sage: # This worksheet was developed in the first lecture of Spring 2016.
sage: # It is modified to prepare for Spring 2017.  The most important change
sage: # is the introduction of pi.n(digits=20) which is the quickest way to
sage: # evaluate the decimal expansion of a symbolic constant such as pi.
sage: # Sage knows the symbol pi
sage: pi
sage: sin(pi)
0
sage: # we can ask to see an approximation of pi
sage: print float(pi)
3.14159265359
sage: # as in Python, every symbol has a type
sage: type(pi)
<type 'sage.symbolic.expression.Expression'>
sage: # We are not limited by the hardware machine arithmetic.
sage: pi20 = pi.n(digits=20)
sage: print pi20
sage: print sin(pi20)
sage: print type(pi20)
3.1415926535897932385
6.5640070857470010853e-22
<type 'sage.rings.real_mpfr.RealNumber'>
sage: # we can make a plot of a sine function
sage: plot(sin,(-pi,+pi))
sage: # let us make a more interesting plot
sage: var('x')
sage: plot(exp(-x^2)*sin(8*x),(-pi,pi))
sage: # Sage consists of many systems, in the command below we save the command as a string
sage: # and then execute the command with the eval.
sage: # The argument to RealField is the number of bits.
sage: # For the number of decimal places (in our decimal number system),
sage: # we multiply with the logarithm of 10.
sage: # Suppose we wish to see pi with 30 decimal places
sage: cmd = 'RealField(30*log(10))(pi)'
sage: print eval(cmd)
3.1415926535897932385
sage: # to see which system executed the command
sage: from sage.misc.citation import get_systems
sage: get_systems(cmd)
['MPFI', 'MPFR', 'ginac']
sage: # to see the dictionary of systems Sage recognizes:
sage: print sage.misc.citation.systems.keys()
['Mathematica', 'M4RI', 'PARI', 'Frobby', 'MuPAD', 'mwrank', 'Axiom', 'LiE', 'Macaulay2', 'MPFI', 'Singular', 'ECM', 'Linbox', 'GAP', 'FLINT', 'matlab', 'Octave', 'povray', 'numpy', 'MPFR', 'qsieve', 'ginac', 'gfan', 'scipy', 'KASH', 'GMP', 'Maxima', 'Symmetrica', 'PolyBoRi', 'Tachyon', 'Magma', 'R', 'NTL', 'Givaro', 'Maple']
sage: # to prepare for the last exercise:
sage: from sage.misc.citation import get_systems
sage: get_systems('pi.n(digits=20)')
['MPFR', 'ginac']