This module is always available. It provides access to the mathematicalfunctions defined by the C standard.
These functions cannot be used with complex numbers; use the functions of thesame name from the cmath module if you require support for complexnumbers. The distinction between functions which support complex numbers andthose which don’t is made since most users do not want to learn quite as muchmathematics as required to understand complex numbers. Receiving an exceptioninstead of a complex result allows earlier detection of the unexpected complexnumber used as a parameter, so that the programmer can determine how and why itwas generated in the first place.
The following functions are provided by this module. Except when explicitlynoted otherwise, all return values are floats.
9.2.1. Number-theoretic and representation functions¶
math.
ceil
(x)¶Return the ceiling of x, the smallest integer greater than or equal to x.If x is not a float, delegates to
x.__ceil__()
, which should return anIntegral value.
math.
copysign
(x, y)¶Return a float with the magnitude (absolute value) of x but the sign ofy. On platforms that support signed zeros,
copysign(1.0, -0.0)
returns -1.0.
math.
fabs
(x)¶Return the absolute value of x.
math.
factorial
(x)¶Return x factorial. Raises ValueError if x is not integral oris negative.
math.
floor
(x)¶Return the floor of x, the largest integer less than or equal to x.If x is not a float, delegates to
x.__floor__()
, which should return anIntegral value.
math.
fmod
(x, y)¶Return
fmod(x, y)
, as defined by the platform C library. Note that thePython expressionx % y
may not return the same result. The intent of the Cstandard is thatfmod(x, y)
be exactly (mathematically; to infiniteprecision) equal tox - n*y
for some integer n such that the result hasthe same sign as x and magnitude less thanabs(y)
. Python’sx % y
returns a result with the sign of y instead, and may not be exactly computablefor float arguments. For example,fmod(-1e-100, 1e100)
is-1e-100
, butthe result of Python’s-1e-100 % 1e100
is1e100-1e-100
, which cannot berepresented exactly as a float, and rounds to the surprising1e100
. Forthis reason, function fmod() is generally preferred when working withfloats, while Python’sx % 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 floatand e is an integer such thatx == m * 2**e
exactly. If x is zero,returns(0.0, 0)
, otherwise0.5 <= abs(m) < 1
. This is used to “pickapart” the internal representation of a float in a portable way.
math.
fsum
(iterable)¶Return an accurate floating point sum of values in the iterable. Avoidsloss 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 thetypical case where the rounding mode is half-even. On some non-Windowsbuilds, the underlying C library uses extended precision addition and mayoccasionally double-round an intermediate sum causing it to be off in itsleast significant bit.
For further discussion and two alternative approaches, see the ASPN cookbookrecipes for accurate floating point summation.
math.
gcd
(a, b)¶Return the greatest common divisor of the integers a and b. If eithera or b is nonzero, then the value of
gcd(a, b)
is the largestpositive integer that divides both a and b.gcd(0, 0)
returns0
.New in version 3.5.
math.
isclose
(a, b, *, rel_tol=1e-09, abs_tol=0.0)¶Return
True
if the values a and b are close to each other andFalse
otherwise.Whether or not two values are considered close is determined according togiven absolute and relative tolerances.
rel_tol is the relative tolerance – it is the maximum allowed differencebetween 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 defaulttolerance is1e-09
, which assures that the two values are the samewithin about 9 decimal digits. rel_tol must be greater than zero.abs_tol is the minimum absolute tolerance – useful for comparisons nearzero. abs_tol must be at least zero.
If no errors occur, the result will be:
abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)
.The IEEE 754 special values of
NaN
,inf
, and-inf
will behandled according to IEEE rules. Specifically,NaN
is not consideredclose to any other value, includingNaN
.inf
and-inf
are onlyconsidered close to themselves.See Alsomath — Mathematical functionsThe Python math Module: Everything You Need to Know – Real PythonPython Math Module - GeeksforGeeksThe Python Math LibraryNew in version 3.5.
See also
PEP 485 – A function for testing approximate equality
math.
isfinite
(x)¶Return
True
if x is neither an infinity nor a NaN, andFalse
otherwise. (Note that0.0
is considered finite.)New in version 3.2.
math.
isinf
(x)¶Return
True
if x is a positive or negative infinity, andFalse
otherwise.
math.
isnan
(x)¶Return
True
if x is a NaN (not a number), andFalse
otherwise.
math.
ldexp
(x, i)¶Return
x * (2**i)
. This is essentially the inverse of functionfrexp().
math.
modf
(x)¶Return the fractional and integer parts of x. Both results carry the signof x and are floats.
math.
trunc
(x)¶Return the Real value x truncated to anIntegral (usually an integer). Delegates to
x.__trunc__()
.
Note that frexp() and modf() have a different call/return patternthan their C equivalents: they take a single argument and return a pair ofvalues, rather than returning their second return value through an ‘outputparameter’ (there is no such thing in Python).
For the ceil(), floor(), and modf() functions, note that allfloating-point numbers of sufficiently large magnitude are exact integers.Python floats typically carry no more than 53 bits of precision (the same as theplatform C double type), in which case any float x with abs(x) >= 2**52
necessarily has no fractional bits.
