math
— 数学函数
¶
此模块提供由 C 标准定义的数学函数的访问。
These functions cannot be used with complex numbers; use the functions of the same name from the
cmath
module if you require support for complex numbers. The distinction between functions which support complex numbers and those which don’t is made since most users do not want to learn quite as much mathematics as required to understand complex numbers. Receiving an exception instead of a complex result allows earlier detection of the unexpected complex number used as a parameter, so that the programmer can determine how and why it was generated in the first place.
The following functions are provided by this module. Except when explicitly noted otherwise, all return values are floats.
math.
ceil
(
x
)
¶
Return the ceiling of
x
, the smallest integer greater than or equal to
x
。若
x
is not a float, delegates to
x.__ceil__()
, which should return an
Integral
值。
math.
comb
(
n
,
k
)
¶
Return the number of ways to choose k items from n items without repetition and without order.
评估为
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 the expression
(1 + x) ** n
.
引发
TypeError
if either of the arguments are not integers. Raises
ValueError
if either of the arguments are negative.
3.8 版新增。
math.
copysign
(
x
,
y
)
¶
Return a float with the magnitude (absolute value) of
x
but the sign of
y
. On platforms that support signed zeros,
copysign(1.0, -0.0)
返回
-1.0
.
math.
fabs
(
x
)
¶
返回绝对值的 x .
math.
factorial
(
x
)
¶
返回
x
factorial as an integer. Raises
ValueError
if
x
不是整型或为负。
从 3.9 版起弃用：
Accepting floats with integral values (like
5.0
) is deprecated.
math.
floor
(
x
)
¶
Return the floor of
x
, the largest integer less than or equal to
x
。若
x
is not a float, delegates to
x.__floor__()
, which should return an
Integral
值。
math.
fmod
(
x
,
y
)
¶
返回
fmod(x, y)
, as defined by the platform C library. Note that the Python expression
x % y
may not return the same result. The intent of the C standard is that
fmod(x, y)
be exactly (mathematically; to infinite precision) equal to
x - n*y
for some integer
n
such that the result has the same sign as
x
and magnitude less than
abs(y)
. Python’s
x % y
returns a result with the sign of
y
instead, and may not be exactly computable for float arguments. For example,
fmod(-1e-100, 1e100)
is
-1e-100
, but the result of Python’s
-1e-100 % 1e100
is
1e100-1e-100
, which cannot be represented exactly as a float, and rounds to the surprising
1e100
. For this reason, function
fmod()
is generally preferred when working with floats, while Python’s
x % y
is preferred when working with integers.
math.
frexp
(
x
)
¶
Return the mantissa and exponent of
x
as the pair
(m, e)
.
m
is a float and
e
is an integer such that
x == m * 2**e
exactly. If
x
is zero, returns
(0.0, 0)
, otherwise
0.5 <= abs(m) < 1
. This is used to “pick apart” the internal representation of a float in a portable way.
math.
fsum
(
iterable
)
¶
Return an accurate floating point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums:
>>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
0.9999999999999999
>>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
1.0
The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the typical case where the rounding mode is half-even. On some non-Windows builds, the underlying C library uses extended precision addition and may occasionally double-round an intermediate sum causing it to be off in its least significant bit.
For further discussion and two alternative approaches, see the ASPN cookbook recipes for accurate floating point summation .
math.
gcd
(
*integers
)
¶
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
.
3.5 版新增。
3.9 版改变： Added support for an arbitrary number of arguments. Formerly, only two arguments were supported.
math.
isclose
(
a
,
b
,
*
,
rel_tol=1e-09
,
abs_tol=0.0
)
¶
返回
True
if the values
a
and
b
are close to each other and
False
否则。
Whether or not two values are considered close is determined according to given absolute and relative tolerances.
rel_tol
is the relative tolerance – it is the maximum allowed difference between
a
and
b
, relative to the larger absolute value of
a
or
b
. For example, to set a tolerance of 5%, pass
rel_tol=0.05
. The default tolerance is
1e-09
, which assures that the two values are the same within about 9 decimal digits.
rel_tol
must be greater than zero.
abs_tol is the minimum absolute tolerance – useful for comparisons near zero. abs_tol must be at least zero.
