Mathematical constants¶

The following standard mathematical constants are defined in Sage, along with support for coercing them into GAP, PARI/GP, KASH, Maxima, Mathematica, Maple, Octave, and Singular:

Sage

sage: pi
pi
sage: e             # base of the natural logarithm
e
sage: NaN           # Not a number
NaN
sage: golden_ratio
golden_ratio
sage: log2          # natural logarithm of the real number 2
log2
sage: euler_gamma   # Euler's gamma constant
euler_gamma
sage: catalan       # the Catalan constant
catalan
sage: khinchin      # Khinchin's constant
khinchin
sage: twinprime
twinprime
sage: mertens
mertens

Python

>>> from sage.all import *
>>> pi
pi
>>> e             # base of the natural logarithm
e
>>> NaN           # Not a number
NaN
>>> golden_ratio
golden_ratio
>>> log2          # natural logarithm of the real number 2
log2
>>> euler_gamma   # Euler's gamma constant
euler_gamma
>>> catalan       # the Catalan constant
catalan
>>> khinchin      # Khinchin's constant
khinchin
>>> twinprime
twinprime
>>> mertens
mertens

Support for coercion into the various systems means that if, e.g., you want to create \(\pi\) in Maxima and Singular, you don’t have to figure out the special notation for each system. You just type the following:

Sage

sage: maxima(pi)
%pi
sage: singular(pi)
pi
sage: gap(pi)
pi
sage: gp(pi)
3.141592653589793238462643383     # 32-bit
3.1415926535897932384626433832795028842   # 64-bit
sage: pari(pi)
3.14159265358979
sage: kash(pi)                    # optional - kash
3.14159265358979323846264338328
sage: mathematica(pi)             # optional - mathematica
Pi
sage: pi._maple_init_()
'Pi'
sage: octave(pi)                  # optional - octave
3.14159

Python

>>> from sage.all import *
>>> maxima(pi)
%pi
>>> singular(pi)
pi
>>> gap(pi)
pi
>>> gp(pi)
3.141592653589793238462643383     # 32-bit
3.1415926535897932384626433832795028842   # 64-bit
>>> pari(pi)
3.14159265358979
>>> kash(pi)                    # optional - kash
3.14159265358979323846264338328
>>> mathematica(pi)             # optional - mathematica
Pi
>>> pi._maple_init_()
'Pi'
>>> octave(pi)                  # optional - octave
3.14159

Arithmetic operations with constants also yield constants, which can be coerced into other systems or evaluated.

Sage

sage: a = pi + e*4/5; a
pi + 4/5*e
sage: maxima(a)
%pi+(4*%e)/5
sage: RealField(15)(a)           # 15 *bits* of precision
5.316
sage: gp(a)
5.316218116357029426750873360              # 32-bit
5.3162181163570294267508733603616328824    # 64-bit
sage: print(mathematica(a))                  # optional - mathematica
 4 E
 --- + Pi
  5

Python

>>> from sage.all import *
>>> a = pi + e*Integer(4)/Integer(5); a
pi + 4/5*e
>>> maxima(a)
%pi+(4*%e)/5
>>> RealField(Integer(15))(a)           # 15 *bits* of precision
5.316
>>> gp(a)
5.316218116357029426750873360              # 32-bit
5.3162181163570294267508733603616328824    # 64-bit
>>> print(mathematica(a))                  # optional - mathematica
 4 E
 --- + Pi
  5

EXAMPLES: Decimal expansions of constants

We can obtain floating point approximations to each of these constants by coercing into the real field with given precision. For example, to 200 binary places we have the following:

Sage

sage: R = RealField(200); R
Real Field with 200 bits of precision

Python

>>> from sage.all import *
>>> R = RealField(Integer(200)); R
Real Field with 200 bits of precision

Sage

sage: R(pi)
3.1415926535897932384626433832795028841971693993751058209749

Python

>>> from sage.all import *
>>> R(pi)
3.1415926535897932384626433832795028841971693993751058209749

Sage

sage: R(e)
2.7182818284590452353602874713526624977572470936999595749670

Python

>>> from sage.all import *
>>> R(e)
2.7182818284590452353602874713526624977572470936999595749670

