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| Meta Title | math ā Mathematical functions ā Python 3.14.4 documentation |
| Meta Description | This module provides access to common mathematical functions and constants, including those defined by the C standard. These functions cannot be used with complex numbers; use the functions of the ... |
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| Boilerpipe Text | This module provides access to common mathematical functions and constants,
including those 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 donāt 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.
Number-theoretic functions
comb(n,
k)
Number of ways to choose
k
items from
n
items without repetition and without order
factorial(n)
n
factorial
gcd(*integers)
Greatest common divisor of the integer arguments
isqrt(n)
Integer square root of a nonnegative integer
n
lcm(*integers)
Least common multiple of the integer arguments
perm(n,
k)
Number of ways to choose
k
items from
n
items without repetition and with order
Floating point arithmetic
ceil(x)
Ceiling of
x
, the smallest integer greater than or equal to
x
fabs(x)
Absolute value of
x
floor(x)
Floor of
x
, the largest integer less than or equal to
x
fma(x,
y,
z)
Fused multiply-add operation:
(x
*
y)
+
z
fmod(x,
y)
Remainder of division
x
/
y
modf(x)
Fractional and integer parts of
x
remainder(x,
y)
Remainder of
x
with respect to
y
trunc(x)
Integer part of
x
Floating point manipulation functions
copysign(x,
y)
Magnitude (absolute value) of
x
with the sign of
y
frexp(x)
Mantissa and exponent of
x
isclose(a,
b,
rel_tol,
abs_tol)
Check if the values
a
and
b
are close to each other
isfinite(x)
Check if
x
is neither an infinity nor a NaN
isinf(x)
Check if
x
is a positive or negative infinity
isnan(x)
Check if
x
is a NaN (not a number)
ldexp(x,
i)
x
*
(2**i)
, inverse of function
frexp()
nextafter(x,
y,
steps)
Floating-point value
steps
steps after
x
towards
y
ulp(x)
Value of the least significant bit of
x
Power, exponential and logarithmic functions
cbrt(x)
Cube root of
x
exp(x)
e
raised to the power
x
exp2(x)
2
raised to the power
x
expm1(x)
e
raised to the power
x
, minus 1
log(x,
base)
Logarithm of
x
to the given base (
e
by default)
log1p(x)
Natural logarithm of
1+x
(base
e
)
log2(x)
Base-2 logarithm of
x
log10(x)
Base-10 logarithm of
x
pow(x,
y)
x
raised to the power
y
sqrt(x)
Square root of
x
Summation and product functions
dist(p,
q)
Euclidean distance between two points
p
and
q
given as an iterable of coordinates
fsum(iterable)
Sum of values in the input
iterable
hypot(*coordinates)
Euclidean norm of an iterable of coordinates
prod(iterable,
start)
Product of elements in the input
iterable
with a
start
value
sumprod(p,
q)
Sum of products from two iterables
p
and
q
Angular conversion
degrees(x)
Convert angle
x
from radians to degrees
radians(x)
Convert angle
x
from degrees to radians
Trigonometric functions
acos(x)
Arc cosine of
x
asin(x)
Arc sine of
x
atan(x)
Arc tangent of
x
atan2(y,
x)
atan(y
/
x)
cos(x)
Cosine of
x
sin(x)
Sine of
x
tan(x)
Tangent of
x
Hyperbolic functions
acosh(x)
Inverse hyperbolic cosine of
x
asinh(x)
Inverse hyperbolic sine of
x
atanh(x)
Inverse hyperbolic tangent of
x
cosh(x)
Hyperbolic cosine of
x
sinh(x)
Hyperbolic sine of
x
tanh(x)
Hyperbolic tangent of
x
Special functions
erf(x)
Error function
at
x
erfc(x)
Complementary error function
at
x
gamma(x)
Gamma function
at
x
lgamma(x)
Natural logarithm of the absolute value of the
Gamma function
at
x
Constants
pi
Ļ
= 3.141592ā¦
e
e
= 2.718281ā¦
tau
Ļ
= 2
Ļ
= 6.283185ā¦
inf
Positive infinity
nan
āNot a numberā (NaN)
Number-theoretic functions
Ā¶
math.
comb
(
n
,
k
)
Ā¶
Return the number of ways to choose
k
items from
n
items without repetition
and without order.
Evaluates to
n!
/
(k!
*
(n
-
k)!)
when
k
<=
n
and evaluates
to zero when
k
>
n
.
Also called the binomial coefficient because it is equivalent
to the coefficient of k-th term in polynomial expansion of
(1
+
x)āæ
.
Raises
TypeError
if either of the arguments are not integers.
Raises
ValueError
if either of the arguments are negative.
Added in version 3.8.
math.
factorial
(
n
)
Ā¶
Return factorial of the nonnegative integer
n
.
Changed in version 3.10:
Floats with integral values (like
5.0
) are no longer accepted.
math.
gcd
(
*
integers
)
Ā¶
Return the greatest common divisor of the specified integer arguments.
If any of the arguments is nonzero, then the returned value is the largest
positive integer that is a divisor of all arguments. If all arguments
are zero, then the returned value is
0
.
gcd()
without arguments
returns
0
.
Added in version 3.5.
Changed in version 3.9:
Added support for an arbitrary number of arguments. Formerly, only two
arguments were supported.
math.
isqrt
(
n
)
Ā¶
Return the integer square root of the nonnegative integer
n
. This is the
floor of the exact square root of
n
, or equivalently the greatest integer
a
such that
a
Ā²Ā ā¤Ā
n
.
For some applications, it may be more convenient to have the least integer
a
such that
n
Ā ā¤Ā
a
Ā², or in other words the ceiling of
the exact square root of
n
. For positive
n
, this can be computed using
a
=
1
+
isqrt(n
-
1)
.
Added in version 3.8.
math.
lcm
(
*
integers
)
Ā¶
Return the least common multiple of the specified integer arguments.
If all arguments are nonzero, then the returned value is the smallest
positive integer that is a multiple of all arguments. If any of the arguments
is zero, then the returned value is
0
.
lcm()
without arguments
returns
1
.
Added in version 3.9.
math.
perm
(
n
,
k
=
None
)
Ā¶
Return the number of ways to choose
k
items from
n
items
without repetition and with order.
Evaluates to
n!
/
(n
-
k)!
when
k
<=
n
and evaluates
to zero when
k
>
n
.
If
k
is not specified or is
None
, then
k
defaults to
n
and the function returns
n!
.
Raises
TypeError
if either of the arguments are not integers.
Raises
ValueError
if either of the arguments are negative.
Added in version 3.8.
Floating point arithmetic
Ā¶
math.
ceil
(
x
)
Ā¶
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.
fabs
(
x
)
Ā¶
Return the absolute value of
x
.
math.
floor
(
x
)
Ā¶
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.
fma
(
x
,
y
,
z
)
Ā¶
Fused multiply-add operation. Return
(x
*
y)
+
z
, computed as though with
infinite precision and range followed by a single round to the
float
format. This operation often provides better accuracy than the direct
expression
(x
*
y)
+
z
.
This function follows the specification of the fusedMultiplyAdd operation
described in the IEEE 754 standard. The standard leaves one case
implementation-defined, namely the result of
fma(0,
inf,
nan)
and
fma(inf,
0,
nan)
. In these cases,
math.fma
returns a NaN,
and does not raise any exception.
Added in version 3.13.
math.
fmod
(
x
,
y
)
Ā¶
Return the floating-point remainder of
x
/
y
,
as defined by the platform C library function
fmod(x,
y)
. 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)
. Pythonās
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 Pythonās
-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 Pythonās
x
%
y
is preferred when working with integers.
math.
modf
(
x
)
Ā¶
Return the fractional and integer parts of
x
. Both results carry the sign
of
x
and are floats.
Note that
modf()
has a different call/return pattern
than its C equivalents: it takes a single argument and return a pair of
values, rather than returning its second return value through an āoutput
parameterā (there is no such thing in Python).
math.
remainder
(
x
,
y
)
Ā¶
Return the IEEE 754-style remainder of
x
with respect to
y
. For
finite
x
and finite nonzero
y
, this is the difference
x
-
n*y
,
where
n
is the closest integer to the exact value of the quotient
x
/
y
. If
x
/
y
is exactly halfway between two consecutive integers, the
nearest
even
integer is used for
n
. The remainder
r
=
remainder(x,
y)
thus always satisfies
abs(r)
<=
0.5
*
abs(y)
.
Special cases follow IEEE 754: in particular,
remainder(x,
math.inf)
is
x
for any finite
x
, and
remainder(x,
0)
and
remainder(math.inf,
x)
raise
ValueError
for any non-NaN
x
.
If the result of the remainder operation is zero, that zero will have
the same sign as
x
.
On platforms using IEEE 754 binary floating point, the result of this
operation is always exactly representable: no rounding error is introduced.
Added in version 3.7.
math.
trunc
(
x
)
Ā¶
Return
x
with the fractional part
removed, leaving the integer part. This rounds toward 0:
trunc()
is
equivalent to
floor()
for positive
x
, and equivalent to
ceil()
for negative
x
. If
x
is not a float, delegates to
x.__trunc__
, which should return an
Integral
value.
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.
Floating point manipulation functions
Ā¶
math.
copysign
(
x
,
y
)
Ā¶
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.
frexp
(
x
)
Ā¶
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 āpick
apartā the internal representation of a float in a portable way.
Note that
frexp()
has a different call/return pattern
than its C equivalents: it takes a single argument and return a pair of
values, rather than returning its second return value through an āoutput
parameterā (there is no such thing in Python).
math.
isclose
(
a
,
b
,
*
,
rel_tol
=
1e-09
,
abs_tol
=
0.0
)
Ā¶
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. If no errors occur, the result will
be:
abs(a-b)
<=
max(rel_tol
*
max(abs(a),
abs(b)),
abs_tol)
.
rel_tol
is the relative tolerance ā 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 nonnegative and less
than
1.0
.
abs_tol
is the absolute tolerance; it defaults to
0.0
and it must be
nonnegative. When comparing
x
to
0.0
,
isclose(x,
0)
is computed
as
abs(x)
<=
rel_tol
Ā
*
abs(x)
, which is
False
for any nonzero
x
and
rel_tol
less than
1.0
. So add an appropriate positive
abs_tol
argument
to the call.
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.
Added in version 3.5.
See also
PEP 485
ā A function for testing approximate equality
math.
isfinite
(
x
)
Ā¶
Return
True
if
x
is neither an infinity nor a NaN, and
False
otherwise. (Note that
0.0
is
considered finite.)
