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# 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: 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
    sage: brun
    brun

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: 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
    3.14159265358979323846264338328
    sage: mathematica(pi)             # optional
    Pi
    sage: maple(pi)                   # optional
    Pi
    sage: octave(pi)                  # optional
    3.14159

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

    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
     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 decimal places we have the following:

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

    sage: R(pi)
    3.1415926535897932384626433832795028841971693993751058209749

    sage: R(e)
    2.7182818284590452353602874713526624977572470936999595749670

    sage: R(NaN)
    NaN

    sage: R(golden_ratio)
    1.6180339887498948482045868343656381177203091798057628621354

    sage: R(log2)
    0.69314718055994530941723212145817656807550013436025525412068

    sage: R(euler_gamma)
    0.57721566490153286060651209008240243104215933593992359880577

    sage: R(catalan)
    0.91596559417721901505460351493238411077414937428167213426650

    sage: R(khinchin)
    2.6854520010653064453097148354817956938203822939944629530512

EXAMPLES: Arithmetic with constants

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

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

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

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

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

TESTS:

Coercing the sum of a bunch of the constants to many different floating point rings:

    sage: a = pi + e + golden_ratio + log2 + euler_gamma + catalan + khinchin + twinprime + mertens; a
    mertens + twinprime + khinchin + log2 + golden_ratio + catalan + euler_gamma + pi + e
    sage: parent(a)
    Symbolic Ring
    sage: RR(a)  # abs tol 1e-13
    13.2713479401972
    sage: RealField(212)(a)
    13.2713479401972493100988191995758139408711068200030748178329712
    sage: RealField(230)(a)
    13.271347940197249310098819199575813940871106820003074817832971189555
    sage: RDF(a)  # abs tol 1e-13
    13.271347940197249
    sage: CC(a)  # abs tol 1e-13
    13.2713479401972
    sage: CDF(a)  # abs tol 1e-13
    13.271347940197249
    sage: ComplexField(230)(a)
    13.271347940197249310098819199575813940871106820003074817832971189555

Check that trac ticket #8237 is fixed:

    sage: maxima('infinity').sage()
    Infinity
    sage: maxima('inf').sage()
    +Infinity
    sage: maxima('minf').sage()
    -Infinity

class `sage.symbolic.constants.``Brun`(name='brun')

  Bases: `sage.symbolic.constants.LimitedPrecisionConstant`

  Brun’s constant is the sum of reciprocals of odd twin primes.

  It is not known to very high precision; calculating the number using twin primes up to \(10^{16}\) (Sebah 2002) gives the number \(1.9021605831040\).

  EXAMPLES:

      sage: float(brun)
      Traceback (most recent call last):
      ...
      NotImplementedError: brun is only available up to 41 bits
      sage: R = RealField(41); R
      Real Field with 41 bits of precision
      sage: R(brun)
      1.90216058310

class `sage.symbolic.constants.``Catalan`(name='catalan')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

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

class `sage.symbolic.constants.``Constant`(name, conversions=None, latex=None, mathml='', domain='complex')

  Bases: `object`

  EXAMPLES:

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

  `domain`()

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

  `expression`()

    Returns an expression for this constant.

    EXAMPLES:

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

  `name`()

    Returns the name of this constant.

    EXAMPLES:

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

class `sage.symbolic.constants.``EulerGamma`(name='euler_gamma')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

      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

class `sage.symbolic.constants.``Glaisher`(name='glaisher')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

      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

class `sage.symbolic.constants.``GoldenRatio`(name='golden_ratio')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

      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

  `minpoly`(bits=None, degree=None, epsilon=0)

    EXAMPLES:

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

class `sage.symbolic.constants.``Khinchin`(name='khinchin')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

      sage: float(khinchin)
      2.6854520010653062
      sage: khinchin.n(digits=60)
      2.68545200106530644530971483548179569382038229399446295305115
      sage: m = mathematica(khinchin); m             # optional
      Khinchin
      sage: m.N(200)                                 # optional
      2.68545200106530644530971483548179569382038229399446295305115234555721885953715200280114117493184769799515346590528809008289767771641096305179253348325966838185231542133211949962603932852204481940961807          # 32-bit
      2.6854520010653064453097148354817956938203822939944629530511523455572188595371520028011411749318476979951534659052880900828976777164109630517925334832596683818523154213321194996260393285220448194096181                # 64-bit

class `sage.symbolic.constants.``LimitedPrecisionConstant`(name, value, **kwds)

  Bases: `sage.symbolic.constants.Constant`

  A class for constants that are only known to a limited precision.

  EXAMPLES:

      sage: from sage.symbolic.constants import LimitedPrecisionConstant
      sage: a = LimitedPrecisionConstant('a', '1.234567891011121314').expression(); a
      a
      sage: RDF(a)
      1.2345678910111213
      sage: RealField(200)(a)
      Traceback (most recent call last):
      ...
      NotImplementedError: a is only available up to 59 bits

class `sage.symbolic.constants.``Log2`(name='log2')

  Bases: `sage.symbolic.constants.Constant`

  The natural logarithm of the real number 2.

  EXAMPLES:

      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

class `sage.symbolic.constants.``Mertens`(name='mertens')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

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

class `sage.symbolic.constants.``NotANumber`(name='NaN')

  Bases: `sage.symbolic.constants.Constant`

  Not a Number

class `sage.symbolic.constants.``Pi`(name='pi')

  Bases: `sage.symbolic.constants.Constant`

  TESTS:

      sage: pi._latex_()
      '\pi'
      sage: latex(pi)
      \pi
      sage: mathml(pi)
      <mi>&pi;</mi>

class `sage.symbolic.constants.``TwinPrime`(name='twinprime')

  Bases: `sage.symbolic.constants.Constant`

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

  EXAMPLES:

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

`sage.symbolic.constants.``unpickle_Constant`(class_name, name, conversions, latex, mathml, domain)

  EXAMPLES:

      sage: from sage.symbolic.constants import unpickle_Constant
      sage: a = unpickle_Constant('Constant', 'a', {}, 'aa', '', 'positive')
      sage: a.domain()
      'positive'
      sage: 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: pi = unpickle_Constant('Pi', 'pi', None, None, None, None)
      sage: pi._maxima_init_()
      '%pi'

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