如果不发生错误，结果将是：
abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)
.
The IEEE 754 special values of
NaN
,
inf
，和
-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.
3.5 版新增。
另请参阅
PEP 485 – 用于测试近似相等的函数
math.
isfinite
(
x
)
¶
返回
True
if
x
is neither an infinity nor a NaN, and
False
otherwise. (Note that
0.0
is
considered finite.)
3.2 版新增。
math.
isinf
(
x
)
¶
返回
True
if
x
is a positive or negative infinity, and
False
否则。
math.
isnan
(
x
)
¶
返回
True
if
x
为 NaN (非数字)，和
False
否则。
math.
isqrt
(
n
)
¶
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 这样 a ² ≤ n .
For some applications, it may be more convenient to have the least integer
a
这样
n
≤
a
², 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)
.
3.8 版新增。
math.
lcm
(
*integers
)
¶
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
.
3.9 版新增。
math.
modf
(
x
)
¶
Return the fractional and integer parts of x . Both results carry the sign of x and are floats.
math.
nextafter
(
x
,
y
)
¶
Return the next floating-point value after x towards y .
若 x 等于 y ，返回 y .
范例：
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.
另请参阅
math.ulp()
.
3.9 版新增。
math.
perm
(
n
,
k=None
)
¶
Return the number of ways to choose k items from n items without repetition and with order.
评估为
n! / (n - k)!
when
k <= n
and evaluates to zero when
k > n
.
若
k
is not specified or is None, then
k
默认为
n
and the function returns
n!
.
引发
TypeError
if either of the arguments are not integers. Raises
ValueError
if either of the arguments are negative.
3.8 版新增。
math.
prod
(
iterable
,
*
,
start=1
)
¶
Calculate the product of all the elements in the input
iterable
。默认
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.
3.8 版新增。
math.
remainder
(
x
,
y
)
¶
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
，其中
n
is the closest integer to the exact value of the quotient
x /
y
。若
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
，和
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.
3.7 版新增。
math.
trunc
(
x
)
¶
返回
Real
value
x
truncated to an
Integral
(usually an integer). Delegates to
x.__trunc__()
.
math.
ulp
(
x
)
¶
Return the value of the least significant bit of the float x :
若 x is a NaN (not a number), return x .
若
x
为负，返回
ulp(-x)
.
若 x is a positive infinity, return x .
若
x
is equal to zero, return the smallest positive
denormalized
representable float (smaller than the minimum positive
normalized
float,
sys.float_info.min
).
若
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 “Unit in the Last Place”.
另请参阅
math.nextafter()
and
sys.float_info.epsilon
.
3.9 版新增。
注意，
frexp()
and
modf()
have a different call/return pattern than their C equivalents: they take a single argument and return a pair of values, rather than returning their second return value through an ‘output parameter’ (there is no such thing in Python).
对于
ceil()
,
floor()
，和
modf()
functions, note that
all
floating-point numbers of sufficiently large magnitude are exact integers. Python floats typically carry no more than 53 bits of precision (the same as the platform C double type), in which case any float
x
with
abs(x) >= 2**52
necessarily has no fractional bits.
math.
exp
(
x
)
¶
返回
e
raised to the power
x
，其中
e
= 2.718281… is the base of natural logarithms. This is usually more accurate than
math.e ** x
or
pow(math.e, x)
.
math.
expm1
(
x
)
¶
返回
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
3.2 版新增。
math.
log
(
x
[
,
base
]
)
¶
With one argument, return the natural logarithm of x (to base e ).
With two arguments, return the logarithm of
x
到给定
base
, calculated as
log(x)/log(base)
.
math.
log1p
(
x
)
¶
Return the natural logarithm of 1+x (base e ). The result is calculated in a way which is accurate for x near zero.
math.
log2
(
x
)
¶
Return the base-2 logarithm of
x
. This is usually more accurate than
log(x, 2)
.
