r"""
Infinity Rings
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::
sage: oo
+Infinity
sage: oo is InfinityRing.0
True
sage: oo is UnsignedInfinityRing.0
False
sage: oo == UnsignedInfinityRing.0
True
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
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: unsigned oo times smaller number not defined
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/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
TESTS::
sage: P = InfinityRing
sage: P == loads(dumps(P))
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
"""
from sage.rings.ring_element import RingElement
from sage.rings.ring import Ring
from sage.structure.element import RingElement, InfinityElement, PlusInfinityElement, MinusInfinityElement
from sage.structure.parent_gens import ParentWithGens
import sage.rings.integer
import sage.rings.rational
from sage.rings.integer_ring import ZZ
_obj = {}
class _uniq(object):
def __new__(cls, *args):
"""
This ensures uniqueness of these objects.
EXAMPLE::
sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class()
True
"""
if _obj.has_key(cls):
return _obj[cls]
_obj[cls] = O = cls.__bases__[-1].__new__(cls, *args)
return O
class AnInfinity:
def _repr_(self):
"""
TESTS::
sage: [x._repr_() for x in [unsigned_infinity, oo, -oo]]
['Infinity', '+Infinity', '-Infinity']
"""
return self._sign_char + "Infinity"
def _giac_init_(self):
"""
TESTS::
sage: [x._repr_() for x in [unsigned_infinity, oo, -oo]]
['Infinity', '+Infinity', '-Infinity']
"""
return self._sign_char + "infinity"
def _maxima_init_(self):
"""
TESTS::
sage: maxima(-oo)
minf
sage: [x._maxima_init_() for x in [unsigned_infinity, oo, -oo]]
['inf', 'inf', 'minf']
"""
if self._sign < 0:
return 'minf'
else:
return 'inf'
def _pari_(self):
"""
Convert self to a Pari object.
This always raises an exception since Pari does not have
infinities.
TESTS::
sage: pari(-oo)
Traceback (most recent call last):
...
TypeError: cannot convert infinity to Pari
sage: pari(oo)
Traceback (most recent call last):
...
TypeError: cannot convert infinity to Pari
"""
raise TypeError, 'cannot convert infinity to Pari'
def _latex_(self):
"""
EXAMPLES::
sage: latex(oo)
+\infty
sage: [x._latex_() for x in [unsigned_infinity, oo, -oo]]
['\\infty', '+\\infty', '-\\infty']
"""
return self._sign_char + "\\infty"
def __cmp__(self, other):
"""
EXAMPLES::
sage: oo == oo
True
sage: oo < oo
False
sage: -oo < oo
True
sage: -oo < 3 < oo
True
sage: unsigned_infinity == 3
False
sage: unsigned_infinity == unsigned_infinity
True
sage: unsigned_infinity == oo
True
"""
if isinstance(other, AnInfinity):
return cmp(self._sign, other._sign)
elif self._sign:
return self._sign
else:
return 1
def __abs__(self):
"""
EXAMPLES::
sage: [abs(x) for x in [UnsignedInfinityRing.gen(), oo, -oo]]
[Infinity, +Infinity, +Infinity]
"""
return -self if self._sign < 0 else self
def _add_(self, other):
"""
EXAMPLES::
sage: -oo + -oo
-Infinity
sage: -oo + 3
-Infinity
sage: oo + -100
+Infinity
sage: oo + -oo
Traceback (most recent call last):
...
SignError: cannot add infinity to minus infinity
sage: unsigned_infinity = UnsignedInfinityRing.gen()
sage: unsigned_infinity + unsigned_infinity
Infinity
sage: unsigned_infinity + 88/3
Infinity
"""
if isinstance(other, AnInfinity):
if self._sign != other._sign:
raise SignError, "cannot add infinity to minus infinity"
return self
def _sub_(self, other):
"""
EXAMPLES::
sage: -oo - oo
-Infinity
sage: oo - -oo
+Infinity
sage: oo - 4
+Infinity
sage: -oo - 1
-Infinity
sage: oo - oo
Traceback (most recent call last):
...
SignError: cannot add infinity to minus infinity
sage: unsigned_infinity - 4
Infinity
sage: unsigned_infinity - unsigned_infinity
Traceback (most recent call last):
...
ValueError: oo - oo not defined
"""
if isinstance(other, AnInfinity):
if self._sign == 0:
raise ValueError, "oo - oo not defined"
elif self._sign == other._sign:
raise SignError, "cannot add infinity to minus infinity"
return self
def _mul_(self, other):
"""
EXAMPLES::
sage: oo * 19
+Infinity
sage: oo * oo
+Infinity
sage: -oo * oo
-Infinity
sage: -oo * 4
-Infinity
sage: -oo * -2/3
+Infinity
sage: -oo * 0
Traceback (most recent call last):
...
