~9\c@ sddlmZmZddlZddlZddlZddlZddlm Z ddl m Z m Z m Z m Z mZddlmZmZddlmZmZddlmZdd lmZmZdd lmZdd lmZmZmZm Z m!Z!m"Z"m#Z#m$Z$dd l%m&Z&ddl'Z'ddl(j)Z*dd l+m,Z,ddl(m-Z-m.Z.m/Z/m0Z0ddl1m2Z2ddl3m4Z5m6Z7m8Z9m:Z;m<Z=m>Z>ddl?m@Z@mAZAddlBmCZCddlDmEZEe*jFZGejHdZIeJdZKdZLieMd6ZNeMdZOdZPdZQdZRe&ddZSyddlmTZUWneVk rdZUnXd ZWd!ZXd"ZYd#ZZd$Z[d%efd&YZ\d'e\fd(YZ]e]e e^YZmd?e!ee\fd@YZnejnZodAe!eefdBYZpejpZqdCefdDYZrdEe!eerfdFYZsejsZtdGe!eerfdHYZuejuZvdIe!eerfdJYZwdKe!eerfdLYZxdMe!eerfdNYZydOe!eerfdPYZzdQe!eefdRYZ{ej{Z|dSZ}e}e ej~|j|jko=t|t|kStdnt ||}t |}|r|dkr|||kS||kSdS( sReturn a bool indicating whether the error between z1 and z2 is <= tol. If ``tol`` is None then True will be returned if there is a significant difference between the numbers: ``abs(z1 - z2)*10**p <= 1/2`` where ``p`` is the lower of the precisions of the values. A comparison of strings will be made if ``z1`` is a Number and a) ``z2`` is a string or b) ``tol`` is '' and ``z2`` is a Number. When ``tol`` is a nonzero value, if z2 is non-zero and ``|z1| > 1`` the error is normalized by ``|z1|``, so if you want to see if the absolute error between ``z1`` and ``z2`` is <= ``tol`` then call this as ``comp(z1 - z2, 0, tol)``. s$when z2 is a str z1 must be a Numbert is_Numberi iics s!|]}t|dtVqdS(R)N(tgetattrtFalse(t.0ti((s1lib/python2.7/site-packages/sympy/core/numbers.pys Dss%exact comparison requires two NumbersN(ttypetstrt isinstancetNumbert ValueErrortTruetNoneR*R+tFloattinttabsR$tmint_prectall(tz1tz2ttoltatbtdifftaz1((s1lib/python2.7/site-packages/sympy/core/numbers.pytcomp's. $%( cC s^|\}}}}|s)|s"tS|Snddlm}t||||||t}|S(seReturn the mpf tuple normalized appropriately for the indicated precision after doing a check to see if zero should be returned or not when the mantissa is 0. ``mpf_normlize`` always assumes that this is zero, but it may not be since the mantissa for mpf's values "+inf", "-inf" and "nan" have a mantissa of zero, too. Note: this is not intended to validate a given mpf tuple, so sending mpf tuples that were not created by mpmath may produce bad results. This is only a wrapper to ``mpf_normalize`` which provides the check for non- zero mpfs that have a 0 for the mantissa. i(R(t _mpf_zerotmpmath.libmp.backendRt mpf_normalizetrnd(tmpftprectsigntmantexpttbcRtrv((s1lib/python2.7/site-packages/sympy/core/numbers.pytmpf_normOs !tdividecC s(td|kr$t|td>> from sympy.core.numbers import igcd >>> igcd(2, 4) 2 >>> igcd(5, 10, 15) 5 is,igcd() takes at least 2 arguments (%s given)i(R[RYR7Rtpoptigcd2(targsR-t args_tempR>R?((s1lib/python2.7/site-packages/sympy/core/numbers.pytigcds%   (tgcdcC sl|jtkr1|jtkr1t||St|t|}}x|rg|||}}qMW|S(s+Compute gcd of two Python integers a and b.(t bit_lengthtBIGBITSt igcd_lehmerR7(R>R?((s1lib/python2.7/site-packages/sympy/core/numbers.pyRns  icC sAtt|tt|}}||krA||}}ndtj}x|j|kr|dkr|j|}t||?t||?}}d\}}}} xtr||dkrPn||||} || ||| | } } | | dkrPn|| }}|| || || f\}}}} || dkrYPn|||| } || ||| |} } | | dkrPn|| }}|| | || | f\}}}} qW|dkr|||}}qQn||||||| |}}qQWx|r<|||}}q"W|S(sComputes greatest common divisor of two integers. Euclid's algorithm for the computation of the greatest common divisor gcd(a, b) of two (positive) integers a and b is based on the division identity a = q*b + r, where the quotient q and the remainder r are integers and 0 <= r < b. Then each common divisor of a and b divides r, and it follows that gcd(a, b) == gcd(b, r). The algorithm works by constructing the sequence r0, r1, r2, ..., where r0 = a, r1 = b, and each rn is the remainder from the division of the two preceding elements. In Python, q = a // b and r = a % b are obtained by the floor division and the remainder operations, respectively. These are the most expensive arithmetic operations, especially for large a and b. Lehmer's algorithm is based on the observation that the quotients qn = r(n-1) // rn are in general small integers even when a and b are very large. Hence the quotients can be usually determined from a relatively small number of most significant bits. The efficiency of the algorithm is further enhanced by not computing each long remainder in Euclid's sequence. The remainders are linear combinations of a and b with integer coefficients derived from the quotients. The coefficients can be computed as far as the quotients can be determined from the chosen most significant parts of a and b. Only then a new pair of consecutive remainders is computed and the algorithm starts anew with this pair. References ========== .. [1] https://en.wikipedia.org/wiki/Lehmer%27s_GCD_algorithm iii(iiii(R7RRtbits_per_digitRsR6R3(R>R?tnbitstntxtytAtBtCtDRVtx_qytB_qDtA_qC((s1lib/python2.7/site-packages/sympy/core/numbers.pyRus@)%  !!  & * ) cG sut|dkr+tdt|nd|kr;dS|d}x)|dD]}|t|||}qPW|S(sComputes integer least common multiple. Examples ======== >>> from sympy.core.numbers import ilcm >>> ilcm(5, 10) 10 >>> ilcm(7, 3) 21 >>> ilcm(5, 10, 15) 30 is,ilcm() takes at least 2 arguments (%s given)ii(R[RYRq(RoR>R?((s1lib/python2.7/site-packages/sympy/core/numbers.pytilcm^s  c C s!| r| rdS|s5d|t|t|fS|sX|t|dt|fS|dkru| d}}nd}|dkr| d}}nd}d\}}}}xY|r ||||}} |||| ||| |||f\}}}}}}qW|||||fS(s3Returns x, y, g such that g = x*a + y*b = gcd(a, b). >>> from sympy.core.numbers import igcdex >>> igcdex(2, 3) (-1, 1, 1) >>> igcdex(10, 12) (-1, 1, 2) >>> x, y, g = igcdex(100, 2004) >>> x, y, g (-20, 1, 4) >>> x*100 + y*2004 4 iii(iii(iiii(R7( R>R?tx_signty_signRyRztrRbtcRV((s1lib/python2.7/site-packages/sympy/core/numbers.pytigcdexxs"   >cC s<d}yit|t|}}|dkrn|dkrnt||\}}}|dkrn||}qnnWntk rt|t|}}|jo|jsttdn|dk}|tj kp|tj kstd|q|rd|}qnX|dkr8td||fn|S(s Return the number c such that, (a * c) = 1 (mod m) where c has the same sign as m. If no such value exists, a ValueError is raised. Examples ======== >>> from sympy import S >>> from sympy.core.numbers import mod_inverse Suppose we wish to find multiplicative inverse x of 3 modulo 11. This is the same as finding x such that 3 * x = 1 (mod 11). One value of x that satisfies this congruence is 4. Because 3 * 4 = 12 and 12 = 1 (mod 11). This is the value return by mod_inverse: >>> mod_inverse(3, 11) 4 >>> mod_inverse(-3, 11) 7 When there is a common factor between the numerators of ``a`` and ``m`` the inverse does not exist: >>> mod_inverse(2, 4) Traceback (most recent call last): ... ValueError: inverse of 2 mod 4 does not exist >>> mod_inverse(S(2)/7, S(5)/2) 7/2 References ========== - https://en.wikipedia.org/wiki/Modular_multiplicative_inverse - https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm iis Expected numbers for arguments; symbolic `mod_inverse` is not implemented