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.
math.
ceil
(x)¶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
Integral
value.
math.
copysign
(x, y)¶Return a float with the magnitude (absolute value) of x but the sign of
y. On platforms that support signed zeros, copysign(1.0, -0.0)
returns -1.0.
math.
fabs
(x)¶Return the absolute value of x.
math.
factorial
(x)¶Return x factorial. Raises ValueError
if x is not integral or
is negative.
math.
floor
(x)¶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
Integral
value.
math.
fmod
(x, y)¶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.
math.
frexp
(x)¶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.
math.
fsum
(iterable)¶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])
0.9999999999999999
>>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
1.0
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.
math.
gcd
(a, b)¶Return the greatest common divisor of the integers a and b. If either
a or b is nonzero, then the value of gcd(a, b)
is the largest
positive integer that divides both a and b. gcd(0, 0)
returns
0
.
New in version 3.5.
math.
isclose
(a, b, *, rel_tol=1e-09, abs_tol=0.0)¶Return True
if the values a and b are close to each other and
False
otherwise.
Whether or not two values are considered close is determined according to given absolute and relative tolerances.
rel_tol is the relative tolerance – it is the maximum allowed difference
between a and b, relative to the larger absolute value of a or b.
For example, to set a tolerance of 5%, pass rel_tol=0.05
. The default
tolerance is 1e-09
, which assures that the two values are the same
within about 9 decimal digits. rel_tol must be greater than zero.
abs_tol is the minimum absolute tolerance – useful for comparisons near zero. abs_tol must be at least zero.
If no errors occur, the result will be:
abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)
.
The IEEE 754 special values of NaN
, inf
, and -inf
will be
handled according to IEEE rules. Specifically, NaN
is not considered
close to any other value, including NaN
. inf
and -inf
are only
considered close to themselves.
New in version 3.5.
See also
PEP 485 – A function for testing approximate equality
math.
isfinite
(x)¶Return True
if x is neither an infinity nor a NaN, and
False
otherwise. (Note that 0.0
is considered finite.)
New in version 3.2.
math.
isinf
(x)¶Return True
if x is a positive or negative infinity, and
False
otherwise.
math.
isnan
(x)¶Return True
if x is a NaN (not a number), and False
otherwise.
math.
modf
(x)¶Return the fractional and integer parts of x. Both results carry the sign of x and are floats.
math.
trunc
(x)¶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.
math.
exp
(x)¶Return e**x
.
math.
expm1
(x)¶Return e**x - 1
. For small floats x, the subtraction in exp(x) - 1
can result in a significant loss of precision; the expm1()
function provides a way to compute this quantity to full precision:
>>> from math import exp, expm1
>>> exp(1e-5) - 1 # gives result accurate to 11 places
1.0000050000069649e-05
>>> expm1(1e-5) # result accurate to full precision
1.0000050000166668e-05
New in version 3.2.
math.
log
(x[, base])¶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)
.
math.
log1p
(x)¶Return the natural logarithm of 1+x (base e). The result is calculated in a way which is accurate for x near zero.
math.
log2
(x)¶Return the base-2 logarithm of x. This is usually more accurate than
log(x, 2)
.
New in version 3.3.
See also
int.bit_length()
returns the number of bits necessary to represent
an integer in binary, excluding the sign and leading zeros.
math.
log10
(x)¶Return the base-10 logarithm of x. This is usually more accurate
than log(x, 10)
.
math.
pow
(x, y)¶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
.
Unlike the built-in **
operator, math.pow()
converts both
its arguments to type float
. Use **
or the built-in
pow()
function for computing exact integer powers.
math.
sqrt
(x)¶Return the square root of x.
math.
acos
(x)¶Return the arc cosine of x, in radians.
math.
asin
(x)¶Return the arc sine of x, in radians.
math.
atan
(x)¶Return the arc tangent of x, in radians.
math.
atan2
(y, x)¶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
.
math.
cos
(x)¶Return the cosine of x radians.
math.
hypot
(x, y)¶Return the Euclidean norm, sqrt(x*x + y*y)
. This is the length of the vector
from the origin to point (x, y)
.
math.
sin
(x)¶Return the sine of x radians.
math.
tan
(x)¶Return the tangent of x radians.
math.
degrees
(x)¶Convert angle x from radians to degrees.
math.
radians
(x)¶Convert angle x from degrees to radians.
Hyperbolic functions are analogs of trigonometric functions that are based on hyperbolas instead of circles.
math.
acosh
(x)¶Return the inverse hyperbolic cosine of x.
math.
asinh
(x)¶Return the inverse hyperbolic sine of x.
math.
atanh
(x)¶Return the inverse hyperbolic tangent of x.
math.
cosh
(x)¶Return the hyperbolic cosine of x.
math.
sinh
(x)¶Return the hyperbolic sine of x.
math.
tanh
(x)¶Return the hyperbolic tangent of x.
math.
erf
(x)¶Return the error function at x.
The erf()
function can be used to compute traditional statistical
functions such as the cumulative standard normal distribution:
def phi(x):
'Cumulative distribution function for the standard normal distribution'
return (1.0 + erf(x / sqrt(2.0))) / 2.0
New in version 3.2.
math.
erfc
(x)¶Return the complementary error function at x. The complementary error
function is defined as
1.0 - erf(x)
. It is used for large values of x where a subtraction
from one would cause a loss of significance.
New in version 3.2.
math.
gamma
(x)¶Return the Gamma function at x.
New in version 3.2.
math.
lgamma
(x)¶Return the natural logarithm of the absolute value of the Gamma function at x.
New in version 3.2.
math.
pi
¶The mathematical constant π = 3.141592..., to available precision.
math.
e
¶The mathematical constant e = 2.718281..., to available precision.
math.
tau
¶The mathematical constant τ = 6.283185..., to available precision. Tau is a circle constant equal to 2π, the ratio of a circle’s circumference to its radius. To learn more about Tau, check out Vi Hart’s video Pi is (still) Wrong, and start celebrating Tau day by eating twice as much pie!
New in version 3.6.
math.
inf
¶A floating-point positive infinity. (For negative infinity, use
-math.inf
.) Equivalent to the output of float('inf')
.
New in version 3.5.
math.
nan
¶A floating-point “not a number” (NaN) value. Equivalent to the output of
float('nan')
.
New in version 3.5.
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
hypot(float('nan'), float('inf'))
.
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.
See also
cmath