Added in version 3.2.
math.
isinf
(
x
)
Ā¶
Return
True
if
x
is a positive or negative infinity, and
False
otherwise.
math.
isnan
(
x
)
Ā¶
Return
True
if
x
is a NaN (not a number), and
False
otherwise.
math.
ldexp
(
x
,
i
)
Ā¶
Return
x
*
(2**i)
. This is essentially the inverse of function
frexp()
.
math.
nextafter
(
x
,
y
,
steps
=
1
)
Ā¶
Return the floating-point value
steps
steps after
x
towards
y
.
If
x
is equal to
y
, return
y
, unless
steps
is zero.
Examples:
math.nextafter(x,
math.inf)
goes up: towards positive infinity.
math.nextafter(x,
-math.inf)
goes down: towards minus infinity.
math.nextafter(x,
0.0)
goes towards zero.
math.nextafter(x,
math.copysign(math.inf,
x))
goes away from zero.
See also
math.ulp()
.
Added in version 3.9.
Changed in version 3.12:
Added the
steps
argument.
math.
ulp
(
x
)
Ā¶
Return the value of the least significant bit of the float
x
:
If
x
is a NaN (not a number), return
x
.
If
x
is negative, return
ulp(-x)
.
If
x
is a positive infinity, return
x
.
If
x
is equal to zero, return the smallest positive
denormalized
representable float (smaller than the minimum positive
normalized
float,
sys.float_info.min
).
If
x
is equal to the largest positive representable float,
return the value of the least significant bit of
x
, such that the first
float smaller than
x
is
x
-
ulp(x)
.
Otherwise (
x
is a positive finite number), return the value of the least
significant bit of
x
, such that the first float bigger than
x
is
x
+
ulp(x)
.
ULP stands for āUnit in the Last Placeā.
See also
math.nextafter()
and
sys.float_info.epsilon
.
Added in version 3.9.
Power, exponential and logarithmic functions
Ā¶
math.
cbrt
(
x
)
Ā¶
Return the cube root of
x
.
Added in version 3.11.
math.
exp
(
x
)
Ā¶
Return
e
raised to the power
x
, where
e
= 2.718281ā¦ is the base
of natural logarithms. This is usually more accurate than
math.e
**
x
or
pow(math.e,
x)
.
math.
exp2
(
x
)
Ā¶
Return
2
raised to the power
x
.
Added in version 3.11.
math.
expm1
(
x
)
Ā¶
Return
e
raised to the power
x
, minus 1. Here
e
is the base of natural
logarithms. 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
Added in version 3.2.
math.
log
(
x
[
,
base
]
)
Ā¶
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
)
Ā¶
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
)
Ā¶
Return the base-2 logarithm of
x
. This is usually more accurate than
log(x,
2)
.
Added 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
)
Ā¶
Return the base-10 logarithm of
x
. This is usually more accurate
than
log(x,
10)
.
math.
pow
(
x
,
y
)
Ā¶
Return
x
raised to the power
y
. Exceptional cases follow
the IEEE 754 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.
Changed in version 3.11:
The special cases
pow(0.0,
-inf)
and
pow(-0.0,
-inf)
were
changed to return
inf
instead of raising
ValueError
,
for consistency with IEEE 754.
math.
sqrt
(
x
)
Ā¶
Return the square root of
x
.
Summation and product functions
Ā¶
math.
dist
(
p
,
q
)
Ā¶
Return the Euclidean distance between two points
p
and
q
, each
given as a sequence (or iterable) of coordinates. The two points
must have the same dimension.
Roughly equivalent to:
sqrt
(
sum
((
px
-
qx
)
**
2.0
for
px
,
qx
in
zip
(
p
,
q
)))
Added in version 3.8.
math.
fsum
(
iterable
)
Ā¶
Return an accurate floating-point sum of values in the iterable. Avoids
loss of precision by tracking multiple intermediate partial sums.
The algorithmās 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.
hypot
(
*
coordinates
)
Ā¶
Return the Euclidean norm,
sqrt(sum(x**2
for
x
in
coordinates))
.
This is the length of the vector from the origin to the point
given by the coordinates.
For a two dimensional point
(x,
y)
, this is equivalent to computing
the hypotenuse of a right triangle using the Pythagorean theorem,
sqrt(x*x
+
y*y)
.
Changed in version 3.8:
Added support for n-dimensional points. Formerly, only the two
dimensional case was supported.
Changed in version 3.10:
Improved the algorithmās accuracy so that the maximum error is
under 1 ulp (unit in the last place). More typically, the result
is almost always correctly rounded to within 1/2 ulp.
math.
prod
(
iterable
,
*
,
start
=
1
)
Ā¶
Calculate the product of all the elements in the input
iterable
.
The default
start
value for the product is
1
.
When the iterable is empty, return the start value. This function is
intended specifically for use with numeric values and may reject
non-numeric types.
Added in version 3.8.
math.
sumprod
(
p
,
q
)
Ā¶
Return the sum of products of values from two iterables
p
and
q
.
Raises
ValueError
if the inputs do not have the same length.
Roughly equivalent to:
sum
(
map
(
operator
.
mul
,
p
,
q
,
strict
=
True
))
For float and mixed int/float inputs, the intermediate products
and sums are computed with extended precision.
Added in version 3.12.
Angular conversion
Ā¶
math.
degrees
(
x
)
Ā¶
Convert angle
x
from radians to degrees.
math.
radians
(
x
)
Ā¶
Convert angle
x
from degrees to radians.
Trigonometric functions
Ā¶
math.
acos
(
x
)
Ā¶
Return the arc cosine of
x
, in radians. The result is between
0
and
pi
.
math.
asin
(
x
)
Ā¶
Return the arc sine of
x
, in radians. The result is between
-pi/2
and
pi/2
.
math.
atan
(
x
)
Ā¶
Return the arc tangent of
x
, in radians. The result is between
-pi/2
and
pi/2
.
math.
atan2
(
y
,
x
)
Ā¶
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
)
Ā¶
Return the cosine of
x
radians.
math.
sin
(
x
)
Ā¶
Return the sine of
x
radians.
math.
tan
(
x
)
Ā¶
Return the tangent of
x
radians.
Hyperbolic functions
Ā¶
Hyperbolic functions
are analogs of trigonometric functions that are based on hyperbolas
instead of circles.
math.
acosh
(
x
)
Ā¶
Return the inverse hyperbolic cosine of
x
.
math.
asinh
(
x
)
Ā¶
Return the inverse hyperbolic sine of
x
.
math.
atanh
(
x
)
Ā¶
Return the inverse hyperbolic tangent of
x
.
math.
cosh
(
x
)
Ā¶
Return the hyperbolic cosine of
x
.
math.
sinh
(
x
)
Ā¶
Return the hyperbolic sine of
x
.
math.
tanh
(
x
)
Ā¶
Return the hyperbolic tangent of
x
.
Special functions
Ā¶
math.
erf
(
x
)
Ā¶
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
Added in version 3.2.
math.
erfc
(
x
)
Ā¶
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
.
Added in version 3.2.
math.
gamma
(
x
)
Ā¶
Return the
Gamma function
at
x
.
Added in version 3.2.
math.
lgamma
(
x
)
Ā¶
Return the natural logarithm of the absolute value of the Gamma
function at
x
.
Added in version 3.2.
Constants
Ā¶
math.
pi
Ā¶
The mathematical constant
Ļ
= 3.141592ā¦, to available precision.
math.
e
Ā¶
The mathematical constant
e
= 2.718281ā¦, to available precision.
math.
tau
Ā¶
The mathematical constant
Ļ
= 6.283185ā¦, to available precision.
Tau is a circle constant equal to 2
Ļ
, the ratio of a circleās circumference to
its radius. To learn more about Tau, check out Vi Hartās video
Pi is (still)
Wrong
, and start celebrating
Tau day
by eating twice as much pie!
Added in version 3.6.
math.
inf
Ā¶
A floating-point positive infinity. (For negative infinity, use
-math.inf
.) Equivalent to the output of
float('inf')
.
Added in version 3.5.
math.
nan
Ā¶
A floating-point ānot a numberā (NaN) value. Equivalent to the output of
float('nan')
. Due to the requirements of the
IEEE-754 standard
,
math.nan
and
float('nan')
are
not considered to equal to any other numeric value, including themselves. To check
whether a number is a NaN, use the
isnan()
function to test
for NaNs instead of
is
or
==
.
Example:
>>>
import
math
>>>
math
.
nan
==
math
.
nan
False
>>>
float
(
'nan'
)
==
float
(
'nan'
)
False
>>>
math
.
isnan
(
math
.
nan
)
True
>>>
math
.
isnan
(
float
(
'nan'
))
True
Added in version 3.5.
Changed in version 3.11:
It is now always available.
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
Module
cmath
Complex number versions of many of these functions. |
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### [Table of Contents](https://docs.python.org/3/contents.html)
- [`math` ā Mathematical functions](https://docs.python.org/3/library/math.html)
- [Number-theoretic functions](https://docs.python.org/3/library/math.html#number-theoretic-functions)
- [Floating point arithmetic](https://docs.python.org/3/library/math.html#floating-point-arithmetic)
- [Floating point manipulation functions](https://docs.python.org/3/library/math.html#floating-point-manipulation-functions)
- [Power, exponential and logarithmic functions](https://docs.python.org/3/library/math.html#power-exponential-and-logarithmic-functions)
- [Summation and product functions](https://docs.python.org/3/library/math.html#summation-and-product-functions)
- [Angular conversion](https://docs.python.org/3/library/math.html#angular-conversion)
- [Trigonometric functions](https://docs.python.org/3/library/math.html#trigonometric-functions)
- [Hyperbolic functions](https://docs.python.org/3/library/math.html#hyperbolic-functions)
- [Special functions](https://docs.python.org/3/library/math.html#special-functions)
- [Constants](https://docs.python.org/3/library/math.html#constants)
#### Previous topic
[`numbers` ā Numeric abstract base classes](https://docs.python.org/3/library/numbers.html "previous chapter")
#### Next topic
[`cmath` ā Mathematical functions for complex numbers](https://docs.python.org/3/library/cmath.html "next chapter")
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# `math` ā Mathematical functions[Ā¶](https://docs.python.org/3/library/math.html#module-math "Link to this heading")
***
This module provides access to common mathematical functions and constants, including those defined by the C standard.
