9.2. math — Mathematical functions — Python 3.6.3 documentation (2024)

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 expression x % y may not return the same result. The intent of the Cstandard is that fmod(x, y) be exactly (mathematically; to infiniteprecision) equal to x - n*y for some integer n such that the result hasthe same sign as x and magnitude less than abs(y). Python’s x % yreturns 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 is 1e100-1e-100, which cannot berepresented exactly as a float, and rounds to the surprising 1e100. Forthis reason, function fmod() is generally preferred when working withfloats, 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 floatand 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 “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 is 1e-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, including NaN. inf and -inf are onlyconsidered close to themselves.

See Also

math — Mathematical functionsThe Python math Module: Everything You Need to Know – Real PythonPython Math Module - GeeksforGeeksSimple Python Math Module Guide (22 Examples and 18 Functions)

New in version 3.5.

See also

PEP 485 – A function for testing approximate equality

math.isfinite(x)¶

Return True if x is neither an infinity nor a NaN, andFalse otherwise. (Note that 0.0 is considered finite.)

New in version 3.2.

math.isinf(x)¶

Return True if x is a positive or negative infinity, andFalse otherwise.

math.isnan(x)¶

Return True if x is a NaN (not a number), and False 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 tox.__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**52necessarily 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 in exp(x) - 1can 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 thanlog(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 power y. Exceptional cases followAnnex ‘F’ of the C99 standard as far as possible. In particular,pow(1.0, x) and pow(x, 0.0) always return 1.0, evenwhen 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 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 and pi.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) and atan2(1, 1) are both pi/4, but atan2(-1,-1) is -3*pi/4.

math.cos(x)¶

Return the cosine of x radians.

math.hypot(x, y)¶

Return the Euclidean norm, sqrt(x*x + y*y). This is the length of the 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 as1.0 - 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 of float('inf').

New in version 3.5.

math.nan¶

A floating-point “not a number” (NaN) value. Equivalent to the output offloat('nan').

New in version 3.5.