SignError: cannot multiply infinity by zero
"""
if other < 0:
return -self
if other > 0:
return self
raise SignError, "cannot multiply infinity by zero"
def _div_(self, other):
"""
EXAMPLES::
sage: 1.5 / oo
0
sage: oo / -4
-Infinity
sage: oo / oo
Traceback (most recent call last):
...
SignError: cannot multiply infinity by zero
"""
return self * ~other
def __float__(self):
r"""
Generate a floating-point infinity. The printing of
floating-point infinity varies across platforms.
EXAMPLES::
sage: RDF(infinity)
+infinity
sage: float(infinity) # random
+infinity
sage: CDF(infinity)
+infinity
sage: infinity.__float__() # random
+infinity
sage: RDF(-infinity)
-infinity
sage: float(-infinity) # random
-inf
sage: CDF(-infinity)
-infinity
sage: (-infinity).__float__() # random
-inf
"""
from sage.rings.all import RR
return float(RR(self))
def lcm(self, 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
"""
if x == 0:
return x
else:
return abs(self)
class UnsignedInfinityRing_class(_uniq, Ring):
def __init__(self):
"""
TESTS::
sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class() is UnsignedInfinityRing
True
"""
ParentWithGens.__init__(self, self, names=('oo',), normalize=False)
def ngens(self):
"""
The unsigned infinity ring has one "generator."
EXAMPLES::
sage: UnsignedInfinityRing.ngens()
1
sage: len(UnsignedInfinityRing.gens())
1
"""
return 1
def fraction_field(self):
"""
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
"""
raise TypeError, "infinity 'ring' has no fraction field"
def gen(self, 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
"""
if n == 0:
try:
return self._gen
except AttributeError:
self._gen = UnsignedInfinity()
return self._gen
else:
raise IndexError, "UnsignedInfinityRing only has one generator"
def gens(self):
"""
The "generator" of self is the infinity object.
EXAMPLES::
sage: UnsignedInfinityRing.gens()
[Infinity]
"""
return [self.gen()]
def less_than_infinity(self):
"""
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
"""
try:
return self._less_than_infinity
except AttributeError:
self._less_than_infinity = LessThanInfinity(self)
return self._less_than_infinity
def _repr_(self):
"""
TESTS::
sage: UnsignedInfinityRing._repr_()
'The Unsigned Infinity Ring'
"""
return "The Unsigned Infinity Ring"
def __cmp__(self, right):
"""
TESTS::
sage: infinity == UnsignedInfinityRing.gen()
True
sage: UnsignedInfinityRing(3) == UnsignedInfinityRing(-19.5)
True
"""
if isinstance(right, UnsignedInfinityRing_class):
return 0
return cmp(type(self), type(right))
def _element_constructor_(self, x):
"""
TESTS::
sage: UnsignedInfinityRing(2)
A number less than infinity
sage: UnsignedInfinityRing(I)
A number less than infinity
sage: UnsignedInfinityRing(infinity)
Infinity
"""
if isinstance(x, InfinityElement):
if x.parent() is self:
return x
else:
return self.gen()
elif isinstance(x, RingElement) or isinstance(x, (int,long,float,complex)):
return self.less_than_infinity()
else:
raise TypeError
def _coerce_map_from_(self, R):
"""
EXAMPLES::
sage: UnsignedInfinityRing.has_coerce_map_from(int)
True
sage: UnsignedInfinityRing.has_coerce_map_from(CC)
True
sage: UnsignedInfinityRing.has_coerce_map_from(QuadraticField(-163, 'a'))
True
sage: UnsignedInfinityRing.has_coerce_map_from(QQ^3)
False
sage: UnsignedInfinityRing.has_coerce_map_from(SymmetricGroup(13))
False
"""
return isinstance(R, Ring) or R in (int, long, float, complex)
UnsignedInfinityRing = UnsignedInfinityRing_class()
class LessThanInfinity(_uniq, RingElement):
def __init__(self, parent=UnsignedInfinityRing):
"""
EXAMPLES::
sage: sage.rings.infinity.LessThanInfinity() is UnsignedInfinityRing(5)
True
"""
RingElement.__init__(self, parent)
def _repr_(self):
"""
EXAMPLES::
sage: UnsignedInfinityRing(5)._repr_()
'A number less than infinity'
"""
return "A number less than infinity"
def _latex_(self):
"""
EXAMPLES::
sage: UnsignedInfinityRing(5)._latex_()
'(<\\infty)'
"""
return "(<\\infty)"
def _add_(self, other):
"""
EXAMPLES::
sage: UnsignedInfinityRing(5) + UnsignedInfinityRing(-3)
A number less than infinity
sage: UnsignedInfinityRing(5) + unsigned_infinity
Infinity
"""
if isinstance(other, UnsignedInfinity):
return other
return self
def _sub_(self, other):
"""
EXAMPLES::
sage: UnsignedInfinityRing(5) - UnsignedInfinityRing(-3)
A number less than infinity
sage: UnsignedInfinityRing(5) - unsigned_infinity
Infinity
"""
if isinstance(other, UnsignedInfinity):
return other
return self
def _mul_(self, other):
"""
EXAMPLES::
sage: UnsignedInfinityRing(4) * UnsignedInfinityRing(-3)
A number less than infinity
sage: 5 * unsigned_infinity
Traceback (most recent call last):
...