These functions cannot be used with complex numbers; use the functions of the same name from the [`cmath`](https://docs.python.org/3/library/cmath.html#module-cmath "cmath: Mathematical functions for complex numbers.") module if you require support for complex numbers. The distinction between functions which support complex numbers and those which donāt 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.
| | |
|---|---|
| **Number-theoretic functions** | |
| [`comb(n, k)`](https://docs.python.org/3/library/math.html#math.comb "math.comb") | Number of ways to choose *k* items from *n* items without repetition and without order |
| [`factorial(n)`](https://docs.python.org/3/library/math.html#math.factorial "math.factorial") | *n* factorial |
| [`gcd(*integers)`](https://docs.python.org/3/library/math.html#math.gcd "math.gcd") | Greatest common divisor of the integer arguments |
| [`isqrt(n)`](https://docs.python.org/3/library/math.html#math.isqrt "math.isqrt") | Integer square root of a nonnegative integer *n* |
| [`lcm(*integers)`](https://docs.python.org/3/library/math.html#math.lcm "math.lcm") | Least common multiple of the integer arguments |
| [`perm(n, k)`](https://docs.python.org/3/library/math.html#math.perm "math.perm") | Number of ways to choose *k* items from *n* items without repetition and with order |
| **Floating point arithmetic** | |
| [`ceil(x)`](https://docs.python.org/3/library/math.html#math.ceil "math.ceil") | Ceiling of *x*, the smallest integer greater than or equal to *x* |
| [`fabs(x)`](https://docs.python.org/3/library/math.html#math.fabs "math.fabs") | Absolute value of *x* |
| [`floor(x)`](https://docs.python.org/3/library/math.html#math.floor "math.floor") | Floor of *x*, the largest integer less than or equal to *x* |
| [`fma(x, y, z)`](https://docs.python.org/3/library/math.html#math.fma "math.fma") | Fused multiply-add operation: `(x * y) + z` |
| [`fmod(x, y)`](https://docs.python.org/3/library/math.html#math.fmod "math.fmod") | Remainder of division `x / y` |
| [`modf(x)`](https://docs.python.org/3/library/math.html#math.modf "math.modf") | Fractional and integer parts of *x* |
| [`remainder(x, y)`](https://docs.python.org/3/library/math.html#math.remainder "math.remainder") | Remainder of *x* with respect to *y* |
| [`trunc(x)`](https://docs.python.org/3/library/math.html#math.trunc "math.trunc") | Integer part of *x* |
| **Floating point manipulation functions** | |
| [`copysign(x, y)`](https://docs.python.org/3/library/math.html#math.copysign "math.copysign") | Magnitude (absolute value) of *x* with the sign of *y* |
| [`frexp(x)`](https://docs.python.org/3/library/math.html#math.frexp "math.frexp") | Mantissa and exponent of *x* |
| [`isclose(a, b, rel_tol, abs_tol)`](https://docs.python.org/3/library/math.html#math.isclose "math.isclose") | Check if the values *a* and *b* are close to each other |
| [`isfinite(x)`](https://docs.python.org/3/library/math.html#math.isfinite "math.isfinite") | Check if *x* is neither an infinity nor a NaN |
| [`isinf(x)`](https://docs.python.org/3/library/math.html#math.isinf "math.isinf") | Check if *x* is a positive or negative infinity |
| [`isnan(x)`](https://docs.python.org/3/library/math.html#math.isnan "math.isnan") | Check if *x* is a NaN (not a number) |
| [`ldexp(x, i)`](https://docs.python.org/3/library/math.html#math.ldexp "math.ldexp") | `x * (2**i)`, inverse of function [`frexp()`](https://docs.python.org/3/library/math.html#math.frexp "math.frexp") |
| [`nextafter(x, y, steps)`](https://docs.python.org/3/library/math.html#math.nextafter "math.nextafter") | Floating-point value *steps* steps after *x* towards *y* |
| [`ulp(x)`](https://docs.python.org/3/library/math.html#math.ulp "math.ulp") | Value of the least significant bit of *x* |
| **Power, exponential and logarithmic functions** | |
| [`cbrt(x)`](https://docs.python.org/3/library/math.html#math.cbrt "math.cbrt") | Cube root of *x* |
| [`exp(x)`](https://docs.python.org/3/library/math.html#math.exp "math.exp") | *e* raised to the power *x* |
| [`exp2(x)`](https://docs.python.org/3/library/math.html#math.exp2 "math.exp2") | *2* raised to the power *x* |
| [`expm1(x)`](https://docs.python.org/3/library/math.html#math.expm1 "math.expm1") | *e* raised to the power *x*, minus 1 |
| [`log(x, base)`](https://docs.python.org/3/library/math.html#math.log "math.log") | Logarithm of *x* to the given base (*e* by default) |
| [`log1p(x)`](https://docs.python.org/3/library/math.html#math.log1p "math.log1p") | Natural logarithm of *1+x* (base *e*) |
| [`log2(x)`](https://docs.python.org/3/library/math.html#math.log2 "math.log2") | Base-2 logarithm of *x* |
| [`log10(x)`](https://docs.python.org/3/library/math.html#math.log10 "math.log10") | Base-10 logarithm of *x* |
| [`pow(x, y)`](https://docs.python.org/3/library/math.html#math.pow "math.pow") | *x* raised to the power *y* |
| [`sqrt(x)`](https://docs.python.org/3/library/math.html#math.sqrt "math.sqrt") | Square root of *x* |
| **Summation and product functions** | |
| [`dist(p, q)`](https://docs.python.org/3/library/math.html#math.dist "math.dist") | Euclidean distance between two points *p* and *q* given as an iterable of coordinates |
| [`fsum(iterable)`](https://docs.python.org/3/library/math.html#math.fsum "math.fsum") | Sum of values in the input *iterable* |
| [`hypot(*coordinates)`](https://docs.python.org/3/library/math.html#math.hypot "math.hypot") | Euclidean norm of an iterable of coordinates |
| [`prod(iterable, start)`](https://docs.python.org/3/library/math.html#math.prod "math.prod") | Product of elements in the input *iterable* with a *start* value |
| [`sumprod(p, q)`](https://docs.python.org/3/library/math.html#math.sumprod "math.sumprod") | Sum of products from two iterables *p* and *q* |
| **Angular conversion** | |
| [`degrees(x)`](https://docs.python.org/3/library/math.html#math.degrees "math.degrees") | Convert angle *x* from radians to degrees |
| [`radians(x)`](https://docs.python.org/3/library/math.html#math.radians "math.radians") | Convert angle *x* from degrees to radians |
| **Trigonometric functions** | |
| [`acos(x)`](https://docs.python.org/3/library/math.html#math.acos "math.acos") | Arc cosine of *x* |
| [`asin(x)`](https://docs.python.org/3/library/math.html#math.asin "math.asin") | Arc sine of *x* |
| [`atan(x)`](https://docs.python.org/3/library/math.html#math.atan "math.atan") | Arc tangent of *x* |
| [`atan2(y, x)`](https://docs.python.org/3/library/math.html#math.atan2 "math.atan2") | `atan(y / x)` |
| [`cos(x)`](https://docs.python.org/3/library/math.html#math.cos "math.cos") | Cosine of *x* |
| [`sin(x)`](https://docs.python.org/3/library/math.html#math.sin "math.sin") | Sine of *x* |
| [`tan(x)`](https://docs.python.org/3/library/math.html#math.tan "math.tan") | Tangent of *x* |
| **Hyperbolic functions** | |
| [`acosh(x)`](https://docs.python.org/3/library/math.html#math.acosh "math.acosh") | Inverse hyperbolic cosine of *x* |
| [`asinh(x)`](https://docs.python.org/3/library/math.html#math.asinh "math.asinh") | Inverse hyperbolic sine of *x* |
| [`atanh(x)`](https://docs.python.org/3/library/math.html#math.atanh "math.atanh") | Inverse hyperbolic tangent of *x* |
| [`cosh(x)`](https://docs.python.org/3/library/math.html#math.cosh "math.cosh") | Hyperbolic cosine of *x* |
| [`sinh(x)`](https://docs.python.org/3/library/math.html#math.sinh "math.sinh") | Hyperbolic sine of *x* |
| [`tanh(x)`](https://docs.python.org/3/library/math.html#math.tanh "math.tanh") | Hyperbolic tangent of *x* |
| **Special functions** | |
| [`erf(x)`](https://docs.python.org/3/library/math.html#math.erf "math.erf") | [Error function](https://en.wikipedia.org/wiki/Error_function) at *x* |
| [`erfc(x)`](https://docs.python.org/3/library/math.html#math.erfc "math.erfc") | [Complementary error function](https://en.wikipedia.org/wiki/Error_function) at *x* |
| [`gamma(x)`](https://docs.python.org/3/library/math.html#math.gamma "math.gamma") | [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) at *x* |
| [`lgamma(x)`](https://docs.python.org/3/library/math.html#math.lgamma "math.lgamma") | Natural logarithm of the absolute value of the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) at *x* |
| **Constants** | |
| [`pi`](https://docs.python.org/3/library/math.html#math.pi "math.pi") | *Ļ* = 3.141592ā¦ |
| [`e`](https://docs.python.org/3/library/math.html#math.e "math.e") | *e* = 2.718281ā¦ |
| [`tau`](https://docs.python.org/3/library/math.html#math.tau "math.tau") | *Ļ* = 2*Ļ* = 6.283185ā¦ |
| [`inf`](https://docs.python.org/3/library/math.html#math.inf "math.inf") | Positive infinity |
| [`nan`](https://docs.python.org/3/library/math.html#math.nan "math.nan") | āNot a numberā (NaN) |
## Number-theoretic functions[Ā¶](https://docs.python.org/3/library/math.html#number-theoretic-functions "Link to this heading")
math.comb(*n*, *k*)[Ā¶](https://docs.python.org/3/library/math.html#math.comb "Link to this definition")
Return the number of ways to choose *k* items from *n* items without repetition and without order.
Evaluates to `n! / (k! * (n - k)!)` when `k <= n` and evaluates to zero when `k > n`.
Also called the binomial coefficient because it is equivalent to the coefficient of k-th term in polynomial expansion of `(1 + x)āæ`.
Raises [`TypeError`](https://docs.python.org/3/library/exceptions.html#TypeError "TypeError") if either of the arguments are not integers. Raises [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") if either of the arguments are negative.
Added in version 3.8.
math.factorial(*n*)[Ā¶](https://docs.python.org/3/library/math.html#math.factorial "Link to this definition")
Return factorial of the nonnegative integer *n*.