9.2.2. Power and logarithmic functions¶
math.
exp
(x)¶Return
e**x
.
math.
expm1
(x)¶Return
e**x - 1
. For small floats x, the subtraction inexp(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 places1.0000050000069649e-05>>> expm1(1e-5) # result accurate to full precision1.0000050000166668e-05
New in version 3.2.
math.
log
(x[, base])¶With one argument, return the natural logarithm of x (to base e).
With two arguments, return the logarithm of x to the given base,calculated as
log(x)/log(base)
.
math.
log1p
(x)¶Return the natural logarithm of 1+x (base e). Theresult is calculated in a way which is accurate for x near zero.
math.
log2
(x)¶Return the base-2 logarithm of x. This is usually more accurate than
log(x, 2)
.New in version 3.3.
See also
int.bit_length() returns the number of bits necessary to representan integer in binary, excluding the sign and leading zeros.
math.
log10
(x)¶Return the base-10 logarithm of x. This is usually more accuratethan
log(x, 10)
.
math.
pow
(x, y)¶Return
x
raised to the powery
. Exceptional cases followAnnex ‘F’ of the C99 standard as far as possible. In particular,pow(1.0, x)
andpow(x, 0.0)
always return1.0
, evenwhenx
is a zero or a NaN. If bothx
andy
are finite,x
is negative, andy
is not an integer thenpow(x, y)
is undefined, and raises ValueError.Unlike the built-in
**
operator, math.pow() converts bothits arguments to type float. Use**
or the built-inpow() function for computing exact integer powers.
math.
sqrt
(x)¶Return the square root of x.
9.2.3. Trigonometric functions¶
math.
acos
(x)¶Return the arc cosine of x, in radians.
math.
asin
(x)¶Return the arc sine of x, in radians.
math.
atan
(x)¶Return the arc tangent of x, in radians.
math.
atan2
(y, x)¶Return
atan(y / x)
, in radians. The result is between-pi
andpi
.The vector in the plane from the origin to point(x, y)
makes this anglewith the positive X axis. The point of atan2() is that the signs of bothinputs are known to it, so it can compute the correct quadrant for the angle.For example,atan(1)
andatan2(1, 1)
are bothpi/4
, butatan2(-1,-1)
is-3*pi/4
.
math.
cos
(x)¶Return the cosine of x radians.
math.
hypot
(x, y)¶Return the Euclidean norm,
sqrt(x*x + y*y)
. This is the length of the vectorfrom the origin to point(x, y)
.
math.
sin
(x)¶Return the sine of x radians.
math.
tan
(x)¶Return the tangent of x radians.
9.2.4. Angular conversion¶
math.
degrees
(x)¶Convert angle x from radians to degrees.
math.
radians
(x)¶Convert angle x from degrees to radians.
9.2.5. Hyperbolic functions¶
Hyperbolic functionsare analogs of trigonometric functions that are based on hyperbolasinstead 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.
9.2.6. Special functions¶
math.
erf
(x)¶Return the error function atx.
The erf() function can be used to compute traditional statisticalfunctions such as the cumulative standard normal distribution:
def phi(x): 'Cumulative distribution function for the standard normal distribution' return (1.0 + erf(x / sqrt(2.0))) / 2.0
New in version 3.2.
math.
erfc
(x)¶Return the complementary error function at x. The complementary errorfunction is defined as
1.0 - erf(x)
. It is used for large values of x where a subtractionfrom one would cause a loss of significance.New in version 3.2.
math.
gamma
(x)¶Return the Gamma function atx.
New in version 3.2.
math.
lgamma
(x)¶Return the natural logarithm of the absolute value of the Gammafunction at x.
New in version 3.2.
9.2.7. Constants¶
math.
pi
¶The mathematical constant π = 3.141592..., to available precision.
math.
e
¶The mathematical constant e = 2.718281..., to available precision.
math.
tau
¶The mathematical constant τ = 6.283185..., to available precision.Tau is a circle constant equal to 2π, the ratio of a circle’s circumference toits radius. To learn more about Tau, check out Vi Hart’s video Pi is (still)Wrong, and start celebratingTau day by eating twice as much pie!
New in version 3.6.
math.
inf
¶A floating-point positive infinity. (For negative infinity, use
-math.inf
.) Equivalent to the output offloat('inf')
.New in version 3.5.
math.
nan
¶A floating-point “not a number” (NaN) value. Equivalent to the output of
float('nan')
.New in version 3.5.
CPython implementation detail: The math module consists mostly of thin wrappers around the platform Cmath library functions. Behavior in exceptional cases follows Annex F ofthe C99 standard where appropriate. The current implementation will raiseValueError 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 functionsabove 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) thereare some exceptions to this rule, for example pow(float('nan'), 0.0)
orhypot(float('nan'), float('inf'))
.
Note that Python makes no effort to distinguish signaling NaNs fromquiet 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.
Table Of Contents
- 9.2. math — Mathematical functions
- 9.2.1. Number-theoretic and representation functions
- 9.2.2. Power and logarithmic functions
- 9.2.3. Trigonometric functions
- 9.2.4. Angular conversion
- 9.2.5. Hyperbolic functions
- 9.2.6. Special functions
- 9.2.7. Constants
Previous topic
9.1. numbers — Numeric abstract base classes
Next topic
9.3. cmath — Mathematical functions for complex numbers
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