but symbolic expressions can be handled with the similar function, sympy.polys.polytools.inverts*m > 1 did not evaluate; try to simplify %ss%inverse of %s (mod %s) does not existN( R4RRR2Rt is_numberRYR&Rttruetfalse(R>tmRRyRztgtbig((s1lib/python2.7/site-packages/sympy/core/numbers.pyt mod_inverses('    R1cB seZdZeZeZeZgZdZdZ dZ dZ dZ dZ dZdZd Zd Zd Zd Zd ZdZdZdZedZed'dZededZededZ ededZ!ededZ"e"Z#dZ$dZ%dZ&dZ'dZ(dZ)dZ*dZ+d Z,d!Z-e.d"Z/e.d#Z0d$Z1d%Z2d&Z3RS((sRepresents atomic numbers in SymPy. Floating point numbers are represented by the Float class. Rational numbers (of any size) are represented by the Rational class. Integer numbers (of any size) are represented by the Integer class. Float and Rational are subclasses of Number; Integer is a subclass of Rational. For example, ``2/3`` is represented as ``Rational(2, 3)`` which is a different object from the floating point number obtained with Python division ``2/3``. Even for numbers that are exactly represented in binary, there is a difference between how two forms, such as ``Rational(1, 2)`` and ``Float(0.5)``, are used in SymPy. The rational form is to be preferred in symbolic computations. Other kinds of numbers, such as algebraic numbers ``sqrt(2)`` or complex numbers ``3 + 4*I``, are not instances of Number class as they are not atomic. See Also ======== Float, Integer, Rational icG st|dkr|d}nt|tr2|St|trKt|St|trvt|dkrvt|St|ttj t j frt |St|t rt|}t|tr|Std|nd}t|t|jdS(Niiis$String "%s" does not denote a Numbers<expected str|int|long|float|Decimal|Number object but got %r(R[R0R1RR\ttupleR`tfloatRSRGtdecimaltDecimalR5RRR2RYR.t__name__(tclstobjtvaltmsg((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__new__s"  !   cO sBddlm}t|dtr/t||S|||||S(Ni(tinvertR(tsympy.polys.polytoolsRR*R3R(tselftothertgensRoR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs cC sddlm}yt|}Wn?tk rad}t|t|jt|jfnX|swtdn|jr|jr|t|j |j S||}|dkrt |n t |d}|||}|||S(Ni(Rs7unsupported operand type(s) for divmod(): '%s' and '%s'smodulo by zeroi( t containersRR1RYR.RtZeroDivisionErrort is_IntegertdivmodRTR6(RRRRtrattwR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt __divmod__!s , (cC s_yt|}Wn?tk rQd}t|t|jt|jfnXt||S(Ns7unsupported operand type(s) for divmod(): '%s' and '%s'(R1RYR.RR(RRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt __rdivmod__3s  ,cG stt||S(N(troundR(RRo((s1lib/python2.7/site-packages/sympy/core/numbers.pyt __round__;scC std|jjdS(s7Evaluation of mpf tuple accurate to at least prec bits.s%s needs ._as_mpf_val() methodN(tNotImplementedErrort __class__R(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyt _as_mpf_val>scC stj|j||S(N(R5t_newR(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyt _eval_evalfCscC s%t||j}|j||fS(N(tmaxR9R(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyt _as_mpf_opFscC stj|jdS(Ni5(tmlibtto_floatR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt __float__JscC std|jjdS(Ns%s needs .floor() method(RRR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pytfloorMscC std|jjdS(Ns%s needs .ceiling() method(RRR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pytceilingQscC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_eval_conjugateUscG s ddlm}|tj|S(Ni(tOrder(tsympyRRtOne(RtsymbolsR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt _eval_orderXscC s|| kr| S|S(N((Rtoldtnew((s1lib/python2.7/site-packages/sympy/core/numbers.pyt _eval_subs]s cC stS(N(R3(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_eval_is_finitebscC sdS(NiiR1(iiR1((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt class_keyescC s|jddfd|fS(Ni(((R(Rtorder((s1lib/python2.7/site-packages/sympy/core/numbers.pytsort_keyisRcC snt|tr^tdr^|tjkr/tjS|tjkrEtjS|tjkr^tjSntj||S(Ni( R0R1R'RtNaNtInfinitytNegativeInfinityR t__add__(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRms cC snt|tr^tdr^|tjkr/tjS|tjkrEtjS|tjkr^tjSntj||S(Ni( R0R1R'RRRRR t__sub__(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRxs cC st|trtdr|tjkr/tjS|tjkrh|jrNtjS|jr^tjStjSq|tjkr|jrtjS|jrtjStjSqnt|t rt St j ||S(Ni( R0R1R'RRRtis_zerot is_positiveRRtNotImplementedR t__mul__(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs$      cC sgt|trWtdrW|tjkr/tjS|tjksM|tjkrWtjSntj ||S(Ni( R0R1R'RRRRtZeroR t__div__(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs  cC std|jjdS(Ns%s needs .__eq__() method(RRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__eq__scC std|jjdS(Ns%s needs .__ne__() method(RRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__ne__scC sTyt|}Wn'tk r9td||fnXtd|jjdS(NsInvalid comparison %s < %ss%s needs .__lt__() method(RRRYRRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__lt__s  cC sTyt|}Wn'tk r9td||fnXtd|jjdS(NsInvalid comparison %s <= %ss%s needs .__le__() method(RRRYRRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__le__s  cC sMyt|}Wn'tk r9td||fnXt|j|S(NsInvalid comparison %s > %s(RRRYR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__gt__s  cC sMyt|}Wn'tk r9td||fnXt|j|S(NsInvalid comparison %s >= %s(RRRYR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt__ge__s  cC stt|jS(N(tsuperR1t__hash__(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscO stS(N(R3(Rtwrttflags((s1lib/python2.7/site-packages/sympy/core/numbers.pyt is_constantscO sS|js|jdt r)|tfS|jrCtj| ffStj|ffS(Ntrational(t is_RationalRmR3Rt is_negativeRt NegativeOneR(Rtdepstkwargs((s1lib/python2.7/site-packages/sympy/core/numbers.pyt as_coeff_muls   cG s&|jr|tfStj|ffS(N(RRRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt as_coeff_adds  cC s=|r|j rtj|fS|r0|tjfStj|fS(s2Efficiently extract the coefficient of a product. (RRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt as_coeff_Muls cC s |s|tjfStj|fS(s4Efficiently extract the coefficient of a summation. (RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt as_coeff_Adds cC sddlm}|||S(s#Compute GCD of `self` and `other`. i(Rr(t sympy.polysRr(RRRr((s1lib/python2.7/site-packages/sympy/core/numbers.pyRrscC sddlm}|||S(s#Compute LCM of `self` and `other`. i(tlcm(RR(RRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC sddlm}|||S(s1Compute GCD and cofactors of `self` and `other`. i(t cofactors(RR(RRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRsN(4Rt __module__t__doc__R3tis_commutativeRR)t __slots__R9RRRRRRRRRRRRRRRt classmethodRR R4RR RRRRRt __truediv__RRRRRRRRRRR+RRRrRR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1sX                                R5cB s eZdZddgZd*Zd*ZeZeZ eZ d*d*d*dZ e dZ dZdZdZdZd Zed Zd Zd Zd ZdZdZdZdZdZdZeZdZe de!dZ"e de!dZ#e de!dZ$e de!dZ%e%Z&e de!dZ'e