initialize self.

EXAMPLES:

sage: sage.rings.infinity.LessThanInfinity() is UnsignedInfinityRing(5)
True

sign()

Raise an error because the sign of self is not well defined.

EXAMPLES:

sage: sign(UnsignedInfinityRing(2))
Traceback (most recent call last):
...
NotImplementedError: sign of number < oo is not well defined
sage: sign(UnsignedInfinityRing(0))
Traceback (most recent call last):
...
NotImplementedError: sign of number < oo is not well defined
sage: sign(UnsignedInfinityRing(-2))
Traceback (most recent call last):
...
NotImplementedError: sign of number < oo is not well defined

class sage.rings.infinity.MinusInfinity(*args)

Bases: _uniq, AnInfinity, InfinityElement

Initialize self.

sqrt()

EXAMPLES:

sage: (-oo).sqrt()
Traceback (most recent call last):
...
SignError: cannot take square root of negative infinity

class sage.rings.infinity.PlusInfinity(*args)

Bases: _uniq, AnInfinity, InfinityElement

Initialize self.

sqrt()

The square root of self.

The square root of infinity is infinity.

EXAMPLES:

sage: oo.sqrt()
+Infinity

exception sage.rings.infinity.SignError

Bases: ArithmeticError

Sign error exception.

class sage.rings.infinity.UnsignedInfinity(*args)

Bases: _uniq, AnInfinity, InfinityElement

Initialize self.

class sage.rings.infinity.UnsignedInfinityRing_class

Bases: Singleton, CommutativeRing

Initialize self.

fraction_field()

The unsigned infinity ring isn’t an integral domain.

EXAMPLES:

sage: UnsignedInfinityRing.fraction_field()
Traceback (most recent call last):
...
TypeError: infinity 'ring' has no fraction field

gen(n=0)

The “generator” of self is the infinity object.

EXAMPLES:

sage: UnsignedInfinityRing.gen()
Infinity
sage: UnsignedInfinityRing.gen(1)
Traceback (most recent call last):
...
IndexError: UnsignedInfinityRing only has one generator

gens()

The “generator” of self is the infinity object.

EXAMPLES:

sage: UnsignedInfinityRing.gens()
(Infinity,)

less_than_infinity()

This is the element that represents a finite value.

EXAMPLES:

sage: UnsignedInfinityRing.less_than_infinity()
A number less than infinity
sage: UnsignedInfinityRing(5) is UnsignedInfinityRing.less_than_infinity()
True

ngens()

The unsigned infinity ring has one “generator.”

EXAMPLES:

sage: UnsignedInfinityRing.ngens()
1
sage: len(UnsignedInfinityRing.gens())
1

sage.rings.infinity.is_Infinite(x)

This is a type check for infinity elements.

EXAMPLES:

sage: sage.rings.infinity.is_Infinite(oo)
True
sage: sage.rings.infinity.is_Infinite(-oo)
True
sage: sage.rings.infinity.is_Infinite(unsigned_infinity)
True
sage: sage.rings.infinity.is_Infinite(3)
False
sage: sage.rings.infinity.is_Infinite(RR(infinity))
False
sage: sage.rings.infinity.is_Infinite(ZZ)
False

sage.rings.infinity.test_comparison(ring)

Check comparison with infinity

INPUT:

• ring – a sub-ring of the real numbers

OUTPUT:

Various attempts are made to generate elements of ring. An assertion is triggered if one of these elements does not compare correctly with plus/minus infinity.

EXAMPLES:

sage: from sage.rings.infinity import test_comparison
sage: rings = [ZZ, QQ, RDF]
sage: rings += [RR, RealField(200)]                                             # needs sage.rings.real_mpfr
sage: rings += [RLF, RIF]                                                       # needs sage.rings.real_interval_field
sage: for R in rings:
....:     print('testing {}'.format(R))
....:     test_comparison(R)
testing Integer Ring
testing Rational Field
testing Real Double Field...
sage: test_comparison(AA)                                                       # needs sage.rings.number_field

Comparison with number fields does not work:

sage: x = polygen(ZZ, 'x')
sage: K.<sqrt3> = NumberField(x^2 - 3)                                          # needs sage.rings.number_field
sage: (-oo < 1 + sqrt3) and (1 + sqrt3 < oo)    # known bug                     # needs sage.rings.number_field
False

The symbolic ring handles its own infinities, but answers False (meaning: cannot decide) already for some very elementary comparisons:

sage: test_comparison(SR)               # known bug                             # needs sage.symbolic
Traceback (most recent call last):
...
AssertionError: testing -1000.0 in Symbolic Ring: id = ...

sage.rings.infinity.test_signed_infinity(pos_inf)

Test consistency of infinity representations.

There are different possible representations of infinity in Sage. These are all consistent with the infinity ring, that is, compare with infinity in the expected way. See also github issue #14045

INPUT:

• pos_inf – a representation of positive infinity.

OUTPUT:

An assertion error is raised if the representation is not consistent with the infinity ring.

Check that github issue #14045 is fixed:

sage: InfinityRing(float('+inf'))
+Infinity
sage: InfinityRing(float('-inf'))
-Infinity
sage: oo > float('+inf')
False
sage: oo == float('+inf')
True

EXAMPLES:

sage: from sage.rings.infinity import test_signed_infinity
sage: test_signed_infinity(oo)
sage: test_signed_infinity(float('+inf'))
sage: test_signed_infinity(RLF(oo))                                             # needs sage.rings.real_interval_field
sage: test_signed_infinity(RIF(oo))                                             # needs sage.rings.real_interval_field
sage: test_signed_infinity(SR(oo))                                              # needs sage.symbolic