Changed in version 3.10: Floats with integral values (like `5.0`) are no longer accepted.
math.gcd(*\*integers*)[Ā¶](https://docs.python.org/3/library/math.html#math.gcd "Link to this definition")
Return the greatest common divisor of the specified integer arguments. If any of the arguments is nonzero, then the returned value is the largest positive integer that is a divisor of all arguments. If all arguments are zero, then the returned value is `0`. `gcd()` without arguments returns `0`.
Added in version 3.5.
Changed in version 3.9: Added support for an arbitrary number of arguments. Formerly, only two arguments were supported.
math.isqrt(*n*)[Ā¶](https://docs.python.org/3/library/math.html#math.isqrt "Link to this definition")
Return the integer square root of the nonnegative integer *n*. This is the floor of the exact square root of *n*, or equivalently the greatest integer *a* such that *a*Ā² ā¤ *n*.
For some applications, it may be more convenient to have the least integer *a* such that *n* ā¤ *a*Ā², or in other words the ceiling of the exact square root of *n*. For positive *n*, this can be computed using `a = 1 + isqrt(n - 1)`.
Added in version 3.8.
math.lcm(*\*integers*)[Ā¶](https://docs.python.org/3/library/math.html#math.lcm "Link to this definition")
Return the least common multiple of the specified integer arguments. If all arguments are nonzero, then the returned value is the smallest positive integer that is a multiple of all arguments. If any of the arguments is zero, then the returned value is `0`. `lcm()` without arguments returns `1`.
Added in version 3.9.
math.perm(*n*, *k\=None*)[Ā¶](https://docs.python.org/3/library/math.html#math.perm "Link to this definition")
Return the number of ways to choose *k* items from *n* items without repetition and with order.
Evaluates to `n! / (n - k)!` when `k <= n` and evaluates to zero when `k > n`.
If *k* is not specified or is `None`, then *k* defaults to *n* and the function returns `n!`.
Raises [`TypeError`](https://docs.python.org/3/library/exceptions.html#TypeError "TypeError") if either of the arguments are not integers. Raises [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") if either of the arguments are negative.
Added in version 3.8.
## Floating point arithmetic[Ā¶](https://docs.python.org/3/library/math.html#floating-point-arithmetic "Link to this heading")
math.ceil(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.ceil "Link to this definition")
Return the ceiling of *x*, the smallest integer greater than or equal to *x*. If *x* is not a float, delegates to [`x.__ceil__`](https://docs.python.org/3/reference/datamodel.html#object.__ceil__ "object.__ceil__"), which should return an [`Integral`](https://docs.python.org/3/library/numbers.html#numbers.Integral "numbers.Integral") value.
math.fabs(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.fabs "Link to this definition")
Return the absolute value of *x*.
math.floor(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.floor "Link to this definition")
Return the floor of *x*, the largest integer less than or equal to *x*. If *x* is not a float, delegates to [`x.__floor__`](https://docs.python.org/3/reference/datamodel.html#object.__floor__ "object.__floor__"), which should return an [`Integral`](https://docs.python.org/3/library/numbers.html#numbers.Integral "numbers.Integral") value.
math.fma(*x*, *y*, *z*)[Ā¶](https://docs.python.org/3/library/math.html#math.fma "Link to this definition")
Fused multiply-add operation. Return `(x * y) + z`, computed as though with infinite precision and range followed by a single round to the `float` format. This operation often provides better accuracy than the direct expression `(x * y) + z`.
This function follows the specification of the fusedMultiplyAdd operation described in the IEEE 754 standard. The standard leaves one case implementation-defined, namely the result of `fma(0, inf, nan)` and `fma(inf, 0, nan)`. In these cases, `math.fma` returns a NaN, and does not raise any exception.
Added in version 3.13.
math.fmod(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.fmod "Link to this definition")
Return the floating-point remainder of `x / y`, as defined by the platform C library function `fmod(x, y)`. 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)`. Pythonās `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 Pythonās `-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 Pythonās `x % y` is preferred when working with integers.
math.modf(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.modf "Link to this definition")
Return the fractional and integer parts of *x*. Both results carry the sign of *x* and are floats.
Note that `modf()` has a different call/return pattern than its C equivalents: it takes a single argument and return a pair of values, rather than returning its second return value through an āoutput parameterā (there is no such thing in Python).
math.remainder(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.remainder "Link to this definition")
Return the IEEE 754-style remainder of *x* with respect to *y*. For finite *x* and finite nonzero *y*, this is the difference `x - n*y`, where `n` is the closest integer to the exact value of the quotient . If `x / y` is exactly halfway between two consecutive integers, the nearest *even* integer is used for `n`. The remainder thus always satisfies `abs(r) <= 0.5 * abs(y)`.
Special cases follow IEEE 754: in particular, `remainder(x, math.inf)` is *x* for any finite *x*, and `remainder(x, 0)` and `remainder(math.inf, x)` raise [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") for any non-NaN *x*. If the result of the remainder operation is zero, that zero will have the same sign as *x*.
On platforms using IEEE 754 binary floating point, the result of this operation is always exactly representable: no rounding error is introduced.
Added in version 3.7.
math.trunc(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.trunc "Link to this definition")
Return *x* with the fractional part removed, leaving the integer part. This rounds toward 0: `trunc()` is equivalent to [`floor()`](https://docs.python.org/3/library/math.html#math.floor "math.floor") for positive *x*, and equivalent to [`ceil()`](https://docs.python.org/3/library/math.html#math.ceil "math.ceil") for negative *x*. If *x* is not a float, delegates to [`x.__trunc__`](https://docs.python.org/3/reference/datamodel.html#object.__trunc__ "object.__trunc__"), which should return an [`Integral`](https://docs.python.org/3/library/numbers.html#numbers.Integral "numbers.Integral") value.
For the [`ceil()`](https://docs.python.org/3/library/math.html#math.ceil "math.ceil"), [`floor()`](https://docs.python.org/3/library/math.html#math.floor "math.floor"), and [`modf()`](https://docs.python.org/3/library/math.html#math.modf "math.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.
## Floating point manipulation functions[Ā¶](https://docs.python.org/3/library/math.html#floating-point-manipulation-functions "Link to this heading")
math.copysign(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.copysign "Link to this definition")
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.frexp(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.frexp "Link to this definition")
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 āpick apartā the internal representation of a float in a portable way.
Note that `frexp()` has a different call/return pattern than its C equivalents: it takes a single argument and return a pair of values, rather than returning its second return value through an āoutput parameterā (there is no such thing in Python).
math.isclose(*a*, *b*, *\**, *rel\_tol\=1e-09*, *abs\_tol\=0\.0*)[Ā¶](https://docs.python.org/3/library/math.html#math.isclose "Link to this definition")
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. If no errors occur, the result will be: `abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)`.
*rel\_tol* is the relative tolerance ā 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 nonnegative and less than `1.0`.
*abs\_tol* is the absolute tolerance; it defaults to `0.0` and it must be nonnegative. When comparing `x` to `0.0`, `isclose(x, 0)` is computed as `abs(x) <= rel_tolĀ * abs(x)`, which is `False` for any nonzero `x` and *rel\_tol* less than `1.0`. So add an appropriate positive *abs\_tol* argument to the call.
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.
Added in version 3.5.
See also
[**PEP 485**](https://peps.python.org/pep-0485/) ā A function for testing approximate equality
math.isfinite(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.isfinite "Link to this definition")
Return `True` if *x* is neither an infinity nor a NaN, and `False` otherwise. (Note that `0.0` *is* considered finite.)
Added in version 3.2.
math.isinf(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.isinf "Link to this definition")
Return `True` if *x* is a positive or negative infinity, and `False` otherwise.
math.isnan(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.isnan "Link to this definition")
Return `True` if *x* is a NaN (not a number), and `False` otherwise.
math.ldexp(*x*, *i*)[Ā¶](https://docs.python.org/3/library/math.html#math.ldexp "Link to this definition")
Return `x * (2**i)`. This is essentially the inverse of function [`frexp()`](https://docs.python.org/3/library/math.html#math.frexp "math.frexp").
math.nextafter(*x*, *y*, *steps\=1*)[Ā¶](https://docs.python.org/3/library/math.html#math.nextafter "Link to this definition")
Return the floating-point value *steps* steps after *x* towards *y*.
If *x* is equal to *y*, return *y*, unless *steps* is zero.
Examples:
- `math.nextafter(x, math.inf)` goes up: towards positive infinity.
- `math.nextafter(x, -math.inf)` goes down: towards minus infinity.
- `math.nextafter(x, 0.0)` goes towards zero.
- `math.nextafter(x, math.copysign(math.inf, x))` goes away from zero.
See also [`math.ulp()`](https://docs.python.org/3/library/math.html#math.ulp "math.ulp").
Added in version 3.9.
Changed in version 3.12: Added the *steps* argument.
math.ulp(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.ulp "Link to this definition")
Return the value of the least significant bit of the float *x*:
- If *x* is a NaN (not a number), return *x*.
- If *x* is negative, return `ulp(-x)`.
- If *x* is a positive infinity, return *x*.
- If *x* is equal to zero, return the smallest positive *denormalized* representable float (smaller than the minimum positive *normalized* float, [`sys.float_info.min`](https://docs.python.org/3/library/sys.html#sys.float_info "sys.float_info")).
- If *x* is equal to the largest positive representable float, return the value of the least significant bit of *x*, such that the first float smaller than *x* is `x - ulp(x)`.
- Otherwise (*x* is a positive finite number), return the value of the least significant bit of *x*, such that the first float bigger than *x* is `x + ulp(x)`.
ULP stands for āUnit in the Last Placeā.
See also [`math.nextafter()`](https://docs.python.org/3/library/math.html#math.nextafter "math.nextafter") and [`sys.float_info.epsilon`](https://docs.python.org/3/library/sys.html#sys.float_info "sys.float_info").
Added in version 3.9.
## Power, exponential and logarithmic functions[Ā¶](https://docs.python.org/3/library/math.html#power-exponential-and-logarithmic-functions "Link to this heading")
math.cbrt(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.cbrt "Link to this definition")
Return the cube root of *x*.
Added in version 3.11.
math.exp(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.exp "Link to this definition")
Return *e* raised to the power *x*, where *e* = 2.718281ā¦ is the base of natural logarithms. This is usually more accurate than `math.e ** x` or `pow(math.e, x)`.
math.exp2(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.exp2 "Link to this definition")
Return *2* raised to the power *x*.