Sage

sage: R(NaN)
NaN

Python

>>> from sage.all import *
>>> R(NaN)
NaN

Sage

sage: R(golden_ratio)
1.6180339887498948482045868343656381177203091798057628621354

Python

>>> from sage.all import *
>>> R(golden_ratio)
1.6180339887498948482045868343656381177203091798057628621354

Sage

sage: R(log2)
0.69314718055994530941723212145817656807550013436025525412068

Python

>>> from sage.all import *
>>> R(log2)
0.69314718055994530941723212145817656807550013436025525412068

Sage

sage: R(euler_gamma)
0.57721566490153286060651209008240243104215933593992359880577

Python

>>> from sage.all import *
>>> R(euler_gamma)
0.57721566490153286060651209008240243104215933593992359880577

Sage

sage: R(catalan)
0.91596559417721901505460351493238411077414937428167213426650

Python

>>> from sage.all import *
>>> R(catalan)
0.91596559417721901505460351493238411077414937428167213426650

Sage

sage: R(khinchin)
2.6854520010653064453097148354817956938203822939944629530512

Python

>>> from sage.all import *
>>> R(khinchin)
2.6854520010653064453097148354817956938203822939944629530512

EXAMPLES: Arithmetic with constants

Sage

sage: f = I*(e+1); f
I*e + I
sage: f^2
(I*e + I)^2
sage: _.expand()
-e^2 - 2*e - 1

Python

>>> from sage.all import *
>>> f = I*(e+Integer(1)); f
I*e + I
>>> f**Integer(2)
(I*e + I)^2
>>> _.expand()
-e^2 - 2*e - 1

Sage

sage: pp = pi+pi; pp
2*pi
sage: R(pp)
6.2831853071795864769252867665590057683943387987502116419499

Python

>>> from sage.all import *
>>> pp = pi+pi; pp
2*pi
>>> R(pp)
6.2831853071795864769252867665590057683943387987502116419499

Sage

sage: s = (1 + e^pi); s
e^pi + 1
sage: R(s)
24.140692632779269005729086367948547380266106242600211993445
sage: R(s-1)
23.140692632779269005729086367948547380266106242600211993445

Python

>>> from sage.all import *
>>> s = (Integer(1) + e**pi); s
e^pi + 1
>>> R(s)
24.140692632779269005729086367948547380266106242600211993445
>>> R(s-Integer(1))
23.140692632779269005729086367948547380266106242600211993445

Sage

sage: l = (1-log2)/(1+log2); l
-(log2 - 1)/(log2 + 1)
sage: R(l)
0.18123221829928249948761381864650311423330609774776013488056

Python

>>> from sage.all import *
>>> l = (Integer(1)-log2)/(Integer(1)+log2); l
-(log2 - 1)/(log2 + 1)
>>> R(l)
0.18123221829928249948761381864650311423330609774776013488056

Sage

sage: pim = maxima(pi)
sage: maxima.eval('fpprec : 100')
'100'
sage: pim.bfloat()
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068b0

Python

>>> from sage.all import *
>>> pim = maxima(pi)
>>> maxima.eval('fpprec : 100')
'100'
>>> pim.bfloat()
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068b0

AUTHORS:

Alex Clemesha (2006-01-15)
William Stein
Alex Clemesha, William Stein (2006-02-20): added new constants; removed todos
Didier Deshommes (2007-03-27): added constants from RQDF (deprecated)

class sage.symbolic.constants.Catalan(name='catalan')[source]¶

Bases: Constant

A number appearing in combinatorics defined as the Dirichlet beta function evaluated at the number 2.

EXAMPLES:

Sage

sage: catalan^2 + mertens
mertens + catalan^2

Python

>>> from sage.all import *
>>> catalan**Integer(2) + mertens
mertens + catalan^2

class sage.symbolic.constants.Constant(name, conversions=None, latex=None, mathml='', domain='complex')[source]¶

Bases: object

EXAMPLES:

Sage

sage: from sage.symbolic.constants import Constant
sage: p = Constant('p')
sage: loads(dumps(p))
p

Python

>>> from sage.all import *
>>> from sage.symbolic.constants import Constant
>>> p = Constant('p')
>>> loads(dumps(p))
p

domain()[source]¶

Return the domain of this constant. This is either positive, real, or complex, and is used by Pynac to make inferences about expressions containing this constant.