de!dZ(dZ)dZ*dZ+e+Z,dZ-d Z.d!Z/d"Z0d#Z1d$Z2d%Z3d&d'Z4d(Z5d)Z6RS(+sRepresent a floating-point number of arbitrary precision. Examples ======== >>> from sympy import Float >>> Float(3.5) 3.50000000000000 >>> Float(3) 3.00000000000000 Creating Floats from strings (and Python ``int`` and ``long`` types) will give a minimum precision of 15 digits, but the precision will automatically increase to capture all digits entered. >>> Float(1) 1.00000000000000 >>> Float(10**20) 100000000000000000000. >>> Float('1e20') 100000000000000000000. However, *floating-point* numbers (Python ``float`` types) retain only 15 digits of precision: >>> Float(1e20) 1.00000000000000e+20 >>> Float(1.23456789123456789) 1.23456789123457 It may be preferable to enter high-precision decimal numbers as strings: Float('1.23456789123456789') 1.23456789123456789 The desired number of digits can also be specified: >>> Float('1e-3', 3) 0.00100 >>> Float(100, 4) 100.0 Float can automatically count significant figures if a null string is sent for the precision; space are also allowed in the string. (Auto- counting is only allowed for strings, ints and longs). >>> Float('123 456 789 . 123 456', '') 123456789.123456 >>> Float('12e-3', '') 0.012 >>> Float(3, '') 3. If a number is written in scientific notation, only the digits before the exponent are considered significant if a decimal appears, otherwise the "e" signifies only how to move the decimal: >>> Float('60.e2', '') # 2 digits significant 6.0e+3 >>> Float('60e2', '') # 4 digits significant 6000. >>> Float('600e-2', '') # 3 digits significant 6.00 Notes ===== Floats are inexact by their nature unless their value is a binary-exact value. >>> approx, exact = Float(.1, 1), Float(.125, 1) For calculation purposes, evalf needs to be able to change the precision but this will not increase the accuracy of the inexact value. The following is the most accurate 5-digit approximation of a value of 0.1 that had only 1 digit of precision: >>> approx.evalf(5) 0.099609 By contrast, 0.125 is exact in binary (as it is in base 10) and so it can be passed to Float or evalf to obtain an arbitrary precision with matching accuracy: >>> Float(exact, 5) 0.12500 >>> exact.evalf(20) 0.12500000000000000000 Trying to make a high-precision Float from a float is not disallowed, but one must keep in mind that the *underlying float* (not the apparent decimal value) is being obtained with high precision. For example, 0.3 does not have a finite binary representation. The closest rational is the fraction 5404319552844595/2**54. So if you try to obtain a Float of 0.3 to 20 digits of precision you will not see the same thing as 0.3 followed by 19 zeros: >>> Float(0.3, 20) 0.29999999999999998890 If you want a 20-digit value of the decimal 0.3 (not the floating point approximation of 0.3) you should send the 0.3 as a string. The underlying representation is still binary but a higher precision than Python's float is used: >>> Float('0.3', 20) 0.30000000000000000000 Although you can increase the precision of an existing Float using Float it will not increase the accuracy -- the underlying value is not changed: >>> def show(f): # binary rep of Float ... from sympy import Mul, Pow ... s, m, e, b = f._mpf_ ... v = Mul(int(m), Pow(2, int(e), evaluate=False), evaluate=False) ... print('%s at prec=%s' % (v, f._prec)) ... >>> t = Float('0.3', 3) >>> show(t) 4915/2**14 at prec=13 >>> show(Float(t, 20)) # higher prec, not higher accuracy 4915/2**14 at prec=70 >>> show(Float(t, 2)) # lower prec 307/2**10 at prec=10 The same thing happens when evalf is used on a Float: >>> show(t.evalf(20)) 4915/2**14 at prec=70 >>> show(t.evalf(2)) 307/2**10 at prec=10 Finally, Floats can be instantiated with an mpf tuple (n, c, p) to produce the number (-1)**n*c*2**p: >>> n, c, p = 1, 5, 0 >>> (-1)**n*c*2**p -5 >>> Float((1, 5, 0)) -5.00000000000000 An actual mpf tuple also contains the number of bits in c as the last element of the tuple: >>> _._mpf_ (1, 5, 0, 3) This is not needed for instantiation and is not the same thing as the precision. 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R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s     %cC syt|}Wn'tk r9td||fnX|jrP|j|S|jr|j r||j9}t|j}n|j r|j }n|j r|t j k rtttj|j|j|jStj||S(NsInvalid comparison %s < %s(RRRYR4RRRRVRTR9RR)RRRgRtmpf_ltRRRR9R R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs     %cC syt|}Wn'tk r9td||fnX|jrP|j|S|jr|j r||j9}t|j}n|j r|j }n|j r|t j k rtttj|j|j|jStj||S(NsInvalid comparison %s <= %s(RRRYR4RRRRVRTR9RR)RRRgRtmpf_leRRRR9R R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR.s     %cC stt|jS(N(RR5R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR?ss1e-15cC st||t|kS(N(R7R5(RRtepsilon((s1lib/python2.7/site-packages/sympy/core/numbers.pyt epsilon_eqBscC s"ddlj}|jt|S(Ni(tsage.allR:t RealNumberR/(Rtsage((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_sage_EscC sttjt||S(N(tformatRRR/(Rt format_spec((s1lib/python2.7/site-packages/sympy/core/numbers.pyt __format__IsN(7RRRRR4t is_rationalR5R3Rtis_realR6RRRRRRRRtpropertyRRRRRRRRRRt__bool__RR 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Examples ======== >>> from sympy import Rational, nsimplify, S, pi >>> Rational(1, 2) 1/2 Rational is unprejudiced in accepting input. If a float is passed, the underlying value of the binary representation will be returned: >>> Rational(.5) 1/2 >>> Rational(.2) 3602879701896397/18014398509481984 If the simpler representation of the float is desired then consider limiting the denominator to the desired value or convert the float to a string (which is roughly equivalent to limiting the denominator to 10**12): >>> Rational(str(.2)) 1/5 >>> Rational(.2).limit_denominator(10**12) 1/5 An arbitrarily precise Rational is obtained when a string literal is passed: >>> Rational("1.23") 123/100 >>> Rational('1e-2') 1/100 >>> Rational(".1") 1/10 >>> Rational('1e-2/3.2') 1/320 The conversion of other types of strings can be handled by the sympify() function, and conversion of floats to expressions or simple fractions can be handled with nsimplify: >>> S('.[3]') # repeating digits in brackets 1/3 >>> S('3**2/10') # general expressions 9/10 >>> nsimplify(.3) # numbers that have a simple form 3/10 But if the input does not reduce to a literal Rational, an error will be raised: >>> Rational(pi) Traceback (most recent call last): ... TypeError: invalid input: pi Low-level --------- Access numerator and denominator as .p and .q: >>> r = Rational(3, 4) >>> r 3/4 >>> r.p 3 >>> r.q 4 Note that p and q return integers (not SymPy Integers) so some care is needed when using them in expressions: >>> r.p/r.q 0.75 See Also ======== sympify, sympy.simplify.simplify.nsimplify RTRVcC s|dkrt|tr|St|tr1nUt|ttfrVtt|St|tsyt|}Wqdt t fk rqdXn|j ddkrt d|n|j dd}|jdd}t|dkr'|\}}tj|}tj|}||}nytj|}Wntk rMnXt|j|jdSt|tst d|nd}d}nt|}t|}t|tr||j9}|j}nt|tr||j9}|j}n|dkr?