Added in version 3.11.
math.expm1(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.expm1 "Link to this definition")
Return *e* raised to the power *x*, minus 1. Here *e* is the base of natural logarithms. For small floats *x*, the subtraction in `exp(x) - 1` can result in a [significant loss of precision](https://en.wikipedia.org/wiki/Loss_of_significance); the `expm1()` function provides a way to compute this quantity to full precision:
Copy
```
>>> 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
```
Added in version 3.2.
math.log(*x*\[, *base*\])[Ā¶](https://docs.python.org/3/library/math.html#math.log "Link to this definition")
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*)[Ā¶](https://docs.python.org/3/library/math.html#math.log1p "Link to this definition")
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*)[Ā¶](https://docs.python.org/3/library/math.html#math.log2 "Link to this definition")
Return the base-2 logarithm of *x*. This is usually more accurate than `log(x, 2)`.
Added in version 3.3.
See also
[`int.bit_length()`](https://docs.python.org/3/library/stdtypes.html#int.bit_length "int.bit_length") returns the number of bits necessary to represent an integer in binary, excluding the sign and leading zeros.
math.log10(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.log10 "Link to this definition")
Return the base-10 logarithm of *x*. This is usually more accurate than `log(x, 10)`.
math.pow(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.pow "Link to this definition")
Return *x* raised to the power *y*. Exceptional cases follow the IEEE 754 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`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError").
Unlike the built-in `**` operator, [`math.pow()`](https://docs.python.org/3/library/math.html#math.pow "math.pow") converts both its arguments to type [`float`](https://docs.python.org/3/library/functions.html#float "float"). Use `**` or the built-in `pow()` function for computing exact integer powers.
Changed in version 3.11: The special cases `pow(0.0, -inf)` and `pow(-0.0, -inf)` were changed to return `inf` instead of raising [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError"), for consistency with IEEE 754.
math.sqrt(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.sqrt "Link to this definition")
Return the square root of *x*.
## Summation and product functions[Ā¶](https://docs.python.org/3/library/math.html#summation-and-product-functions "Link to this heading")
math.dist(*p*, *q*)[Ā¶](https://docs.python.org/3/library/math.html#math.dist "Link to this definition")
Return the Euclidean distance between two points *p* and *q*, each given as a sequence (or iterable) of coordinates. The two points must have the same dimension.
Roughly equivalent to:
Copy
```
sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))
```
Added in version 3.8.
math.fsum(*iterable*)[Ā¶](https://docs.python.org/3/library/math.html#math.fsum "Link to this definition")
Return an accurate floating-point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums.
The algorithmās 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](https://code.activestate.com/recipes/393090-binary-floating-point-summation-accurate-to-full-p/).
math.hypot(*\*coordinates*)[Ā¶](https://docs.python.org/3/library/math.html#math.hypot "Link to this definition")
Return the Euclidean norm, `sqrt(sum(x**2 for x in coordinates))`. This is the length of the vector from the origin to the point given by the coordinates.
For a two dimensional point `(x, y)`, this is equivalent to computing the hypotenuse of a right triangle using the Pythagorean theorem, `sqrt(x*x + y*y)`.
Changed in version 3.8: Added support for n-dimensional points. Formerly, only the two dimensional case was supported.
Changed in version 3.10: Improved the algorithmās accuracy so that the maximum error is under 1 ulp (unit in the last place). More typically, the result is almost always correctly rounded to within 1/2 ulp.
math.prod(*iterable*, *\**, *start\=1*)[Ā¶](https://docs.python.org/3/library/math.html#math.prod "Link to this definition")
Calculate the product of all the elements in the input *iterable*. The default *start* value for the product is `1`.
When the iterable is empty, return the start value. This function is intended specifically for use with numeric values and may reject non-numeric types.
Added in version 3.8.
math.sumprod(*p*, *q*)[Ā¶](https://docs.python.org/3/library/math.html#math.sumprod "Link to this definition")
Return the sum of products of values from two iterables *p* and *q*.
Raises [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") if the inputs do not have the same length.
Roughly equivalent to:
Copy
```
sum(map(operator.mul, p, q, strict=True))
```
For float and mixed int/float inputs, the intermediate products and sums are computed with extended precision.
Added in version 3.12.
## Angular conversion[Ā¶](https://docs.python.org/3/library/math.html#angular-conversion "Link to this heading")
math.degrees(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.degrees "Link to this definition")
Convert angle *x* from radians to degrees.
math.radians(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.radians "Link to this definition")
Convert angle *x* from degrees to radians.
## Trigonometric functions[Ā¶](https://docs.python.org/3/library/math.html#trigonometric-functions "Link to this heading")
math.acos(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.acos "Link to this definition")
Return the arc cosine of *x*, in radians. The result is between `0` and `pi`.
math.asin(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.asin "Link to this definition")
Return the arc sine of *x*, in radians. The result is between `-pi/2` and `pi/2`.
math.atan(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.atan "Link to this definition")
Return the arc tangent of *x*, in radians. The result is between `-pi/2` and `pi/2`.
math.atan2(*y*, *x*)[Ā¶](https://docs.python.org/3/library/math.html#math.atan2 "Link to this definition")
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 is `-3*pi/4`.
math.cos(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.cos "Link to this definition")
Return the cosine of *x* radians.
math.sin(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.sin "Link to this definition")
Return the sine of *x* radians.
math.tan(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.tan "Link to this definition")
Return the tangent of *x* radians.
## Hyperbolic functions[Ā¶](https://docs.python.org/3/library/math.html#hyperbolic-functions "Link to this heading")
[Hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_functions) are analogs of trigonometric functions that are based on hyperbolas instead of circles.
math.acosh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.acosh "Link to this definition")
Return the inverse hyperbolic cosine of *x*.
math.asinh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.asinh "Link to this definition")
Return the inverse hyperbolic sine of *x*.
math.atanh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.atanh "Link to this definition")
Return the inverse hyperbolic tangent of *x*.
math.cosh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.cosh "Link to this definition")
Return the hyperbolic cosine of *x*.
math.sinh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.sinh "Link to this definition")
Return the hyperbolic sine of *x*.
math.tanh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.tanh "Link to this definition")
Return the hyperbolic tangent of *x*.
## Special functions[Ā¶](https://docs.python.org/3/library/math.html#special-functions "Link to this heading")
math.erf(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.erf "Link to this definition")
Return the [error function](https://en.wikipedia.org/wiki/Error_function) at *x*.
The `erf()` function can be used to compute traditional statistical functions such as the [cumulative standard normal distribution](https://en.wikipedia.org/wiki/Cumulative_distribution_function):
Copy
```
def phi(x):
'Cumulative distribution function for the standard normal distribution'
return (1.0 + erf(x / sqrt(2.0))) / 2.0
```
Added in version 3.2.
math.erfc(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.erfc "Link to this definition")
Return the complementary error function at *x*. The [complementary error function](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Loss_of_significance).
Added in version 3.2.
math.gamma(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.gamma "Link to this definition")
Return the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) at *x*.
Added in version 3.2.
math.lgamma(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.lgamma "Link to this definition")
Return the natural logarithm of the absolute value of the Gamma function at *x*.
Added in version 3.2.
## Constants[Ā¶](https://docs.python.org/3/library/math.html#constants "Link to this heading")
math.pi[Ā¶](https://docs.python.org/3/library/math.html#math.pi "Link to this definition")
The mathematical constant *Ļ* = 3.141592ā¦, to available precision.
math.e[Ā¶](https://docs.python.org/3/library/math.html#math.e "Link to this definition")
The mathematical constant *e* = 2.718281ā¦, to available precision.
math.tau[Ā¶](https://docs.python.org/3/library/math.html#math.tau "Link to this definition")
The mathematical constant *Ļ* = 6.283185ā¦, to available precision. Tau is a circle constant equal to 2*Ļ*, the ratio of a circleās circumference to its radius. To learn more about Tau, check out Vi Hartās video [Pi is (still) Wrong](https://vimeo.com/147792667), and start celebrating [Tau day](https://tauday.com/) by eating twice as much pie\!
Added in version 3.6.
math.inf[Ā¶](https://docs.python.org/3/library/math.html#math.inf "Link to this definition")
A floating-point positive infinity. (For negative infinity, use `-math.inf`.) Equivalent to the output of `float('inf')`.
Added in version 3.5.
math.nan[Ā¶](https://docs.python.org/3/library/math.html#math.nan "Link to this definition")
A floating-point ānot a numberā (NaN) value. Equivalent to the output of `float('nan')`. Due to the requirements of the [IEEE-754 standard](https://en.wikipedia.org/wiki/IEEE_754), `math.nan` and `float('nan')` are not considered to equal to any other numeric value, including themselves. To check whether a number is a NaN, use the [`isnan()`](https://docs.python.org/3/library/math.html#math.isnan "math.isnan") function to test for NaNs instead of `is` or `==`. Example:
Copy
```
>>> import math
>>> math.nan == math.nan
False
>>> float('nan') == float('nan')
False
>>> math.isnan(math.nan)
True
>>> math.isnan(float('nan'))
True
```
Added in version 3.5.
Changed in version 3.11: It is now always available.
**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`](https://docs.python.org/3/library/exceptions.html#ValueError "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`](https://docs.python.org/3/library/exceptions.html#OverflowError "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
Module [`cmath`](https://docs.python.org/3/library/cmath.html#module-cmath "cmath: Mathematical functions for complex numbers.")
Complex number versions of many of these functions.
### [Table of Contents](https://docs.python.org/3/contents.html)
- [`math` ā Mathematical functions](https://docs.python.org/3/library/math.html)
- [Number-theoretic functions](https://docs.python.org/3/library/math.html#number-theoretic-functions)
- [Floating point arithmetic](https://docs.python.org/3/library/math.html#floating-point-arithmetic)
- [Floating point manipulation functions](https://docs.python.org/3/library/math.html#floating-point-manipulation-functions)
- [Power, exponential and logarithmic functions](https://docs.python.org/3/library/math.html#power-exponential-and-logarithmic-functions)
- [Summation and product functions](https://docs.python.org/3/library/math.html#summation-and-product-functions)
- [Angular conversion](https://docs.python.org/3/library/math.html#angular-conversion)
- [Trigonometric functions](https://docs.python.org/3/library/math.html#trigonometric-functions)
- [Hyperbolic functions](https://docs.python.org/3/library/math.html#hyperbolic-functions)
- [Special functions](https://docs.python.org/3/library/math.html#special-functions)
- [Constants](https://docs.python.org/3/library/math.html#constants)
#### Previous topic
[`numbers` ā Numeric abstract base classes](https://docs.python.org/3/library/numbers.html "previous chapter")
#### Next topic
[`cmath` ā Mathematical functions for complex numbers](https://docs.python.org/3/library/cmath.html "next chapter")
### This page
- [Report a bug](https://docs.python.org/3/bugs.html)
- [Improve this page](https://docs.python.org/3/improve-page.html?pagetitle=math+%E2%80%94+Mathematical+functions&pageurl=https%3A%2F%2Fdocs.python.org%2F3%2Flibrary%2Fmath.html&pagesource=library%2Fmath.rst)
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This module provides access to common mathematical functions and constants, including those defined by the C standard.