ValueError: oo times number < oo not defined
sage: unsigned_infinity * unsigned_infinity
Infinity
"""
if isinstance(other, UnsignedInfinity):
raise ValueError, "oo times number < oo not defined"
return self
def _div_(self, other):
"""
Can't eliminate possibility of zero division....
EXAMPLES::
sage: UnsignedInfinityRing(2) / UnsignedInfinityRing(5)
Traceback (most recent call last):
...
ValueError: quotient of number < oo by number < oo not defined
sage: 1 / unsigned_infinity
0
"""
if isinstance(other, UnsignedInfinity):
return ZZ(0)
raise ValueError, "quotient of number < oo by number < oo not defined"
def __cmp__(self, other):
"""
EXAMPLES::
sage: 1 == unsigned_infinity
False
"""
if isinstance(other, UnsignedInfinity):
return -1
return 0
class UnsignedInfinity(_uniq, AnInfinity, InfinityElement):
_sign = 0
_sign_char = ''
def __init__(self):
"""
TESTS::
sage: sage.rings.infinity.UnsignedInfinity() is sage.rings.infinity.UnsignedInfinity() is unsigned_infinity
True
"""
InfinityElement.__init__(self, UnsignedInfinityRing)
def _mul_(self, other):
"""
Can't rule out an attempt at multiplication by 0.
EXAMPLES::
sage: unsigned_infinity * unsigned_infinity
Infinity
sage: unsigned_infinity * 0
Traceback (most recent call last):
...
ValueError: unsigned oo times smaller number not defined
sage: unsigned_infinity * 3
Traceback (most recent call last):
...
ValueError: unsigned oo times smaller number not defined
"""
if isinstance(other, UnsignedInfinity):
return self
raise ValueError, "unsigned oo times smaller number not defined"
unsigned_infinity = UnsignedInfinityRing.gen(0)
less_than_infinity = UnsignedInfinityRing.less_than_infinity()
def 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
"""
return isinstance(x, InfinityElement)
class SignError(Exception):
pass
class InfinityRing_class(_uniq, Ring):
def __init__(self):
"""
TEST::
sage: sage.rings.infinity.InfinityRing_class() is sage.rings.infinity.InfinityRing_class() is InfinityRing
True
"""
ParentWithGens.__init__(self, self, names=('oo',), normalize=False)
def fraction_field(self):
"""
This isn't really a ring, let alone an integral domain.
TEST::
sage: InfinityRing.fraction_field()
Traceback (most recent call last):
...
TypeError: infinity 'ring' has no fraction field
"""
raise TypeError, "infinity 'ring' has no fraction field"
def ngens(self):
"""
The two generators are plus and minus infinity.
EXAMPLES::
sage: InfinityRing.ngens()
2
sage: len(InfinityRing.gens())
2
"""
return 2
def gen(self, 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
"""
try:
if n == 0:
return self._gen0
elif n == 1:
return self._gen1
else:
raise IndexError, "n must be 0 or 1"
except AttributeError:
if n == 0:
self._gen0 = PlusInfinity()
return self._gen0
elif n == 1:
self._gen1 = MinusInfinity()
return self._gen1
def gens(self):
"""
The two generators are plus and minus infinity.
