9.2. math — Mathematical functions
This module is always available. It provides access to the mathematical
functions defined by the C standard.
These functions cannot be used with complex numbers; use the functions of the
same name from the cmath module if you require support for complex
numbers. The distinction between functions which support complex numbers and
those which don’t is made since most users do not want to learn quite as much
mathematics as required to understand complex numbers. Receiving an exception
instead of a complex result allows earlier detection of the unexpected complex
number used as a parameter, so that the programmer can determine how and why it
was generated in the first place.
The following functions are provided by this module. Except when explicitly
noted otherwise, all return values are floats.
9.2.1. Number-theoretic and representation functions
- Return the ceiling of x, the smallest integer greater than or equal to x.
If x is not a float, delegates to x.__ceil__(), which should return an
- Return x with the sign of y. On a platform that supports
signed zeros, copysign(1.0, -0.0) returns -1.0.
- Return the absolute value of x.
- Return x factorial. Raises ValueError if x is not integral or
- Return the floor of x, the largest integer less than or equal to x.
If x is not a float, delegates to x.__floor__(), which should return an
- Return fmod(x, y), as defined by the platform C library. Note that the
Python expression x % y may not return the same result. The intent of the C
standard is that fmod(x, y) be exactly (mathematically; to infinite
precision) equal to x - n*y for some integer n such that the result has
the same sign as x and magnitude less than abs(y). Python’s x % y
returns a result with the sign of y instead, and may not be exactly computable
for float arguments. For example, fmod(-1e-100, 1e100) is -1e-100, but
the result of Python’s -1e-100 % 1e100 is 1e100-1e-100, which cannot be
represented exactly as a float, and rounds to the surprising 1e100. For
this reason, function fmod() is generally preferred when working with
floats, while Python’s x % y is preferred when working with integers.
- Return the mantissa and exponent of x as the pair (m, e). m is a float
and e is an integer such that x == m * 2**e exactly. If x is zero,
returns (0.0, 0), otherwise 0.5 <= abs(m) < 1. This is used to “pick
apart” the internal representation of a float in a portable way.
Return an accurate floating point sum of values in the iterable. Avoids
loss of precision by tracking multiple intermediate partial sums:
>>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
>>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the
typical case where the rounding mode is half-even. On some non-Windows
builds, the underlying C library uses extended precision addition and may
occasionally double-round an intermediate sum causing it to be off in its
least significant bit.
For further discussion and two alternative approaches, see the ASPN cookbook
recipes for accurate floating point summation.
- Check if the float x is positive or negative infinity.
- Check if the float x is a NaN (not a number). For more information
on NaNs, see the IEEE 754 standards.
- Return x * (2**i). This is essentially the inverse of function
- Return the fractional and integer parts of x. Both results carry the sign
of x and are floats.
- Return the Real value x truncated to an Integral (usually
an integer). Delegates to x.__trunc__().
Note that frexp() and modf() have a different call/return pattern
than their C equivalents: they take a single argument and return a pair of
values, rather than returning their second return value through an ‘output
parameter’ (there is no such thing in Python).
For the ceil(), floor(), and modf() functions, note that all
floating-point numbers of sufficiently large magnitude are exact integers.
Python floats typically carry no more than 53 bits of precision (the same as the
platform C double type), in which case any float x with abs(x) >= 2**52
necessarily has no fractional bits.
9.2.2. Power and logarithmic functions
- Return e**x.
With one argument, return the natural logarithm of x (to base e).
With two arguments, return the logarithm of x to the given base,
calculated as log(x)/log(base).
- Return the natural logarithm of 1+x (base e). The
result is calculated in a way which is accurate for x near zero.
- Return the base-10 logarithm of x. This is usually more accurate
than log(x, 10).
- Return x raised to the power y. Exceptional cases follow
Annex ‘F’ of the C99 standard as far as possible. In particular,
pow(1.0, x) and pow(x, 0.0) always return 1.0, even
when x is a zero or a NaN. If both x and y are finite,
x is negative, and y is not an integer then pow(x, y)
is undefined, and raises ValueError.
- Return the square root of x.
9.2.3. Trigonometric functions
- Return the arc cosine of x, in radians.
- Return the arc sine of x, in radians.
- Return the arc tangent of x, in radians.
- Return atan(y / x), in radians. The result is between -pi and pi.
The vector in the plane from the origin to point (x, y) makes this angle
with the positive X axis. The point of atan2() is that the signs of both
inputs are known to it, so it can compute the correct quadrant for the angle.
For example, atan(1) and atan2(1, 1) are both pi/4, but atan2(-1,
-1) is -3*pi/4.
- Return the cosine of x radians.
- Return the Euclidean norm, sqrt(x*x + y*y). This is the length of the vector
from the origin to point (x, y).
- Return the sine of x radians.
- Return the tangent of x radians.
9.2.4. Angular conversion
- Converts angle x from radians to degrees.
- Converts angle x from degrees to radians.
9.2.5. Hyperbolic functions
- Return the inverse hyperbolic cosine of x.
- Return the inverse hyperbolic sine of x.
- Return the inverse hyperbolic tangent of x.
- Return the hyperbolic cosine of x.
- Return the hyperbolic sine of x.
- Return the hyperbolic tangent of x.
- The mathematical constant π = 3.141592..., to available precision.
- The mathematical constant e = 2.718281..., to available precision.
CPython implementation detail: The math module consists mostly of thin wrappers around the platform C
math library functions. Behavior in exceptional cases follows Annex F of
the C99 standard where appropriate. The current implementation will raise
ValueError for invalid operations like sqrt(-1.0) or log(0.0)
(where C99 Annex F recommends signaling invalid operation or divide-by-zero),
and OverflowError for results that overflow (for example,
exp(1000.0)). A NaN will not be returned from any of the functions
above unless one or more of the input arguments was a NaN; in that case,
most functions will return a NaN, but (again following C99 Annex F) there
are some exceptions to this rule, for example pow(float('nan'), 0.0) or
Note that Python makes no effort to distinguish signaling NaNs from
quiet NaNs, and behavior for signaling NaNs remains unspecified.
Typical behavior is to treat all NaNs as though they were quiet.
- Module cmath
- Complex number versions of many of these functions.