These functions cannot be used with complex numbers; use the functions of the same name from the [`cmath`](https://docs.python.org/3/library/cmath.html#module-cmath "cmath: Mathematical functions for complex numbers.") module if you require support for complex numbers. The distinction between functions which support complex numbers and those which donāt 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.
| | |
|---|---|
| **Number-theoretic functions** | |
| [`comb(n, k)`](https://docs.python.org/3/library/math.html#math.comb "math.comb") | Number of ways to choose *k* items from *n* items without repetition and without order |
| [`factorial(n)`](https://docs.python.org/3/library/math.html#math.factorial "math.factorial") | *n* factorial |
| [`gcd(*integers)`](https://docs.python.org/3/library/math.html#math.gcd "math.gcd") | Greatest common divisor of the integer arguments |
| [`isqrt(n)`](https://docs.python.org/3/library/math.html#math.isqrt "math.isqrt") | Integer square root of a nonnegative integer *n* |
| [`lcm(*integers)`](https://docs.python.org/3/library/math.html#math.lcm "math.lcm") | Least common multiple of the integer arguments |
| [`perm(n, k)`](https://docs.python.org/3/library/math.html#math.perm "math.perm") | Number of ways to choose *k* items from *n* items without repetition and with order |
| **Floating point arithmetic** | |
| [`ceil(x)`](https://docs.python.org/3/library/math.html#math.ceil "math.ceil") | Ceiling of *x*, the smallest integer greater than or equal to *x* |
| [`fabs(x)`](https://docs.python.org/3/library/math.html#math.fabs "math.fabs") | Absolute value of *x* |
| [`floor(x)`](https://docs.python.org/3/library/math.html#math.floor "math.floor") | Floor of *x*, the largest integer less than or equal to *x* |
| [`fma(x, y, z)`](https://docs.python.org/3/library/math.html#math.fma "math.fma") | Fused multiply-add operation: `(x * y) + z` |
| [`fmod(x, y)`](https://docs.python.org/3/library/math.html#math.fmod "math.fmod") | Remainder of division `x / y` |
| [`modf(x)`](https://docs.python.org/3/library/math.html#math.modf "math.modf") | Fractional and integer parts of *x* |
| [`remainder(x, y)`](https://docs.python.org/3/library/math.html#math.remainder "math.remainder") | Remainder of *x* with respect to *y* |
| [`trunc(x)`](https://docs.python.org/3/library/math.html#math.trunc "math.trunc") | Integer part of *x* |
| **Floating point manipulation functions** | |
| [`copysign(x, y)`](https://docs.python.org/3/library/math.html#math.copysign "math.copysign") | Magnitude (absolute value) of *x* with the sign of *y* |
| [`frexp(x)`](https://docs.python.org/3/library/math.html#math.frexp "math.frexp") | Mantissa and exponent of *x* |
| [`isclose(a, b, rel_tol, abs_tol)`](https://docs.python.org/3/library/math.html#math.isclose "math.isclose") | Check if the values *a* and *b* are close to each other |
| [`isfinite(x)`](https://docs.python.org/3/library/math.html#math.isfinite "math.isfinite") | Check if *x* is neither an infinity nor a NaN |
| [`isinf(x)`](https://docs.python.org/3/library/math.html#math.isinf "math.isinf") | Check if *x* is a positive or negative infinity |
| [`isnan(x)`](https://docs.python.org/3/library/math.html#math.isnan "math.isnan") | Check if *x* is a NaN (not a number) |
| [`ldexp(x, i)`](https://docs.python.org/3/library/math.html#math.ldexp "math.ldexp") | `x * (2**i)`, inverse of function [`frexp()`](https://docs.python.org/3/library/math.html#math.frexp "math.frexp") |
| [`nextafter(x, y, steps)`](https://docs.python.org/3/library/math.html#math.nextafter "math.nextafter") | Floating-point value *steps* steps after *x* towards *y* |
| [`ulp(x)`](https://docs.python.org/3/library/math.html#math.ulp "math.ulp") | Value of the least significant bit of *x* |
| **Power, exponential and logarithmic functions** | |
| [`cbrt(x)`](https://docs.python.org/3/library/math.html#math.cbrt "math.cbrt") | Cube root of *x* |
| [`exp(x)`](https://docs.python.org/3/library/math.html#math.exp "math.exp") | *e* raised to the power *x* |
| [`exp2(x)`](https://docs.python.org/3/library/math.html#math.exp2 "math.exp2") | *2* raised to the power *x* |
| [`expm1(x)`](https://docs.python.org/3/library/math.html#math.expm1 "math.expm1") | *e* raised to the power *x*, minus 1 |
| [`log(x, base)`](https://docs.python.org/3/library/math.html#math.log "math.log") | Logarithm of *x* to the given base (*e* by default) |
| [`log1p(x)`](https://docs.python.org/3/library/math.html#math.log1p "math.log1p") | Natural logarithm of *1+x* (base *e*) |
| [`log2(x)`](https://docs.python.org/3/library/math.html#math.log2 "math.log2") | Base-2 logarithm of *x* |
| [`log10(x)`](https://docs.python.org/3/library/math.html#math.log10 "math.log10") | Base-10 logarithm of *x* |
| [`pow(x, y)`](https://docs.python.org/3/library/math.html#math.pow "math.pow") | *x* raised to the power *y* |
| [`sqrt(x)`](https://docs.python.org/3/library/math.html#math.sqrt "math.sqrt") | Square root of *x* |
| **Summation and product functions** | |
| [`dist(p, q)`](https://docs.python.org/3/library/math.html#math.dist "math.dist") | Euclidean distance between two points *p* and *q* given as an iterable of coordinates |
| [`fsum(iterable)`](https://docs.python.org/3/library/math.html#math.fsum "math.fsum") | Sum of values in the input *iterable* |
| [`hypot(*coordinates)`](https://docs.python.org/3/library/math.html#math.hypot "math.hypot") | Euclidean norm of an iterable of coordinates |
| [`prod(iterable, start)`](https://docs.python.org/3/library/math.html#math.prod "math.prod") | Product of elements in the input *iterable* with a *start* value |
| [`sumprod(p, q)`](https://docs.python.org/3/library/math.html#math.sumprod "math.sumprod") | Sum of products from two iterables *p* and *q* |
| **Angular conversion** | |
| [`degrees(x)`](https://docs.python.org/3/library/math.html#math.degrees "math.degrees") | Convert angle *x* from radians to degrees |
| [`radians(x)`](https://docs.python.org/3/library/math.html#math.radians "math.radians") | Convert angle *x* from degrees to radians |
| **Trigonometric functions** | |
| [`acos(x)`](https://docs.python.org/3/library/math.html#math.acos "math.acos") | Arc cosine of *x* |
| [`asin(x)`](https://docs.python.org/3/library/math.html#math.asin "math.asin") | Arc sine of *x* |
| [`atan(x)`](https://docs.python.org/3/library/math.html#math.atan "math.atan") | Arc tangent of *x* |
| [`atan2(y, x)`](https://docs.python.org/3/library/math.html#math.atan2 "math.atan2") | `atan(y / x)` |
| [`cos(x)`](https://docs.python.org/3/library/math.html#math.cos "math.cos") | Cosine of *x* |
| [`sin(x)`](https://docs.python.org/3/library/math.html#math.sin "math.sin") | Sine of *x* |
| [`tan(x)`](https://docs.python.org/3/library/math.html#math.tan "math.tan") | Tangent of *x* |
| **Hyperbolic functions** | |
| [`acosh(x)`](https://docs.python.org/3/library/math.html#math.acosh "math.acosh") | Inverse hyperbolic cosine of *x* |
| [`asinh(x)`](https://docs.python.org/3/library/math.html#math.asinh "math.asinh") | Inverse hyperbolic sine of *x* |
| [`atanh(x)`](https://docs.python.org/3/library/math.html#math.atanh "math.atanh") | Inverse hyperbolic tangent of *x* |
| [`cosh(x)`](https://docs.python.org/3/library/math.html#math.cosh "math.cosh") | Hyperbolic cosine of *x* |
| [`sinh(x)`](https://docs.python.org/3/library/math.html#math.sinh "math.sinh") | Hyperbolic sine of *x* |
| [`tanh(x)`](https://docs.python.org/3/library/math.html#math.tanh "math.tanh") | Hyperbolic tangent of *x* |
| **Special functions** | |
| [`erf(x)`](https://docs.python.org/3/library/math.html#math.erf "math.erf") | [Error function](https://en.wikipedia.org/wiki/Error_function) at *x* |
| [`erfc(x)`](https://docs.python.org/3/library/math.html#math.erfc "math.erfc") | [Complementary error function](https://en.wikipedia.org/wiki/Error_function) at *x* |
| [`gamma(x)`](https://docs.python.org/3/library/math.html#math.gamma "math.gamma") | [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) at *x* |
| [`lgamma(x)`](https://docs.python.org/3/library/math.html#math.lgamma "math.lgamma") | Natural logarithm of the absolute value of the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) at *x* |
| **Constants** | |
| [`pi`](https://docs.python.org/3/library/math.html#math.pi "math.pi") | *Ļ* = 3.141592ā¦ |
| [`e`](https://docs.python.org/3/library/math.html#math.e "math.e") | *e* = 2.718281ā¦ |
| [`tau`](https://docs.python.org/3/library/math.html#math.tau "math.tau") | *Ļ* = 2*Ļ* = 6.283185ā¦ |
| [`inf`](https://docs.python.org/3/library/math.html#math.inf "math.inf") | Positive infinity |
| [`nan`](https://docs.python.org/3/library/math.html#math.nan "math.nan") | āNot a numberā (NaN) |
## Number-theoretic functions[Ā¶](https://docs.python.org/3/library/math.html#number-theoretic-functions "Link to this heading")
math.comb(*n*, *k*)[Ā¶](https://docs.python.org/3/library/math.html#math.comb "Link to this definition")
Return the number of ways to choose *k* items from *n* items without repetition and without order.