|dkr8tdr.td q8tjSntjS|dkr\| }| }n|sztt||}n|dkr||}||}n|dkrt|S|dkr|dkrtjStj |}||_||_|S( Nt/isinvalid input: %sRRiiROsIndeterminate 0/0(!R4R0R`RRR5RWRRRt SyntaxErrortcountRYRtrsplitR[t fractionstFractionR2t numeratort denominatorRVRTRPRRtComplexInfinityRqR7R\tHalfR R(RRTRVRrtpqtfptfqR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRsx                        i@BcC s:tj|j|j}t|jtjt|S(sClosest Rational to self with denominator at most max_denominator. >>> from sympy import Rational >>> Rational('3.141592653589793').limit_denominator(10) 22/7 >>> Rational('3.141592653589793').limit_denominator(100) 311/99 (RPRQRTRVR`tlimit_denominatorR6(Rtmax_denominatorRj((s1lib/python2.7/site-packages/sympy/core/numbers.pyRYs cC s|j|jfS(N(RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s|j|jfS(N(RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s |jdkS(Ni(RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s |jdkS(Ni(RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC st|j |jS(N(R`RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRsRcC stdrt|tr=t|j|j|j|jdSt|tr{t|j|j|j|j|j|jSt|tr||Stj||Sntj||S(Nii( R'R0R\R`RTRVR5R1R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs $/cC stdrt|tr=t|j|j|j|jdSt|tr{t|j|j|j|j|j|jSt|tr| |Stj||Sntj||S(Nii( R'R0R\R`RTRVR5R1R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR&s $/ cC stdrt|tr=t|j|j|j|jdSt|tr{t|j|j|j|j|j|jSt|tr| |Stj||Sntj||S(Nii( R'R0R\R`RVRTR5R1t__rsub__(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR[2s $/ cC stdrt|trEt|j|j|jt|j|jSt|trt|j|j|j|jt|j|jt|j|jSt|tr||Stj ||Sntj ||S(Ni( R'R0R\R`RTRVRqR5R1R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR>s ,FcC stdrt|trj|jr;|jtjkr;tjSt|j|j|jt |j|jSqt|trt|j|j|j|jt |j|jt |j|jSt|t r|d|St j ||Snt j ||S(Nii( R'R0R\RTRRRTR`RVRqR5R1R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRLs /F cC stdrt|trEt|j|j|jt|j|jSt|trt|j|j|j|jt|j|jt|j|jSt|tr|d|Stj ||Sntj ||S(Nii( R'R0R\R`RTRVRqR5R1t__rdiv__(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR\[s ,F cC stdrt|trj|j|j|j|j}t|j|j||j|j|j|jSt|trt|jt|d|jStj||Stj||S(NiR ( R'R0R`RTRVR5R"R9R1(RRRx((s1lib/python2.7/site-packages/sympy/core/numbers.pyR"is 3 cC s/t|trtj||Stj||S(N(R0R`R"R1R$(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR$vscC st|trt|tr2|j|j|S|jr| }|tjkrdt|j |j S|jrtj |t|j |j |St|j |j |Sn|tj kr|j |j krtj S|j |j krtj tj tj StjSt|tr4t|j |j |j |j dSt|tr|j dkrut|j |t|j | St|j t|j |j d|j t|j t|j Sn|jr|jr| |SdS(Ni(R0R1R5RR9RRRR`RVRTRRR,RR\tis_even(RRKtne((s1lib/python2.7/site-packages/sympy/core/numbers.pyR)|s6  #$# cC stj|j|j|tS(N(Rt from_rationalRTRVRF(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s%tjtj|j|j||S(N(RStmake_mpfRR_RTRV(RRHRF((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_mpmath_scC stt|j|jS(N(R`R7RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1scC s=|j|j}}|dkr/t| | St||S(Ni(RTRVR6(RRTRV((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2s cC st|j|jS(N(R\RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC st|j |j S(N(R\RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC syt|}Wntk r$tSX|jrH|jr;tS|j|S|jr|jr||j |j ko{|j |j kS|j rt j |j|j|jSntS(N(RRRR4R5R+RR)RRTRVR6RR3RR9RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs      " "cC s ||k S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s yt|}Wn'tk r9td||fnX|jrP|j|S|}|jr|jrtt|j|j |j |jkS|j rttt j |j |j|jSn2|jr|jrt|j|j |}}ntj||S(NsInvalid comparison %s > %s(RRRYR4RR)RRgRTRVR6RR:RR9RRRRHR\R R(RRtexpr((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs      *  " cC s yt|}Wn'tk r9td||fnX|jrP|j|S|}|jr|jrtt|j|j |j |jkS|j rttt j |j |j|jSn2|jr|jrt|j|j |}}ntj||S(NsInvalid comparison %s >= %s(RRRYR4RR)RRgRTRVR6RR;RR9RRRRHR\R R(RRRb((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs      *  " cC s yt|}Wn'tk r9td||fnX|jrP|j|S|}|jr|jrtt|j|j |j |jkS|j rttt j |j |j|jSn2|jr|jrt|j|j |}}ntj||S(NsInvalid comparison %s < %s(RRRYR4RR)RRgRTRVR6RR<RR9RRRRHR\R R(RRRb((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs      *  " cC s yt|}Wn'tk r9td||fnX|}|jrV|j|S|jr|jrtt|j|j |j |jkS|j rttt j |j |j|jSn2|jr|jrt|j|j |}}ntj||S(NsInvalid comparison %s <= %s(RRRYR4RR)RRgRTRVR6RR=RR9RRRRHR\R R(RRRb((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs      *  " cC stt|jS(N(RR`R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRsc C s>ddlm}||d|d|d|d|d|jS(sA wrapper to factorint which return factors of self that are smaller than limit (or cheap to compute). Special methods of factoring are disabled by default so that only trial division is used. i(t factorrattlimitt use_trialtuse_rhotuse_pm1tverbose(t sympy.ntheoryRctcopy(RRdReRfRgRhtvisualRc((s1lib/python2.7/site-packages/sympy/core/numbers.pytfactorss cC sit|trY|tjkr"|Sttt|j|jtt|j|jSt j ||S(N( R0R`RRR\RqRTRRVR1Rr(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRr%scC sXt|trHt|jt|j|j|jt|j|jStj||S(N(R0R`RTRqRVR1R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR/s  cC st|jt|jfS(N(R\RTRV(R((s1lib/python2.7/site-packages/sympy/core/numbers.pytas_numer_denom7scC s/ddlj}|j|j|j|jS(Ni(R@R:R\RTRV(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRC:scC s7|r*|jr|tjfS| tjfStj|fS(s-Return the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import S >>> (S(-3)/2).as_content_primitive() (3/2, -1) See docstring of Expr.as_content_primitive for more examples. (RRRR(Rtradicaltclear((s1lib/python2.7/site-packages/sympy/core/numbers.pytas_content_primitive>s   cC s |tjfS(s2Efficiently extract the coefficient of a product. (RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRRscC s |tjfS(s4Efficiently extract the coefficient of a summation. (RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRVsN(8RRRR3RHR+t is_integerRGRRRR R4RRYRRRRRR RRt__radd__RR[Rt__rmul__RR\RR"R$R)RRaR1R2RKRRRRRRRRRRlRrRRmRCRpRR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR`TsdR J            &                   R\cB smeZdZdZeZeZeZdgZdZ dZ e dZ dZ dZeZdZd Zd Zd Zd Zd ZdZdZdZdZdZdZdZdZdZdZdZ dZ!dZ"dZ#dZ$dZ%dZ&dZ'd Z(d!Z)d"Z*d#Z+d$Z,RS(%sjRepresents integer numbers of any size. Examples ======== >>> from sympy import Integer >>> Integer(3) 3 If a float or a rational is passed to Integer, the fractional part will be discarded; the effect is of rounding toward zero. >>> Integer(3.8) 3 >>> Integer(-3.8) -3 A