EXAMPLES:

Sage

sage: p = pi.pyobject(); p
pi
sage: type(_)
<class 'sage.symbolic.constants.Pi'>
sage: p.domain()
'positive'

Python

>>> from sage.all import *
>>> p = pi.pyobject(); p
pi
>>> type(_)
<class 'sage.symbolic.constants.Pi'>
>>> p.domain()
'positive'

expression()[source]¶

Return an expression for this constant.

EXAMPLES:

Sage

sage: a = pi.pyobject()
sage: pi2 = a.expression()
sage: pi2
pi
sage: pi2 + 2
pi + 2
sage: pi - pi2
0

Python

>>> from sage.all import *
>>> a = pi.pyobject()
>>> pi2 = a.expression()
>>> pi2
pi
>>> pi2 + Integer(2)
pi + 2
>>> pi - pi2
0

name()[source]¶

Return the name of this constant.

EXAMPLES:

Sage

sage: from sage.symbolic.constants import Constant
sage: c = Constant('c')
sage: c.name()
'c'

Python

>>> from sage.all import *
>>> from sage.symbolic.constants import Constant
>>> c = Constant('c')
>>> c.name()
'c'

class sage.symbolic.constants.EulerGamma(name='euler_gamma')[source]¶

Bases: Constant

The limiting difference between the harmonic series and the natural logarithm.

EXAMPLES:

Sage

sage: R = RealField()
sage: R(euler_gamma)
0.577215664901533
sage: R = RealField(200); R
Real Field with 200 bits of precision
sage: R(euler_gamma)
0.57721566490153286060651209008240243104215933593992359880577
sage: eg = euler_gamma + euler_gamma; eg
2*euler_gamma
sage: R(eg)
1.1544313298030657212130241801648048620843186718798471976115

Python

>>> from sage.all import *
>>> R = RealField()
>>> R(euler_gamma)
0.577215664901533
>>> R = RealField(Integer(200)); R
Real Field with 200 bits of precision
>>> R(euler_gamma)
0.57721566490153286060651209008240243104215933593992359880577
>>> eg = euler_gamma + euler_gamma; eg
2*euler_gamma
>>> R(eg)
1.1544313298030657212130241801648048620843186718798471976115

class sage.symbolic.constants.Glaisher(name='glaisher')[source]¶

Bases: Constant

The Glaisher-Kinkelin constant \(A = \exp(\frac{1}{12}-\zeta'(-1))\).

EXAMPLES:

Sage

sage: float(glaisher)
1.2824271291006226
sage: glaisher.n(digits=60)
1.28242712910062263687534256886979172776768892732500119206374
sage: a = glaisher + 2
sage: a
glaisher + 2
sage: parent(a)
Symbolic Ring

Python

>>> from sage.all import *
>>> float(glaisher)
1.2824271291006226
>>> glaisher.n(digits=Integer(60))
1.28242712910062263687534256886979172776768892732500119206374
>>> a = glaisher + Integer(2)
>>> a
glaisher + 2
>>> parent(a)
Symbolic Ring

class sage.symbolic.constants.GoldenRatio(name='golden_ratio')[source]¶

Bases: Constant

The number (1+sqrt(5))/2.

EXAMPLES:

Sage

sage: gr = golden_ratio
sage: RR(gr)
1.61803398874989
sage: R = RealField(200)
sage: R(gr)
1.6180339887498948482045868343656381177203091798057628621354
sage: grm = maxima(golden_ratio);grm
(sqrt(5)+1)/2
sage: grm + grm
sqrt(5)+1
sage: float(grm + grm)
3.23606797749979

Python

>>> from sage.all import *
>>> gr = golden_ratio
>>> RR(gr)
1.61803398874989
>>> R = RealField(Integer(200))
>>> R(gr)
1.6180339887498948482045868343656381177203091798057628621354
>>> grm = maxima(golden_ratio);grm
(sqrt(5)+1)/2
>>> grm + grm
sqrt(5)+1
>>> float(grm + grm)
3.23606797749979

minpoly(bits=None, degree=None, epsilon=0)[source]¶

EXAMPLES:

Sage

sage: golden_ratio.minpoly()
x^2 - x - 1

Python

>>> from sage.all import *
>>> golden_ratio.minpoly()
x^2 - x - 1

class sage.symbolic.constants.Khinchin(name='khinchin')[source]¶

Bases: Constant

The geometric mean of the continued fraction expansion of any (almost any) real number.