Evaluates to `n! / (k! * (n - k)!)` when `k <= n` and evaluates to zero when `k > n`.
Also called the binomial coefficient because it is equivalent to the coefficient of k-th term in polynomial expansion of `(1 + x)āæ`.
Raises [`TypeError`](https://docs.python.org/3/library/exceptions.html#TypeError "TypeError") if either of the arguments are not integers. Raises [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") if either of the arguments are negative.
Added in version 3.8.
math.factorial(*n*)[Ā¶](https://docs.python.org/3/library/math.html#math.factorial "Link to this definition")
Return factorial of the nonnegative integer *n*.
Changed in version 3.10: Floats with integral values (like `5.0`) are no longer accepted.
math.gcd(*\*integers*)[Ā¶](https://docs.python.org/3/library/math.html#math.gcd "Link to this definition")
Return the greatest common divisor of the specified integer arguments. If any of the arguments is nonzero, then the returned value is the largest positive integer that is a divisor of all arguments. If all arguments are zero, then the returned value is `0`. `gcd()` without arguments returns `0`.
Added in version 3.5.
Changed in version 3.9: Added support for an arbitrary number of arguments. Formerly, only two arguments were supported.
math.isqrt(*n*)[Ā¶](https://docs.python.org/3/library/math.html#math.isqrt "Link to this definition")
Return the integer square root of the nonnegative integer *n*. This is the floor of the exact square root of *n*, or equivalently the greatest integer *a* such that *a*Ā² ā¤ *n*.
For some applications, it may be more convenient to have the least integer *a* such that *n* ā¤ *a*Ā², or in other words the ceiling of the exact square root of *n*. For positive *n*, this can be computed using `a = 1 + isqrt(n - 1)`.
Added in version 3.8.
math.lcm(*\*integers*)[Ā¶](https://docs.python.org/3/library/math.html#math.lcm "Link to this definition")
Return the least common multiple of the specified integer arguments. If all arguments are nonzero, then the returned value is the smallest positive integer that is a multiple of all arguments. If any of the arguments is zero, then the returned value is `0`. `lcm()` without arguments returns `1`.
Added in version 3.9.
math.perm(*n*, *k\=None*)[Ā¶](https://docs.python.org/3/library/math.html#math.perm "Link to this definition")
Return the number of ways to choose *k* items from *n* items without repetition and with order.
Evaluates to `n! / (n - k)!` when `k <= n` and evaluates to zero when `k > n`.
If *k* is not specified or is `None`, then *k* defaults to *n* and the function returns `n!`.
Raises [`TypeError`](https://docs.python.org/3/library/exceptions.html#TypeError "TypeError") if either of the arguments are not integers. Raises [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") if either of the arguments are negative.
Added in version 3.8.
## Floating point arithmetic[Ā¶](https://docs.python.org/3/library/math.html#floating-point-arithmetic "Link to this heading")
math.ceil(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.ceil "Link to this definition")
Return the ceiling of *x*, the smallest integer greater than or equal to *x*. If *x* is not a float, delegates to [`x.__ceil__`](https://docs.python.org/3/reference/datamodel.html#object.__ceil__ "object.__ceil__"), which should return an [`Integral`](https://docs.python.org/3/library/numbers.html#numbers.Integral "numbers.Integral") value.
math.fabs(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.fabs "Link to this definition")
Return the absolute value of *x*.
math.floor(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.floor "Link to this definition")
Return the floor of *x*, the largest integer less than or equal to *x*. If *x* is not a float, delegates to [`x.__floor__`](https://docs.python.org/3/reference/datamodel.html#object.__floor__ "object.__floor__"), which should return an [`Integral`](https://docs.python.org/3/library/numbers.html#numbers.Integral "numbers.Integral") value.
math.fma(*x*, *y*, *z*)[Ā¶](https://docs.python.org/3/library/math.html#math.fma "Link to this definition")
Fused multiply-add operation. Return `(x * y) + z`, computed as though with infinite precision and range followed by a single round to the `float` format. This operation often provides better accuracy than the direct expression `(x * y) + z`.
This function follows the specification of the fusedMultiplyAdd operation described in the IEEE 754 standard. The standard leaves one case implementation-defined, namely the result of `fma(0, inf, nan)` and `fma(inf, 0, nan)`. In these cases, `math.fma` returns a NaN, and does not raise any exception.
Added in version 3.13.
math.fmod(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.fmod "Link to this definition")
Return the floating-point remainder of `x / y`, as defined by the platform C library function `fmod(x, y)`. 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)`. Pythonās `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 Pythonās `-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 Pythonās `x % y` is preferred when working with integers.
math.modf(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.modf "Link to this definition")
Return the fractional and integer parts of *x*. Both results carry the sign of *x* and are floats.
Note that `modf()` has a different call/return pattern than its C equivalents: it takes a single argument and return a pair of values, rather than returning its second return value through an āoutput parameterā (there is no such thing in Python).
math.remainder(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.remainder "Link to this definition")
Return the IEEE 754-style remainder of *x* with respect to *y*. For finite *x* and finite nonzero *y*, this is the difference `x - n*y`, where `n` is the closest integer to the exact value of the quotient . If `x / y` is exactly halfway between two consecutive integers, the nearest *even* integer is used for `n`. The remainder thus always satisfies `abs(r) <= 0.5 * abs(y)`.
Special cases follow IEEE 754: in particular, `remainder(x, math.inf)` is *x* for any finite *x*, and `remainder(x, 0)` and `remainder(math.inf, x)` raise [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") for any non-NaN *x*. If the result of the remainder operation is zero, that zero will have the same sign as *x*.
On platforms using IEEE 754 binary floating point, the result of this operation is always exactly representable: no rounding error is introduced.
Added in version 3.7.
math.trunc(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.trunc "Link to this definition")
Return *x* with the fractional part removed, leaving the integer part. This rounds toward 0: `trunc()` is equivalent to [`floor()`](https://docs.python.org/3/library/math.html#math.floor "math.floor") for positive *x*, and equivalent to [`ceil()`](https://docs.python.org/3/library/math.html#math.ceil "math.ceil") for negative *x*. If *x* is not a float, delegates to [`x.__trunc__`](https://docs.python.org/3/reference/datamodel.html#object.__trunc__ "object.__trunc__"), which should return an [`Integral`](https://docs.python.org/3/library/numbers.html#numbers.Integral "numbers.Integral") value.
For the [`ceil()`](https://docs.python.org/3/library/math.html#math.ceil "math.ceil"), [`floor()`](https://docs.python.org/3/library/math.html#math.floor "math.floor"), and [`modf()`](https://docs.python.org/3/library/math.html#math.modf "math.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.
## Floating point manipulation functions[Ā¶](https://docs.python.org/3/library/math.html#floating-point-manipulation-functions "Link to this heading")
math.copysign(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.copysign "Link to this definition")
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.frexp(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.frexp "Link to this definition")
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 āpick apartā the internal representation of a float in a portable way.
Note that `frexp()` has a different call/return pattern than its C equivalents: it takes a single argument and return a pair of values, rather than returning its second return value through an āoutput parameterā (there is no such thing in Python).
math.isclose(*a*, *b*, *\**, *rel\_tol\=1e-09*, *abs\_tol\=0\.0*)[Ā¶](https://docs.python.org/3/library/math.html#math.isclose "Link to this definition")
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. If no errors occur, the result will be: `abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)`.
*rel\_tol* is the relative tolerance ā 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 nonnegative and less than `1.0`.
*abs\_tol* is the absolute tolerance; it defaults to `0.0` and it must be nonnegative. When comparing `x` to `0.0`, `isclose(x, 0)` is computed as `abs(x) <= rel_tolĀ * abs(x)`, which is `False` for any nonzero `x` and *rel\_tol* less than `1.0`. So add an appropriate positive *abs\_tol* argument to the call.
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.
Added in version 3.5.
See also
[**PEP 485**](https://peps.python.org/pep-0485/) ā A function for testing approximate equality
math.isfinite(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.isfinite "Link to this definition")
Return `True` if *x* is neither an infinity nor a NaN, and `False` otherwise. (Note that `0.0` *is* considered finite.)
Added in version 3.2.
math.isinf(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.isinf "Link to this definition")
Return `True` if *x* is a positive or negative infinity, and `False` otherwise.
math.isnan(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.isnan "Link to this definition")
Return `True` if *x* is a NaN (not a number), and `False` otherwise.
math.ldexp(*x*, *i*)[Ā¶](https://docs.python.org/3/library/math.html#math.ldexp "Link to this definition")
Return `x * (2**i)`. This is essentially the inverse of function [`frexp()`](https://docs.python.org/3/library/math.html#math.frexp "math.frexp").
math.nextafter(*x*, *y*, *steps\=1*)[Ā¶](https://docs.python.org/3/library/math.html#math.nextafter "Link to this definition")
Return the floating-point value *steps* steps after *x* towards *y*.
If *x* is equal to *y*, return *y*, unless *steps* is zero.
Examples:
- `math.nextafter(x, math.inf)` goes up: towards positive infinity.
- `math.nextafter(x, -math.inf)` goes down: towards minus infinity.
- `math.nextafter(x, 0.0)` goes towards zero.
- `math.nextafter(x, math.copysign(math.inf, x))` goes away from zero.
See also [`math.ulp()`](https://docs.python.org/3/library/math.html#math.ulp "math.ulp").
Added in version 3.9.
Changed in version 3.12: Added the *steps* argument.
math.ulp(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.ulp "Link to this definition")
Return the value of the least significant bit of the float *x*:
- If *x* is a NaN (not a number), return *x*.
- If *x* is negative, return `ulp(-x)`.
- If *x* is a positive infinity, return *x*.
- If *x* is equal to zero, return the smallest positive *denormalized* representable float (smaller than the minimum positive *normalized* float, [`sys.float_info.min`](https://docs.python.org/3/library/sys.html#sys.float_info "sys.float_info")).
- If *x* is equal to the largest positive representable float, return the value of the least significant bit of *x*, such that the first float smaller than *x* is `x - ulp(x)`.
- Otherwise (*x* is a positive finite number), return the value of the least significant bit of *x*, such that the first float bigger than *x* is `x + ulp(x)`.
ULP stands for āUnit in the Last Placeā.