EXAMPLES::
sage: InfinityRing.gens()
[+Infinity, -Infinity]
"""
return [self.gen(0), self.gen(1)]
def _repr_(self):
"""
TEST::
sage: InfinityRing._repr_()
'The Infinity Ring'
"""
return "The Infinity Ring"
def __cmp__(self, right):
"""
TESTS::
sage: InfinityRing == InfinityRing
True
sage: InfinityRing == UnsignedInfinityRing
False
"""
if isinstance(right, InfinityRing_class):
return 0
return cmp(type(self), type(right))
def _element_constructor_(self, x):
"""
TESTS::
sage: InfinityRing(-oo)
-Infinity
sage: InfinityRing(3)
A positive finite number
sage: InfinityRing(-1.5)
A negative finite number
sage: [InfinityRing(a) for a in [-2..2]]
[A negative finite number, A negative finite number, Zero, A positive finite number, A positive finite number]
sage: K.<a> = QuadraticField(3)
sage: InfinityRing(a)
A positive finite number
sage: InfinityRing(a - 2)
A negative finite number
"""
if isinstance(x, PlusInfinityElement):
return self.gen(0)
elif isinstance(x, MinusInfinityElement):
return self.gen(1)
elif isinstance(x, InfinityElement):
return self.gen(0)
elif (isinstance(x, (sage.rings.integer.Integer,
sage.rings.rational.Rational,
sage.rings.real_double.RealDoubleElement,
sage.rings.real_mpfr.RealNumber))
or isinstance(x, (int,long,float))
or sage.rings.real_mpfr.RealField(sage.rings.real_mpfr.mpfr_prec_min())(x)):
if x < 0:
return FiniteNumber(self, -1)
elif x > 0:
return FiniteNumber(self, 1)
elif x == 0:
return FiniteNumber(self, 0)
else:
raise TypeError
else:
raise TypeError
def _coerce_map_from_(self, R):
"""
There is a coercion from anything that has a coercion into the reals.
EXAMPLES::
sage: InfinityRing.has_coerce_map_from(int)
True
sage: InfinityRing.has_coerce_map_from(AA)
True
sage: InfinityRing.has_coerce_map_from(RDF)
True
sage: InfinityRing.has_coerce_map_from(CC)
False
"""
return sage.rings.real_mpfr.RealField(sage.rings.real_mpfr.mpfr_prec_min()).has_coerce_map_from(R)
class FiniteNumber(RingElement):
def __init__(self, parent, x):
"""
TESTS::
sage: sage.rings.infinity.FiniteNumber(InfinityRing, 1)
A positive finite number
sage: sage.rings.infinity.FiniteNumber(InfinityRing, -1)
A negative finite number
sage: sage.rings.infinity.FiniteNumber(InfinityRing, 0)
Zero
"""
RingElement.__init__(self, parent)
self.value = x
def __cmp__(self, other):
"""
EXAMPLES::
sage: P = InfinityRing
sage: -oo < P(-5) < P(0) < P(1.5) < oo
True
sage: P(1) < P(100)
False
sage: P(-1) == P(-100)
True
"""
if isinstance(other, PlusInfinity):
return -1
if isinstance(other, MinusInfinity):
return 1
return cmp(self.value, other.value)
def _add_(self, other):
"""
EXAMPLES::
sage: P = InfinityRing
sage: 4 + oo
+Infinity
sage: P(4) + P(2)
A positive finite number
sage: P(-1) + P(1)
Traceback (most recent call last):
...
SignError: cannot add positive finite value to negative finite value
"""
if isinstance(other, InfinityElement):
return other
if self.value * other.value < 0:
raise SignError, "cannot add positive finite value to negative finite value"
return FiniteNumber(self.parent(), self.value)
def _mul_(self, other):
"""
EXAMPLES::
sage: P = InfinityRing
sage: 0 * oo
Traceback (most recent call last):
...
SignError: cannot multiply infinity by zero
sage: -1 * oo
-Infinity
sage: -2 * oo
-Infinity
sage: 3 * oo
+Infinity
sage: -oo * oo
-Infinity
sage: P(0) * 3
0
sage: P(-3) * P(2/3)
A negative finite number
"""
if other.is_zero():
if isinstance(self, InfinityElement):
raise SignError, "cannot multiply infinity by zero"
return ZZ(0)
if self.value < 0:
if isinstance(other, InfinityElement):
return -other
return FiniteNumber(self.parent(), self.value * other.value)
if self.value > 0:
if isinstance(other, InfinityElement):
return other
return FiniteNumber(self.parent(), self.value * other.value)
if self.value == 0:
if isinstance(other, InfinityElement):
raise SignError, "cannot multiply infinity by zero"
return ZZ(0)
def _div_(self, other):
"""
EXAMPLES::
sage: P = InfinityRing
sage: 1 / oo
0
sage: oo / 4
+Infinity
sage: oo / -4
-Infinity
sage: P(1) / P(-4)
A negative finite number
"""
return self * ~other
def _sub_(self, other):
"""
EXAMPLES::
sage: P = InfinityRing
sage: 4 - oo
-Infinity
sage: 5 - -oo
+Infinity
sage: P(44) - P(4)
Traceback (most recent call last):
...