EXAMPLES:

Sage

sage: float(khinchin)
2.6854520010653062
sage: khinchin.n(digits=60)
2.68545200106530644530971483548179569382038229399446295305115
sage: m = mathematica(khinchin); m             # optional - mathematica
Khinchin
sage: m.N(200)                                 # optional - mathematica
2.685452001065306445309714835481795693820382293...32852204481940961807

Python

>>> from sage.all import *
>>> float(khinchin)
2.6854520010653062
>>> khinchin.n(digits=Integer(60))
2.68545200106530644530971483548179569382038229399446295305115
>>> m = mathematica(khinchin); m             # optional - mathematica
Khinchin
>>> m.N(Integer(200))                                 # optional - mathematica
2.685452001065306445309714835481795693820382293...32852204481940961807

class sage.symbolic.constants.Log2(name='log2')[source]¶

Bases: Constant

The natural logarithm of the real number 2.

EXAMPLES:

Sage

sage: log2
log2
sage: float(log2)
0.6931471805599453
sage: RR(log2)
0.693147180559945
sage: R = RealField(200); R
Real Field with 200 bits of precision
sage: R(log2)
0.69314718055994530941723212145817656807550013436025525412068
sage: l = (1-log2)/(1+log2); l
-(log2 - 1)/(log2 + 1)
sage: R(l)
0.18123221829928249948761381864650311423330609774776013488056
sage: maxima(log2)
log(2)
sage: maxima(log2).float()
0.6931471805599453
sage: gp(log2)
0.6931471805599453094172321215             # 32-bit
0.69314718055994530941723212145817656807   # 64-bit
sage: RealField(150)(2).log()
0.69314718055994530941723212145817656807550013

Python

>>> from sage.all import *
>>> log2
log2
>>> float(log2)
0.6931471805599453
>>> RR(log2)
0.693147180559945
>>> R = RealField(Integer(200)); R
Real Field with 200 bits of precision
>>> R(log2)
0.69314718055994530941723212145817656807550013436025525412068
>>> l = (Integer(1)-log2)/(Integer(1)+log2); l
-(log2 - 1)/(log2 + 1)
>>> R(l)
0.18123221829928249948761381864650311423330609774776013488056
>>> maxima(log2)
log(2)
>>> maxima(log2).float()
0.6931471805599453
>>> gp(log2)
0.6931471805599453094172321215             # 32-bit
0.69314718055994530941723212145817656807   # 64-bit
>>> RealField(Integer(150))(Integer(2)).log()
0.69314718055994530941723212145817656807550013

class sage.symbolic.constants.Mertens(name='mertens')[source]¶

Bases: Constant

The Mertens constant is related to the Twin Primes constant and appears in Mertens’ second theorem.

EXAMPLES:

Sage

sage: float(mertens)
0.26149721284764277
sage: mertens.n(digits=60)
0.261497212847642783755426838608695859051566648261199206192064

Python

>>> from sage.all import *
>>> float(mertens)
0.26149721284764277
>>> mertens.n(digits=Integer(60))
0.261497212847642783755426838608695859051566648261199206192064

class sage.symbolic.constants.NotANumber(name='NaN')[source]¶

Bases: Constant

Not a Number

class sage.symbolic.constants.Pi(name='pi')[source]¶: Bases: Constant

class sage.symbolic.constants.TwinPrime(name='twinprime')[source]¶

Bases: Constant

The Twin Primes constant is defined as \(\prod 1 - 1/(p-1)^2\) for primes \(p > 2\).