See also [`math.nextafter()`](https://docs.python.org/3/library/math.html#math.nextafter "math.nextafter") and [`sys.float_info.epsilon`](https://docs.python.org/3/library/sys.html#sys.float_info "sys.float_info").
Added in version 3.9.
## Power, exponential and logarithmic functions[Ā¶](https://docs.python.org/3/library/math.html#power-exponential-and-logarithmic-functions "Link to this heading")
math.cbrt(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.cbrt "Link to this definition")
Return the cube root of *x*.
Added in version 3.11.
math.exp(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.exp "Link to this definition")
Return *e* raised to the power *x*, where *e* = 2.718281ā¦ is the base of natural logarithms. This is usually more accurate than `math.e ** x` or `pow(math.e, x)`.
math.exp2(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.exp2 "Link to this definition")
Return *2* raised to the power *x*.
Added in version 3.11.
math.expm1(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.expm1 "Link to this definition")
Return *e* raised to the power *x*, minus 1. Here *e* is the base of natural logarithms. For small floats *x*, the subtraction in `exp(x) - 1` can result in a [significant loss of precision](https://en.wikipedia.org/wiki/Loss_of_significance); 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
```
Added in version 3.2.
math.log(*x*\[, *base*\])[Ā¶](https://docs.python.org/3/library/math.html#math.log "Link to this definition")
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*)[Ā¶](https://docs.python.org/3/library/math.html#math.log1p "Link to this definition")
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*)[Ā¶](https://docs.python.org/3/library/math.html#math.log2 "Link to this definition")
Return the base-2 logarithm of *x*. This is usually more accurate than `log(x, 2)`.
Added in version 3.3.
See also
[`int.bit_length()`](https://docs.python.org/3/library/stdtypes.html#int.bit_length "int.bit_length") returns the number of bits necessary to represent an integer in binary, excluding the sign and leading zeros.
math.log10(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.log10 "Link to this definition")
Return the base-10 logarithm of *x*. This is usually more accurate than `log(x, 10)`.
math.pow(*x*, *y*)[Ā¶](https://docs.python.org/3/library/math.html#math.pow "Link to this definition")
Return *x* raised to the power *y*. Exceptional cases follow the IEEE 754 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`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError").
Unlike the built-in `**` operator, [`math.pow()`](https://docs.python.org/3/library/math.html#math.pow "math.pow") converts both its arguments to type [`float`](https://docs.python.org/3/library/functions.html#float "float"). Use `**` or the built-in `pow()` function for computing exact integer powers.
Changed in version 3.11: The special cases `pow(0.0, -inf)` and `pow(-0.0, -inf)` were changed to return `inf` instead of raising [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError"), for consistency with IEEE 754.
math.sqrt(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.sqrt "Link to this definition")
Return the square root of *x*.
## Summation and product functions[Ā¶](https://docs.python.org/3/library/math.html#summation-and-product-functions "Link to this heading")
math.dist(*p*, *q*)[Ā¶](https://docs.python.org/3/library/math.html#math.dist "Link to this definition")
Return the Euclidean distance between two points *p* and *q*, each given as a sequence (or iterable) of coordinates. The two points must have the same dimension.
Roughly equivalent to:
```
sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))
```
Added in version 3.8.
math.fsum(*iterable*)[Ā¶](https://docs.python.org/3/library/math.html#math.fsum "Link to this definition")
Return an accurate floating-point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums.
The algorithmās 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](https://code.activestate.com/recipes/393090-binary-floating-point-summation-accurate-to-full-p/).
math.hypot(*\*coordinates*)[Ā¶](https://docs.python.org/3/library/math.html#math.hypot "Link to this definition")
Return the Euclidean norm, `sqrt(sum(x**2 for x in coordinates))`. This is the length of the vector from the origin to the point given by the coordinates.
For a two dimensional point `(x, y)`, this is equivalent to computing the hypotenuse of a right triangle using the Pythagorean theorem, `sqrt(x*x + y*y)`.
Changed in version 3.8: Added support for n-dimensional points. Formerly, only the two dimensional case was supported.
Changed in version 3.10: Improved the algorithmās accuracy so that the maximum error is under 1 ulp (unit in the last place). More typically, the result is almost always correctly rounded to within 1/2 ulp.
math.prod(*iterable*, *\**, *start\=1*)[Ā¶](https://docs.python.org/3/library/math.html#math.prod "Link to this definition")
Calculate the product of all the elements in the input *iterable*. The default *start* value for the product is `1`.
When the iterable is empty, return the start value. This function is intended specifically for use with numeric values and may reject non-numeric types.
Added in version 3.8.
math.sumprod(*p*, *q*)[Ā¶](https://docs.python.org/3/library/math.html#math.sumprod "Link to this definition")
Return the sum of products of values from two iterables *p* and *q*.
Raises [`ValueError`](https://docs.python.org/3/library/exceptions.html#ValueError "ValueError") if the inputs do not have the same length.
Roughly equivalent to:
```
sum(map(operator.mul, p, q, strict=True))
```
For float and mixed int/float inputs, the intermediate products and sums are computed with extended precision.
Added in version 3.12.
## Angular conversion[Ā¶](https://docs.python.org/3/library/math.html#angular-conversion "Link to this heading")
math.degrees(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.degrees "Link to this definition")
Convert angle *x* from radians to degrees.
math.radians(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.radians "Link to this definition")
Convert angle *x* from degrees to radians.
## Trigonometric functions[Ā¶](https://docs.python.org/3/library/math.html#trigonometric-functions "Link to this heading")
math.acos(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.acos "Link to this definition")
Return the arc cosine of *x*, in radians. The result is between `0` and `pi`.
math.asin(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.asin "Link to this definition")
Return the arc sine of *x*, in radians. The result is between `-pi/2` and `pi/2`.
math.atan(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.atan "Link to this definition")
Return the arc tangent of *x*, in radians. The result is between `-pi/2` and `pi/2`.
math.atan2(*y*, *x*)[Ā¶](https://docs.python.org/3/library/math.html#math.atan2 "Link to this definition")
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 is `-3*pi/4`.
math.cos(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.cos "Link to this definition")
Return the cosine of *x* radians.
math.sin(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.sin "Link to this definition")
Return the sine of *x* radians.
math.tan(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.tan "Link to this definition")
Return the tangent of *x* radians.
## Hyperbolic functions[Ā¶](https://docs.python.org/3/library/math.html#hyperbolic-functions "Link to this heading")
[Hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_functions) are analogs of trigonometric functions that are based on hyperbolas instead of circles.
math.acosh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.acosh "Link to this definition")
Return the inverse hyperbolic cosine of *x*.
math.asinh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.asinh "Link to this definition")
Return the inverse hyperbolic sine of *x*.
math.atanh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.atanh "Link to this definition")
Return the inverse hyperbolic tangent of *x*.
math.cosh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.cosh "Link to this definition")
Return the hyperbolic cosine of *x*.
math.sinh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.sinh "Link to this definition")
Return the hyperbolic sine of *x*.
math.tanh(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.tanh "Link to this definition")
Return the hyperbolic tangent of *x*.
## Special functions[Ā¶](https://docs.python.org/3/library/math.html#special-functions "Link to this heading")
math.erf(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.erf "Link to this definition")
Return the [error function](https://en.wikipedia.org/wiki/Error_function) at *x*.
The `erf()` function can be used to compute traditional statistical functions such as the [cumulative standard normal distribution](https://en.wikipedia.org/wiki/Cumulative_distribution_function):
```
def phi(x):
'Cumulative distribution function for the standard normal distribution'
return (1.0 + erf(x / sqrt(2.0))) / 2.0
```
Added in version 3.2.
math.erfc(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.erfc "Link to this definition")
Return the complementary error function at *x*. The [complementary error function](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Loss_of_significance).
Added in version 3.2.
math.gamma(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.gamma "Link to this definition")
Return the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) at *x*.
Added in version 3.2.
math.lgamma(*x*)[Ā¶](https://docs.python.org/3/library/math.html#math.lgamma "Link to this definition")
Return the natural logarithm of the absolute value of the Gamma function at *x*.
Added in version 3.2.
## Constants[Ā¶](https://docs.python.org/3/library/math.html#constants "Link to this heading")
math.pi[Ā¶](https://docs.python.org/3/library/math.html#math.pi "Link to this definition")
The mathematical constant *Ļ* = 3.141592ā¦, to available precision.
math.e[Ā¶](https://docs.python.org/3/library/math.html#math.e "Link to this definition")
The mathematical constant *e* = 2.718281ā¦, to available precision.
math.tau[Ā¶](https://docs.python.org/3/library/math.html#math.tau "Link to this definition")
The mathematical constant *Ļ* = 6.283185ā¦, to available precision. Tau is a circle constant equal to 2*Ļ*, the ratio of a circleās circumference to its radius. To learn more about Tau, check out Vi Hartās video [Pi is (still) Wrong](https://vimeo.com/147792667), and start celebrating [Tau day](https://tauday.com/) by eating twice as much pie\!
Added in version 3.6.
math.inf[Ā¶](https://docs.python.org/3/library/math.html#math.inf "Link to this definition")
A floating-point positive infinity. (For negative infinity, use `-math.inf`.) Equivalent to the output of `float('inf')`.
Added in version 3.5.
math.nan[Ā¶](https://docs.python.org/3/library/math.html#math.nan "Link to this definition")
A floating-point ānot a numberā (NaN) value. Equivalent to the output of `float('nan')`. Due to the requirements of the [IEEE-754 standard](https://en.wikipedia.org/wiki/IEEE_754), `math.nan` and `float('nan')` are not considered to equal to any other numeric value, including themselves. To check whether a number is a NaN, use the [`isnan()`](https://docs.python.org/3/library/math.html#math.isnan "math.isnan") function to test for NaNs instead of `is` or `==`. Example:
```
>>> import math
>>> math.nan == math.nan
False
>>> float('nan') == float('nan')
False
>>> math.isnan(math.nan)
True
>>> math.isnan(float('nan'))
True
```
Added in version 3.5.
Changed in version 3.11: It is now always available.
**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`](https://docs.python.org/3/library/exceptions.html#ValueError "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`](https://docs.python.org/3/library/exceptions.html#OverflowError "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
Module [`cmath`](https://docs.python.org/3/library/cmath.html#module-cmath "cmath: Mathematical functions for complex numbers.")
Complex number versions of many of these functions. |
| Shard | 16 (laksa) |
| Root Hash | 10954876678907435016 |
| Unparsed URL | org,python!docs,/3/library/math.html s443 |