\section{\module{math} ---         Mathematical functions} \declaremodule{builtin}{math}\modulesynopsis{Mathematical functions (\function{sin()} etc.).} This module is always available.  It provides access to themathematical functions defined by the C standard. These functions cannot be used with complex numbers; use the functionsof the same name from the \refmodule{cmath} module if you requiresupport for complex numbers.  The distinction between functions whichsupport complex numbers and those which don't is made since most usersdo not want to learn quite as much mathematics as required tounderstand complex numbers.  Receiving an exception instead of acomplex result allows earlier detection of the unexpected complexnumber used as a parameter, so that the programmer can determine howand why it was generated in the first place. The following functions are provided by this module.  Exceptwhen explicitly noted otherwise, all return values are floats. Number-theoretic and representation functions: \begin{funcdesc}{ceil}{x}Return the ceiling of \var{x} as a float, the smallest integer valuegreater than or equal to \var{x}.\end{funcdesc} \begin{funcdesc}{fabs}{x}Return the absolute value of \var{x}.\end{funcdesc} \begin{funcdesc}{floor}{x}Return the floor of \var{x} as a float, the largest integer valueless than or equal to \var{x}.\end{funcdesc} \begin{funcdesc}{fmod}{x, y}Return \code{fmod(\var{x}, \var{y})}, as defined by the platform C library.Note that the Python expression \code{\var{x} \%\ \var{y}} may not returnthe same result.  The intent of the C standard is that\code{fmod(\var{x}, \var{y})} be exactly (mathematically; to infiniteprecision) equal to \code{\var{x} - \var{n}*\var{y}} for some integer\var{n} such that the result has the same sign as \var{x} andmagnitude less than \code{abs(\var{y})}.  Python's\code{\var{x} \%\ \var{y}} returns a result with the sign of\var{y} instead, and may not be exactly computable for float arguments.For example, \code{fmod(-1e-100, 1e100)} is \code{-1e-100}, but theresult of Python's \code{-1e-100 \%\ 1e100} is \code{1e100-1e-100}, whichcannot be represented exactly as a float, and rounds to the surprising\code{1e100}.  For this reason, function \function{fmod()} is generallypreferred when working with floats, while Python's\code{\var{x} \%\ \var{y}} is preferred when working with integers.\end{funcdesc} \begin{funcdesc}{frexp}{x}Return the mantissa and exponent of \var{x} as the pair\code{(\var{m}, \var{e})}.  \var{m} is a float and \var{e} is aninteger such that \code{\var{x} == \var{m} * 2**\var{e}} exactly.If \var{x} is zero, returns \code{(0.0, 0)}, otherwise\code{0.5 <= abs(\var{m}) < 1}.  This is used to "pick apart" theinternal representation of a float in a portable way.\end{funcdesc} \begin{funcdesc}{ldexp}{x, i}Return \code{\var{x} * (2**\var{i})}.  This is essentially the inverse offunction \function{frexp()}.\end{funcdesc} \begin{funcdesc}{modf}{x}Return the fractional and integer parts of \var{x}.  Both resultscarry the sign of \var{x}, and both are floats.\end{funcdesc} Note that \function{frexp()} and \function{modf()} have a differentcall/return pattern than their C equivalents: they take a singleargument and return a pair of values, rather than returning theirsecond return value through an `output parameter' (there is no suchthing in Python). For the \function{ceil()}, \function{floor()}, and \function{modf()}functions, note that \emph{all} floating-point numbers of sufficientlylarge magnitude are exact integers.  Python floats typically carry no morethan 53 bits of precision (the same as the platform C double type), inwhich case any float \var{x} with \code{abs(\var{x}) >= 2**52}necessarily has no fractional bits.  Power and logarithmic functions: \begin{funcdesc}{exp}{x}Return \code{e**\var{x}}.\end{funcdesc} \begin{funcdesc}{log}{x\optional{, base}}Return the logarithm of \var{x} to the given \var{base}.If the \var{base} is not specified, return the natural logarithm of \var{x}(that is, the logarithm to base \emph{e}).\versionchanged[\var{base} argument added]{2.3}\end{funcdesc} \begin{funcdesc}{log10}{x}Return the base-10 logarithm of \var{x}.\end{funcdesc} \begin{funcdesc}{pow}{x, y}Return \code{\var{x}**\var{y}}.\end{funcdesc} \begin{funcdesc}{sqrt}{x}Return the square root of \var{x}.\end{funcdesc} Trigonometric functions: \begin{funcdesc}{acos}{x}Return the arc cosine of \var{x}, in radians.\end{funcdesc} \begin{funcdesc}{asin}{x}Return the arc sine of \var{x}, in radians.\end{funcdesc} \begin{funcdesc}{atan}{x}Return the arc tangent of \var{x}, in radians.\end{funcdesc} \begin{funcdesc}{atan2}{y, x}Return \code{atan(\var{y} / \var{x})}, in radians.The result is between \code{-pi} and \code{pi}.The vector in the plane from the origin to point \code{(\var{x}, \var{y})}makes this angle with the positive X axis.The point of \function{atan2()} is that the signs of both inputs areknown to it, so it can compute the correct quadrant for the angle.For example, \code{atan(1}) and \code{atan2(1, 1)} are both \code{pi/4},but \code{atan2(-1, -1)} is \code{-3*pi/4}.\end{funcdesc} \begin{funcdesc}{cos}{x}Return the cosine of \var{x} radians.\end{funcdesc} \begin{funcdesc}{hypot}{x, y}Return the Euclidean norm, \code{sqrt(\var{x}*\var{x} + \var{y}*\var{y})}.This is the length of the vector from the origin to point\code{(\var{x}, \var{y})}.\end{funcdesc} \begin{funcdesc}{sin}{x}Return the sine of \var{x} radians.\end{funcdesc} \begin{funcdesc}{tan}{x}Return the tangent of \var{x} radians.\end{funcdesc} Angular conversion: \begin{funcdesc}{degrees}{x}Converts angle \var{x} from radians to degrees.\end{funcdesc} \begin{funcdesc}{radians}{x}Converts angle \var{x} from degrees to radians.\end{funcdesc} Hyperbolic functions: \begin{funcdesc}{cosh}{x}Return the hyperbolic cosine of \var{x}.\end{funcdesc} \begin{funcdesc}{sinh}{x}Return the hyperbolic sine of \var{x}.\end{funcdesc} \begin{funcdesc}{tanh}{x}Return the hyperbolic tangent of \var{x}.\end{funcdesc} The module also defines two mathematical constants: \begin{datadesc}{pi}The mathematical constant \emph{pi}.\end{datadesc} \begin{datadesc}{e}The mathematical constant \emph{e}.\end{datadesc} \begin{notice}  The \module{math} module consists mostly of thin wrappers around  the platform C math library functions.  Behavior in exceptional cases is  loosely specified by the C standards, and Python inherits much of its  math-function error-reporting behavior from the platform C  implementation.  As a result,  the specific exceptions raised in error cases (and even whether some  arguments are considered to be exceptional at all) are not defined in any  useful cross-platform or cross-release way.  For example, whether  \code{math.log(0)} returns \code{-Inf} or raises \exception{ValueError} or  \exception{OverflowError} isn't defined, and in  cases where \code{math.log(0)} raises \exception{OverflowError},  \code{math.log(0L)} may raise \exception{ValueError} instead.\end{notice} \begin{seealso}  \seemodule{cmath}{Complex number versions of many of these functions.}\end{seealso}