EXAMPLES:

Sage

sage: float(twinprime)
0.6601618158468696
sage: twinprime.n(digits=60)
0.660161815846869573927812110014555778432623360284733413319448

Python

>>> from sage.all import *
>>> float(twinprime)
0.6601618158468696
>>> twinprime.n(digits=Integer(60))
0.660161815846869573927812110014555778432623360284733413319448

sage.symbolic.constants.pi = pi[source]¶

The formal square root of -1.

EXAMPLES:

Sage

sage: SR.I()
I
sage: SR.I()^2
-1

Python

>>> from sage.all import *
>>> SR.I()
I
>>> SR.I()**Integer(2)
-1

Note that conversions to real fields will give TypeErrors:

Sage

sage: float(SR.I())
Traceback (most recent call last):
...
TypeError: unable to simplify to float approximation
sage: gp(SR.I())
I
sage: RR(SR.I())
Traceback (most recent call last):
...
TypeError: unable to convert '1.00000000000000*I' to a real number

Python

>>> from sage.all import *
>>> float(SR.I())
Traceback (most recent call last):
...
TypeError: unable to simplify to float approximation
>>> gp(SR.I())
I
>>> RR(SR.I())
Traceback (most recent call last):
...
TypeError: unable to convert '1.00000000000000*I' to a real number

Expressions involving I that are real-valued can be converted to real fields:

Sage

sage: float(I*I)
-1.0
sage: RR(I*I)
-1.00000000000000

Python

>>> from sage.all import *
>>> float(I*I)
-1.0
>>> RR(I*I)
-1.00000000000000

We can convert to complex fields:

Sage

sage: C = ComplexField(200); C
Complex Field with 200 bits of precision
sage: C(SR.I())
1.0000000000000000000000000000000000000000000000000000000000*I
sage: SR.I()._complex_mpfr_field_(ComplexField(53))
1.00000000000000*I

sage: SR.I()._complex_double_(CDF)
1.0*I
sage: CDF(SR.I())
1.0*I

sage: z = SR.I() + I; z
2*I
sage: C(z)
2.0000000000000000000000000000000000000000000000000000000000*I
sage: 1e8*SR.I()
1.00000000000000e8*I

sage: complex(SR.I())
1j

sage: QQbar(SR.I())
I

sage: abs(SR.I())
1

sage: SR.I().minpoly()
x^2 + 1
sage: maxima(2*SR.I())
2*%i

Python

>>> from sage.all import *
>>> C = ComplexField(Integer(200)); C
Complex Field with 200 bits of precision
>>> C(SR.I())
1.0000000000000000000000000000000000000000000000000000000000*I
>>> SR.I()._complex_mpfr_field_(ComplexField(Integer(53)))
1.00000000000000*I

>>> SR.I()._complex_double_(CDF)
1.0*I
>>> CDF(SR.I())
1.0*I

>>> z = SR.I() + I; z
2*I
>>> C(z)
2.0000000000000000000000000000000000000000000000000000000000*I
>>> RealNumber('1e8')*SR.I()
1.00000000000000e8*I

>>> complex(SR.I())
1j

>>> QQbar(SR.I())
I

>>> abs(SR.I())
1

>>> SR.I().minpoly()
x^2 + 1
>>> maxima(Integer(2)*SR.I())
2*%i

sage.symbolic.constants.unpickle_Constant(class_name, name, conversions, latex, mathml, domain)[source]¶

EXAMPLES:

Sage

sage: from sage.symbolic.constants import unpickle_Constant
sage: a = unpickle_Constant('Constant', 'a', {}, 'aa', '', 'positive')
sage: a.domain()
'positive'
sage: latex(a)
aa

Python

>>> from sage.all import *
>>> from sage.symbolic.constants import unpickle_Constant
>>> a = unpickle_Constant('Constant', 'a', {}, 'aa', '', 'positive')
>>> a.domain()
'positive'
>>> latex(a)
aa

Note that if the name already appears in the constants_name_table, then that will be returned instead of constructing a new object:

Sage

sage: pi = unpickle_Constant('Pi', 'pi', None, None, None, None)
sage: pi._maxima_init_()
'%pi'

Python

>>> from sage.all import *
>>> pi = unpickle_Constant('Pi', 'pi', None, None, None, None)
>>> pi._maxima_init_()
'%pi'