SignError: cannot add positive finite value to negative finite value
sage: P(44) - P(-1)
A positive finite number
"""
return self._add_(-other)
def __invert__(self):
"""
EXAMPLES::
sage: P = InfinityRing
sage: ~P(2)
A positive finite number
sage: ~P(-7)
A negative finite number
sage: ~P(0)
Traceback (most recent call last):
...
ZeroDivisionError: Cannot divide by zero
"""
if self.value == 0:
raise ZeroDivisionError, "Cannot divide by zero"
return self
def _neg_(self):
"""
EXAMPLES::
sage: a = InfinityRing(5); a
A positive finite number
sage: -a
A negative finite number
sage: -(-a) == a
True
sage: -InfinityRing(0)
Zero
"""
return FiniteNumber(self.parent(), -self.value)
def _repr_(self):
"""
EXAMPLES::
sage: InfinityRing(-2)._repr_()
'A negative finite number'
sage: InfinityRing(7)._repr_()
'A positive finite number'
sage: InfinityRing(0)._repr_()
'Zero'
"""
if self.value < 0:
return "A negative finite number"
if self.value > 0:
return "A positive finite number"
return "Zero"
def _latex_(self):
"""
TESTS::
sage: a = InfinityRing(pi); a
A positive finite number
sage: a._latex_()
'A positive finite number'
sage: [latex(InfinityRing(a)) for a in [-2..2]]
[A negative finite number, A negative finite number, Zero, A positive finite number, A positive finite number]
"""
return self._repr_()
def __abs__(self):
"""
EXAMPLES::
sage: abs(InfinityRing(-3))
A positive finite number
sage: abs(InfinityRing(3))
A positive finite number
sage: abs(InfinityRing(0))
Zero
"""
if self.value == 0:
return FiniteNumber(self.parent(), 0)
return FiniteNumber(self.parent(), 1)
def sqrt(self):
"""
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
"""
if self.value < 0:
raise SignError, "cannot take square root of a negative number"
return self
class MinusInfinity(_uniq, AnInfinity, MinusInfinityElement):
_sign = -1
_sign_char = '-'
def __init__(self):
"""
TESTS::
sage: sage.rings.infinity.MinusInfinity() is sage.rings.infinity.MinusInfinity() is -oo
True
"""
InfinityElement.__init__(self, InfinityRing)
def _neg_(self):
"""
EXAMPLES::
sage: -(-oo)
+Infinity
"""
return self.parent().gen(0)
def sqrt(self):
"""
EXAMPLES::
sage: (-oo).sqrt()
Traceback (most recent call last):
...
SignError: cannot take square root of negative infinity
"""
raise SignError, "cannot take square root of negative infinity"
def _sympy_(self):
"""
Converts -oo to sympy -oo.
Then you don't have to worry which oo you use, like in these
examples:
EXAMPLE::
sage: import sympy
sage: bool(-oo == -sympy.oo)
True
sage: bool(SR(-oo) == -sympy.oo)
True
sage: bool((-oo)._sympy_() == -sympy.oo)
True
"""
import sympy
return -sympy.oo
class PlusInfinity(_uniq, AnInfinity, PlusInfinityElement):
_sign = 1
_sign_char = '+'
def __init__(self):
"""
TESTS::
sage: sage.rings.infinity.PlusInfinity() is sage.rings.infinity.PlusInfinity() is oo
True
"""
InfinityElement.__init__(self, InfinityRing)
def _neg_(self):
"""
TESTS::
sage: -oo
-Infinity
"""
return self.parent().gen(1)
def sqrt(self):
"""
EXAMPLES::
sage: oo.sqrt()
+Infinity
"""
return self
def _sympy_(self):
"""
Converts oo to sympy oo.
Then you don't have to worry which oo you use, like in these
examples:
EXAMPLE::
sage: import sympy
sage: bool(oo == sympy.oo) # indirect doctest
True
sage: bool(SR(oo) == sympy.oo)
True
"""
import sympy
return sympy.oo
InfinityRing = InfinityRing_class()
infinity = InfinityRing.gen(0)
Infinity = infinity
minus_infinity = InfinityRing.gen(1)