string is acceptable input if it can be parsed as an integer: >>> Integer("9" * 20) 99999999999999999999 It is rarely needed to explicitly instantiate an Integer, because Python integers are automatically converted to Integer when they are used in SymPy expressions. iRTcC stj|j|tS(N(Rtfrom_intRTRF(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyR~scC stj|j|S(N(RSR`R(RRHRF((s1lib/python2.7/site-packages/sympy/core/numbers.pyRascC st|tr$|jdd}nyt|}Wn!tk rWtd|nX|dkrktjS|dkr~tjS|dkrtjSt j |}||_ |S(NRRs6Argument of Integer should be of numeric type, got %s.iii( R0RRR6RYRRRRR RRT(RR-tivalR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs      cC s |jfS(N(RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s|jS(N(RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2scC s t|jS(N(R\RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s t|jS(N(R\RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC st|j S(N(R\RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s%|jdkr|St|j SdS(Ni(RTR\(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1scC sVddlm}t|trBtdrB|t|j|jStj||SdS(Ni(Ri( RRR0R\R'RRTR1R(RRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC sddlm}t|tr?tdr?|t||jSyt|}WnKtk rd}t |j }t |j }t|||fnXtj ||SdS(Ni(Ris7unsupported operand type(s) for divmod(): '%s' and '%s'( RRR0RR'RRTR1RYR.RR(RRRRtonametsname((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs cC stdrt|tr*t|j|St|trMt|j|jSt|trt|j|j|j|jdStj||St||SdS(Nii( R'R0RR\RTR`RVRtAdd(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs $cC s}tdrmt|tr*t||jSt|tr]t|j|j|j|jdStj||Stj||S(Nii(R'R0RR\RTR`RVRr(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRrs $cC stdrt|tr*t|j|St|trMt|j|jSt|trt|j|j|j|jdStj||Stj||S(Nii(R'R0RR\RTR`RVR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs $cC s}tdrmt|tr*t||jSt|tr]t|j|j|j|jdStj||Stj||S(Nii(R'R0RR\RTR`RVR[(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR[s $cC stdrt|tr*t|j|St|trMt|j|jSt|trt|j|j|jt|j|jStj||Stj||S(Ni( R'R0RR\RTR`RVRqR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs ,cC stdrut|tr*t||jSt|tret|j|j|jt|j|jStj||Stj||S(Ni( R'R0RR\RTR`RVRqRs(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRss ,cC smtdr]t|tr*t|j|St|trMt|j|jStj||Stj||S(Ni(R'R0RR\RTR`R"(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR" s cC smtdr]t|tr*t||jSt|trMt|j|jStj||Stj||S(Ni(R'R0RR\RTR`R$(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR$s cC sKt|tr|j|kSt|tr;|j|jkStj||S(N(R0RRTR\R`R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs  cC s ||k S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR#scC siyt|}Wn'tk r9td||fnX|jrYt|j|jkStj||S(NsInvalid comparison %s > %s(RRRYRRTR`R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR&s  cC siyt|}Wn'tk r9td||fnX|jrYt|j|jkStj||S(NsInvalid comparison %s < %s(RRRYRRTR`R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR/s  cC siyt|}Wn'tk r9td||fnX|jrYt|j|jkStj||S(NsInvalid comparison %s >= %s(RRRYRRTR`R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR8s  cC siyt|}Wn'tk r9td||fnX|jrYt|j|jkStj||S(NsInvalid comparison %s <= %s(RRRYRRTR`R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRAs  cC s t|jS(N(thashRT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRJscC s|jS(N(RT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt __index__MscC st|jdS(Ni(RgRT(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt _eval_is_oddRscC sddlm}|tjkrM|jtjkr8tjStjtjtjS|tjkrptd|tjSt |t s|j r|j r| |Snt |t rtt|j|St |tsdS|tjkr|j rtjt| |S|j rO| }|j r8tj|td| |Std|j|Sntt|j|j\}}|rt|t|j}|j r|tj|9}n|Stt|j}||}|tk ri|d|d6} nt|jdd } d} d} d} d} i}x| jD]\}}||j9}t||j\}}|dkr| ||9} n|dkr8t||j}|dkr| t|t|||j|9} q|||>> from sympy import S, Integer, zoo >>> Integer(0) is S.Zero True >>> 1/S.Zero zoo References ========== .. [1] https://en.wikipedia.org/wiki/Zero iicC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC st|jr |S|jrtjS|jtkr3tjS|j\}}|jrYtj|S|tjk rp||SdS(N( RRRRTRHR+RRR(RRKRtterms((s1lib/python2.7/site-packages/sympy/core/numbers.pyR) s    cG s|S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC stS(N(R+(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s tj|fS(s4Efficiently extract the coefficient of a summation. (RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s(RRRRTRVR+RRR3RRRt staticmethodR1RR)RRRJR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyRf s   RcB sweZdZeZdZdZgZedZ edZ dZ dZ edeeeeedZRS(s The number one. One is a singleton, and can be accessed by ``S.One``. Examples ======== >>> from sympy import S, Integer >>> Integer(1) is S.One True References ========== .. [1] https://en.wikipedia.org/wiki/1_%28number%29 icC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s|S(N((RRK((s1lib/python2.7/site-packages/sympy/core/numbers.pyR) scG sdS(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s|r tjSiSdS(N(RR(RdReRfRgRhRk((s1lib/python2.7/site-packages/sympy/core/numbers.pyRl sN(RRRR3RRTRVRRR1RR)RR4R+Rl(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s   RcB sMeZdZeZdZdZgZedZ edZ dZ RS(s]The number negative one. NegativeOne is a singleton, and can be accessed by ``S.NegativeOne``. Examples ======== >>> from sympy import S, Integer >>> Integer(-1) is S.NegativeOne True See Also ======== One References ========== .. [1] https://en.wikipedia.org/wiki/%E2%88%921_%28number%29 iicC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s|jrtjS|jr tjSt|trt|trLtd|S|tjkrbtjS|tj ks|tj krtjS|tj krtj St|t r|jdkrtj t|jSt|j|j\}}|r|||t ||jSqndS(Ngi(tis_oddRRR]RR0R1R5RRRRUR,R`RVR\RTR(RRKR-R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR) s(  "( RRRR3RRTRVRRR1RR)(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR sRUcB s5eZdZeZdZdZgZedZ RS(sThe rational number 1/2. Half is a singleton, and can be accessed by ``S.Half``. Examples ======== >>> from sympy import S, Rational >>> Rational(1, 2) is S.Half True References ========== .. [1] https://en.wikipedia.org/wiki/One_half iicC stjS(N(RRU(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1- s( RRRR3RRTRVRRR1(((s1lib/python2.7/site-packages/sympy/core/numbers.pyRU s RcB sLeZdZeZeZeZeZeZ gZ dZ dZ dZ ededZeZededZededZeZededZeZd Zd Zd Zd Zd ZdZdZdZdZdZ dZ!dZ"dZ#e#Z$dZ%dZ&RS(sPositive infinite quantity. In real analysis the symbol `\infty` denotes an unbounded limit: `x\to\infty` means that `x` grows without bound. Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled `+\infty` and `-\infty` can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. Infinity is a singleton, and can be accessed by ``S.Infinity``, or can be imported as ``oo``. Examples ======== >>> from sympy import oo, exp, limit, Symbol >>> 1 + oo oo >>> 42/oo 0 >>> x = Symbol('x') >>> limit(exp(x), x, oo) oo See Also ======== NegativeInfinity, NaN References ========== .. [1] https://en.wikipedia.org/wiki/Infinity cC s tj|S(N(R R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR` scC sdS(Ns\infty((Rtprinter((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_latexc scC s||kr|SdS(N((RRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRf s RcC sqt|trm|tjks-|tjkr4tjS|jrc|tdkrVtjStdSqmtjSntS(Ns-infR%( R0R1RRRR6R5RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRj s   cC sqt|trm|tjks-|tjkr4tjS|jrc|tdkrVtjStdSqmtjSntS(NR%(R0R1RRRR6R5R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRy s   cC st|tr|tjks-|tjkr4tjS|jrs|dkrPtjS|dkrftdStdSq|dkrtjStjSnt S(NiR%s-inf( R0R1RRRR6R5RRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s       cC st|tr|tjks<|tjks<|tjkrCtjS|jr|tdksp|tdkrwtjS|jrtdStdSq|dkrtjStjSnt S(Ns-infR%i( R0R1RRRRR6R5tis_nonnegativeR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s       cC stjS(N(RR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC stjS(N(RR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sddlm}|jr tjS|jr0tjS|tjkrFtjS|tjkr\tjS|j t kr|j r||}|jrtjS|jrtjS|j rtjS||j SdS(s ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``oo ** nan`` ``nan`` ``oo ** -p`` ``0`` ``p`` is number, ``oo`` ================ ======= ============================== See Also ======== Pow NaN NegativeInfinity i(R.N(tsympy.functionsR.RRRRRRRTRHR+RRR(RRKR.t expt_real((s1lib/python2.7/site-packages/sympy/core/numbers.pyR) s$      cC stjS(N(RR(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sddlj}|jS(Ni(R@R:too(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRC scC stt|jS(N(RRR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s |tjkS(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s |tjk S(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sZyt|}Wn'tk r9td||fnX|jrJtjStj||S(NsInvalid comparison %s < %s(RRRYRHRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s  cC syt|}Wn'tk r9td||fnX|jr|js[|tjkrbtjS|jrrtjS|j r|j rtj Snt j ||S(NsInvalid comparison %s <= %s(RRRYRHRXRRRtis_nonpositiveRRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s    cC syt|}Wn'tk r9td||fnX|jr|js[|tjkrbtjS|jrrtjS|j r|j rtj Snt j ||S(NsInvalid comparison %s > %s(RRRYRHRXRRRRRRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s    cC sZyt|}Wn'tk r9td||fnX|jrJtjStj||S(NsInvalid comparison %s >= %s(RRRYRHRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s  cC stjS(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR" scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR# scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR& s('RRRR3RRRRR+RRRRRR RRRrRRRsRRR1RR)RRCRRRRRRRR"R$RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2 sB$       '         RcB sFeZdZeZeZeZeZgZdZ dZ dZ e de dZeZe de dZe de dZeZe de dZeZd Zd Zd Zd Zd ZdZdZdZdZdZdZdZ dZ!e!Z"dZ#dZ$RS(sNegative infinite quantity. NegativeInfinity is a singleton, and can be accessed by ``S.NegativeInfinity``. See Also ======== Infinity cC s tj|S(N(R R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR? scC sdS(Ns-\infty((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRB scC s||kr|SdS(N((RRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRE s RcC stt|trp|tjks-|tjkr4tjS|jrf|tdkrYtdStdSqptjSntS(NR%tnans-inf( R0R1RRRR6R5RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRI s    cC stt|trp|tjks-|tjkr4tjS|jrf|tdkrYtdStdSqptjSntS(Ns-infR(R0R1RRRR6R5R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRX s    cC st|tr|tjks-|tjkr4tjS|jr||tjksU|jr\tjS|jrotdStdSq|jrtj Stj Snt S(Ns-infR%( R0R1RRRR6RRR5RRR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRf s      cC st|tr|tjks<|tjks<|tjkrCtjS|jr|tdks|tdks|tjkrtjS|jrtdStdSq|dkrtjStjSnt S(Ns-infR%i( R0R1RRRRR6R5RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRz s"      cC stjS(N(RR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC stjS(N(RR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s|jr|tjks6|tjks6|tjkr=tjSt|tro|jro|jretjStjSntj |tj|SdS(s^ ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``(-oo) ** nan`` ``nan`` ``(-oo) ** oo`` ``nan`` ``(-oo) ** -oo`` ``nan`` ``(-oo) ** e`` ``oo`` ``e`` is positive even integer ``(-oo) ** o`` ``-oo`` ``o`` is positive odd integer ================ ======= ============================== See Also ======== Infinity Pow NaN N( RRRRRR0R\RRR(RRK((s1lib/python2.7/site-packages/sympy/core/numbers.pyR) s   cC stjS(N(RR (RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sddlj}|j S(Ni(R@R:R(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRC scC stt|jS(N(RRR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s |tjkS(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s |tjk S(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC syt|}Wn'tk r9td||fnX|jr|js[|tjkrbtjS|jrrtjS|j r|j rtj Snt j ||S(NsInvalid comparison %s < %s(RRRYRHRXRRRRRRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s    cC sZyt|}Wn'tk r9td||fnX|jrJtjStj||S(NsInvalid comparison %s <= %s(RRRYRHRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s  cC sZyt|}Wn'tk r9td||fnX|jrJtjStj||S(NsInvalid comparison %s > %s(RRRYRHRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s  cC syt|}Wn'tk r9td||fnX|jr|js[|tjkrbtjS|jrrtjS|j r|j rtj Snt j ||S(NsInvalid comparison %s >= %s(RRRYRHRXRRRRRRRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s    cC stjS(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR" scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s(%RRRR3RRRRRRRRR RRRrRRRsRRR1RR)RRCRRRRRRRR"R$RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR, s@        $         RcB sFeZdZeZdZdZdZdZ dZ e Z dZ dZdZdZdZeZgZdZdZededZededZededZededZeZdZd Zd Zd Z d Z!d Z"dZ#dZ$e%j&Z&e%j'Z'e%j(Z(e%j)Z)RS(s Not a Number. This serves as a place holder for numeric values that are indeterminate. Most operations on NaN, produce another NaN. Most indeterminate forms, such as ``0/0`` or ``oo - oo` produce NaN. Two exceptions are ``0**0`` and ``oo**0``, which all produce ``1`` (this is consistent with Python's float). NaN is loosely related to floating point nan, which is defined in the IEEE 754 floating point standard, and corresponds to the Python ``float('nan')``. Differences are noted below. NaN is mathematically not equal to anything else, even NaN itself. This explains the initially counter-intuitive results with ``Eq`` and ``==`` in the examples below. NaN is not comparable so inequalities raise a TypeError. This is in constrast with floating point nan where all inequalities