3.3 版新增。
另请参阅
int.bit_length()
returns the number of bits necessary to represent an integer in binary, excluding the sign and leading zeros.
math.
log10
(
x
)
¶
Return the base-10 logarithm of
x
. This is usually more accurate than
log(x, 10)
.
math.
pow
(
x
,
y
)
¶
返回
x
raised to the power
y
. Exceptional cases follow Annex ‘F’ of the C99 standard as far as possible. In particular,
pow(1.0, x)
and
pow(x, 0.0)
always return
1.0
, even when
x
is a zero or a NaN. If both
x
and
y
are finite,
x
is negative, and
y
is not an integer then
pow(x, y)
is undefined, and raises
ValueError
.
Unlike the built-in
**
operator,
math.pow()
converts both its arguments to type
float
。使用
**
or the built-in
pow()
function for computing exact integer powers.
math.
sqrt
(
x
)
¶
Return the square root of x .
math.
acos
(
x
)
¶
Return the arc cosine of
x
, in radians. The result is between
0
and
pi
.
math.
asin
(
x
)
¶
Return the arc sine of
x
, in radians. The result is between
-pi/2
and
pi/2
.
math.
atan
(
x
)
¶
Return the arc tangent of
x
, in radians. The result is between
-pi/2
and
pi/2
.
math.
atan2
(
y
,
x
)
¶
返回
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
，但
atan2(-1,
-1)
is
-3*pi/4
.
math.
cos
(
x
)
¶
Return the cosine of x radians.
math.
dist
(
p
,
q
)
¶
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)))
3.8 版新增。
math.
hypot
(
*coordinates
)
¶
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)
.
3.8 版改变： Added support for n-dimensional points. Formerly, only the two dimensional case was supported.
math.
sin
(
x
)
¶
Return the sine of x radians.
math.
tan
(
x
)
¶
Return the tangent of x radians.
math.
degrees
(
x
)
¶
转换角度 x 从弧度到度。
math.
radians
(
x
)
¶
转换角度 x 从度到弧度。
双曲函数 are analogs of trigonometric functions that are based on hyperbolas instead of circles.
math.
acosh
(
x
)
¶
Return the inverse hyperbolic cosine of x .
math.
asinh
(
x
)
¶
Return the inverse hyperbolic sine of x .
math.
atanh
(
x
)
¶
Return the inverse hyperbolic tangent of x .
math.
cosh
(
x
)
¶
Return the hyperbolic cosine of x .
math.
sinh
(
x
)
¶
Return the hyperbolic sine of x .
math.
tanh
(
x
)
¶
Return the hyperbolic tangent of x .
math.
erf
(
x
)
¶
返回 误差函数 at x .
erf()
function can be used to compute traditional statistical functions such as the
累积标准正态分布
:
def phi(x):
'Cumulative distribution function for the standard normal distribution'
return (1.0 + erf(x / sqrt(2.0))) / 2.0
3.2 版新增。
math.
erfc
(
x
)
¶
Return the complementary error function at
x
。
互补误差函数
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
.
3.2 版新增。
math.
lgamma
(
x
)
¶
Return the natural logarithm of the absolute value of the Gamma function at x .
3.2 版新增。
math.
pi
¶
The mathematical constant π = 3.141592…, to available precision.
math.
e
¶
The mathematical constant e = 2.718281…, to available precision.
math.
tau
¶
The mathematical constant τ = 6.283185…, to available precision. Tau is a circle constant equal to 2 π , the ratio of a circle’s circumference to its radius. To learn more about Tau, check out Vi Hart’s video Pi is (still) Wrong , and start celebrating Tau day by eating twice as much pie!
3.6 版新增。
math.
inf
¶
A floating-point positive infinity. (For negative infinity, use
-math.inf
.) Equivalent to the output of
float('inf')
.
3.5 版新增。
math.
nan
¶
A floating-point “not a number” (NaN) value. Equivalent to the output of
float('nan')
.
3.5 版新增。
CPython 实现细节：
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.
另请参阅
cmath
很多这些函数的复数版本。