are false. NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported as ``nan``. Examples ======== >>> from sympy import nan, S, oo, Eq >>> nan is S.NaN True >>> oo - oo nan >>> nan + 1 nan >>> Eq(nan, nan) # mathematical equality False >>> nan == nan # structural equality True References ========== .. [1] https://en.wikipedia.org/wiki/NaN cC s tj|S(N(R R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRD scC sdS(Ns \text{NaN}((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRG sRcC s|S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRJ scC s|S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRN scC s|S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRR scC s|S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRV scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR\ scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR_ scC stS(N(R(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyRb scC sddlj}|jS(Ni(R@R:R(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRCe scC stt|jS(N(RRR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRi scC s |tjkS(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRl scC s |tjk S(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRp scC stjS(N(RR(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_eval_Eqs sN(*RRRR3RR4RHRGRtis_transcendentalRqR+R9RXRRRRRRRRR RRRRRRRRRRCRRRRR RRRR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR sD,             RTcB seZdZeZeZeZeZeZ eZ gZ dZ dZ edZdZdZedZdZdZRS( sComplex infinity. In complex analysis the symbol `\tilde\infty`, called "complex infinity", represents a quantity with infinite magnitude, but undetermined complex phase. ComplexInfinity is a singleton, and can be accessed by ``S.ComplexInfinity``, or can be imported as ``zoo``. Examples ======== >>> from sympy import zoo, oo >>> zoo + 42 zoo >>> 42/zoo 0 >>> zoo + zoo nan >>> zoo*zoo zoo See Also ======== Infinity cC s tj|S(N(R R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sdS(Ns\tilde{\infty}((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s|S(N((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC stjS(N(RRT(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sY|tjkrtjSt|trU|tjkr;tjS|jrKtjStjSndS(N(RRTRR0R1RR(RRK((s1lib/python2.7/site-packages/sympy/core/numbers.pyR) s cC sddlj}|jjS(Ni(R@R:tUnsignedInfinityRingR(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRC s(RRRR3RRRR+Rt is_complexRHRRRRR1RRRR)RC(((s1lib/python2.7/site-packages/sympy/core/numbers.pyRT s      t NumberSymbolcB seZeZeZeZgZeZdZdZ dZ dZ dZ dZ dZdZdZd ZRS( cC s tj|S(N(R R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sdS(s Return an interval with number_cls endpoints that contains the value of NumberSymbol. If not implemented, then return None. N((Rt number_cls((s1lib/python2.7/site-packages/sympy/core/numbers.pyt approximation RcC stj|j||S(N(R5RR(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sOyt|}Wntk r$tSX||kr5tS|jrK|jrKtStS(N(RRRR3R)R5R+(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s  cC s ||k S(N((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s#||krtjStj||S(N(RRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s cC s#||krtjStj||S(N(RRR R(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s cC s tdS(N(R(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2 scC s |jS(N(R2(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRK scC stt|jS(N(RRR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s(RRR3RRXRRR4RRRRRRRR2RKR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s        tExp1cB seZdZeZeZeZeZeZ eZ eZ gZ dZ edZdZdZdZdZdZdZd ZRS( sThe `e` constant. The transcendental number `e = 2.718281828\ldots` is the base of the natural logarithm and of the exponential function, `e = \exp(1)`. Sometimes called Euler's number or Napier's constant. Exp1 is a singleton, and can be accessed by ``S.Exp1``, or can be imported as ``E``. Examples ======== >>> from sympy import exp, log, E >>> E is exp(1) True >>> log(E) 1 References ========== .. [1] https://en.wikipedia.org/wiki/E_%28mathematical_constant%29 cC sdS(NRd((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR, scC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1/ scC sdS(Ni((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR23 scC s t|S(N(R(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyR6 scC s;t|tr%tdtdfSt|tr7ndS(Nii(t issubclassR\R`(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pytapproximation_interval9 scC sddlm}||S(Ni(texp(RR(RRKR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR)? scK s<ddlm}tj}||tjd|||S(Ni(tsini(RRRR,tPi(RRRtI((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_eval_rewrite_as_sinC s cK s<ddlm}tj}|||||tjdS(Ni(tcosi(RRRR,R(RRRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_eval_rewrite_as_cosH s cC sddlj}|jS(Ni(R@R:Rd(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRCM s(RRRR3RHRR+RR5RRRRRRR1R2RRR)RRRC(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s$       RcB szeZdZeZeZeZeZeZ eZ eZ gZ dZ edZdZdZdZdZRS(sThe `\pi` constant. The transcendental number `\pi = 3.141592654\ldots` represents the ratio of a circle's circumference to its diameter, the area of the unit circle, the half-period of trigonometric functions, and many other things in mathematics. Pi is a singleton, and can be accessed by ``S.Pi``, or can be imported as ``pi``. Examples ======== >>> from sympy import S, pi, oo, sin, exp, integrate, Symbol >>> S.Pi pi >>> pi > 3 True >>> pi.is_irrational True >>> x = Symbol('x') >>> sin(x + 2*pi) sin(x) >>> integrate(exp(-x**2), (x, -oo, oo)) sqrt(pi) References ========== .. [1] https://en.wikipedia.org/wiki/Pi cC sdS(Ns\pi((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR~ scC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1 scC sdS(Ni((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2 scC s t|S(N(R(RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sTt|tr%tdtdfSt|trPtddtddfSdS(NiiiiGii(RR\R`(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sddlj}|jS(Ni(R@R:tpi(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRC s(RRRR3RHRR+RR5RRRRRRR1R2RRRC(((s1lib/python2.7/site-packages/sympy/core/numbers.pyRS s    t GoldenRatiocB szeZdZeZeZeZeZeZ eZ eZ gZ dZ dZdZdZdZdZeZRS(s[The golden ratio, `\phi`. `\phi = \frac{1 + \sqrt{5}}{2}` is algebraic number. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities, i.e. their maximum. GoldenRatio is a singleton, and can be accessed by ``S.GoldenRatio``. Examples ======== >>> from sympy import S >>> S.GoldenRatio > 1 True >>> S.GoldenRatio.expand(func=True) 1/2 + sqrt(5)/2 >>> S.GoldenRatio.is_irrational True References ========== .. [1] https://en.wikipedia.org/wiki/Golden_ratio cC sdS(Ns\phi((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sdS(Ni((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2 scC s.tjt|d| d}t||S(Ni (Rt from_man_expRRN(RRHRM((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s!cK s(ddlm}tjtj|dS(Ni(tsqrti(RRRRU(RthintsR((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_eval_expand_func scC s8t|tr"tjtdfSt|tr4ndS(Ni(RR\RRR`(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC sddlj}|jS(Ni(R@R:t golden_ratio(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRC s(RRRR3RHRR+RR5RRRRRR2RRRRCt_eval_rewrite_as_sqrt(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s       tTribonacciConstantcB sqeZdZeZeZeZeZeZ eZ eZ gZ dZ dZdZdZdZeZRS(sThe tribonacci constant. The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial `x^3 - x^2 - x - 1 = 0`, and also satisfies the equation `x + x^{-3} = 2`. TribonacciConstant is a singleton, and can be accessed by ``S.TribonacciConstant``. Examples ======== >>> from sympy import S >>> S.TribonacciConstant > 1 True >>> S.TribonacciConstant.expand(func=True) 1/3 + (19 - 3*sqrt(33))**(1/3)/3 + (3*sqrt(33) + 19)**(1/3)/3 >>> S.TribonacciConstant.is_irrational True >>> S.TribonacciConstant.n(20) 1.8392867552141611326 References ========== .. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers cC sdS(Ns\text{TribonacciConstant}((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC sdS(Ni((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2scC s/|jdtj|d}t|d|S(NtfunctioniR (RR3RR5(RRHRM((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scK sNddlm}m}d|dd|d|dd|ddS(Ni(Rtcbrtiiii!(RRR(RRRR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR scC s8t|tr"tjtdfSt|tr4ndS(Ni(RR\RRR`(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs(RRRR3RHRR+RR5RRRRRR2RRRR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR s      t EulerGammacB s_eZdZeZeZeZdZ eZ gZ dZ dZ dZdZdZRS(sThe Euler-Mascheroni constant. `\gamma = 0.5772157\ldots` (also called Euler's constant) is a mathematical constant recurring in analysis and number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm: .. math:: \gamma = \lim\limits_{n\to\infty} \left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right) EulerGamma is a singleton, and can be accessed by ``S.EulerGamma``. Examples ======== >>> from sympy import S >>> S.EulerGamma.is_irrational >>> S.EulerGamma > 0 True >>> S.EulerGamma > 1 False References ========== .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant cC sdS(Ns\gamma((RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyR?scC sdS(Ni((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2BscC s:tjj|d}tj|| d}t||S(Ni (Rtlibhypert euler_fixedRRN(RRHRRM((s1lib/python2.7/site-packages/sympy/core/numbers.pyREscC sHt|trtjtjfSt|trDtjtddfSdS(Nii(RR\RRRR`RU(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRKscC sddlj}|jS(Ni(R@R:t euler_gamma(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRCQsN(RRRR3RHRR+RR4R5RRRR2RRRC(((s1lib/python2.7/site-packages/sympy/core/numbers.pyRs    tCatalancB sVeZdZeZeZeZdZ eZ gZ dZ dZ dZdZRS(sCatalan's constant. `K = 0.91596559\ldots` is given by the infinite series .. math:: K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2} Catalan is a singleton, and can be accessed by ``S.Catalan``. Examples ======== >>> from sympy import S >>> S.Catalan.is_irrational >>> S.Catalan > 0 True >>> S.Catalan > 1 False References ========== .. [1] https://en.wikipedia.org/wiki/Catalan%27s_constant cC sdS(Ni((R((s1lib/python2.7/site-packages/sympy/core/numbers.pyR2wscC s7tj|d}tj|| d}t||S(Ni (Rt catalan_fixedRRN(RRHRRM((s1lib/python2.7/site-packages/sympy/core/numbers.pyRzscC sHt|trtjtjfSt|trDtddtjfSdS(Ni i (RR\RRRR`(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC sddlj}|jS(Ni(R@R:tcatalan(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRCsN(RRRR3RHRR+RR4R5RRR2RRRC(((s1lib/python2.7/site-packages/sympy/core/numbers.pyRVs   R,cB seZdZeZeZeZeZeZe Z gZ dZ e dZdZdZdZdZdZedZRS( sTThe imaginary unit, `i = \sqrt{-1}`. I is a singleton, and can be accessed by ``S.I``, or can be imported as ``I``. Examples ======== >>> from sympy import I, sqrt >>> sqrt(-1) I >>> I*I -1 >>> 1/I -I References ========== .. [1] https://en.wikipedia.org/wiki/Imaginary_unit cC s |jdS(Ntimaginary_unit_latex(t _settings(RR((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC stjS(N(RR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR1scC s|S(N((RRH((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC stj S(N(RR,(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC stt|trpt|trp|jd}|dkr>tjS|dkrQtjS|dkretj Stj SndS(s b is I = sqrt(-1) e is symbolic object but not equal to 0, 1 I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal I**0 mod 4 -> 1 I**1 mod 4 -> I I**2 mod 4 -> -1 I**3 mod 4 -> -I iiiiN(R0R1R\RTRRR,(RRK((s1lib/python2.7/site-packages/sympy/core/numbers.pyR)s      cC stjtjfS(N(RRRU(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt as_base_expscC sddlj}|jS(Ni(R@R:R(RRB((s1lib/python2.7/site-packages/sympy/core/numbers.pyRCscC stdjtdjfS(Nii(R5RR(R((s1lib/python2.7/site-packages/sympy/core/numbers.pyt_mpc_s(RRRR3Rt is_imaginaryRXRRR+RRRRR1RRR)RRCRIR(((s1lib/python2.7/site-packages/sympy/core/numbers.pyR,s       cC st|j|jdS(Ni(R`RRRS(Rj((s1lib/python2.7/site-packages/sympy/core/numbers.pytsympify_fractionsscC stt|S(N(R\R(Ry((s1lib/python2.7/site-packages/sympy/core/numbers.pyt sympify_mpzscC stt|jt|jS(N(R`RRRRS(Ry((s1lib/python2.7/site-packages/sympy/core/numbers.pyt sympify_mpqscC stj||jjS(N(R t _from_mpmathRRH(Ry((s1lib/python2.7/site-packages/sympy/core/numbers.pytsympify_mpmathscC s|j\}}t||dS(Ni(t_mpq_R`(RyRTRV((s1lib/python2.7/site-packages/sympy/core/numbers.pyRscC s6ttt|j|jf\}}|tj|S(N(RtmapRtrealtimagRR,(R>RR((s1lib/python2.7/site-packages/sympy/core/numbers.pytsympify_complexs'(R(R}(tMul(Rx(t __future__RRRRPtmathR.RhRRRRRRRt singletonRR RbR R t decoratorsR tcacheR RtlogicRtsympy.core.compatibilityRRRRRRRRtsympy.core.cacheRRSt mpmath.libmptlibmpRRDRRRRRt mpmath.ctx_mpRtmpmath.libmp.libmpfRRR RR!RR"RCR#RER$tsympy.utilities.miscR%R&R&R'tsympy.utilities.exceptionsR(t round_nearestRFtlogt_LOG2R4RBRNR+RPRQRWRfRlRqRrRnt ImportErrorRtRuRRRR1R5RRRAR`R\ti_typeRRRRRRRURRRRRRTtzooRRtERRRRRRR,RRRQtgmpy2tgmpyRRR.tmpztmpqRRRtcomplextpowerR(R}tmulRtidentitytaddRx(((s1lib/python2.7/site-packages/sympy/core/numbers.pyts    (: ".  (        + AU y { C/= v L :G A @C<5P