math — Mathematical functions (2024)

This module provides access to the mathematical functions defined by the Cstandard.

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.

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 an Integral value.

math.comb(n, k)

Return the number of ways to choose k items from n items without repetitionand without order.

Evaluates to n! / (k! * (n - k)!) when k <= n and evaluatesto zero when k > n.

Also called the binomial coefficient because it is equivalentto the coefficient of k-th term in polynomial expansion of(1 + x)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.

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(n)

Return n factorial as an integer. Raises ValueError if n is not integral oris negative.

Deprecated since version 3.9: 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. Ifx is not a float, delegates to x.__floor__, whichshould return an Integral 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.

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(*integers)

Return the greatest common divisor of the specified integer arguments.If any of the arguments is nonzero, then the returned value is the largestpositive integer that is a divisor of all arguments. If all argumentsare zero, then the returned value is 0. gcd() without argumentsreturns 0.

Added in version 3.5.

Changed in version 3.9: Added support for an arbitrary number of arguments. Formerly, only twoarguments were supported.

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.

Added 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.)

Added 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.isqrt(n)

Return the integer square root of the nonnegative integer n. This is thefloor of the exact square root of n, or equivalently the greatest integera such that a²≤n.

For some applications, it may be more convenient to have the least integera such that na², or in other words the ceiling ofthe exact square root of n. For positive n, this can be computed usinga = 1 + isqrt(n - 1).

Added in version 3.8.

math.lcm(*integers)

Return the least common multiple of the specified integer arguments.If all arguments are nonzero, then the returned value is the smallestpositive integer that is a multiple of all arguments. If any of the argumentsis zero, then the returned value is 0. lcm() without argumentsreturns 1.

Added in version 3.9.

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.nextafter(x, y, steps=1)

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.perm(n, k=None)

Return the number of ways to choose k items from n itemswithout repetition and with order.

Evaluates to n! / (n - k)! when k <= n and evaluatesto zero when k > n.

If k is not specified or is None, then k defaults to nand 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.

math.prod(iterable, *, start=1)

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 isintended specifically for use with numeric values and may rejectnon-numeric types.

Added in version 3.8.

math.remainder(x, y)

Return the IEEE 754-style remainder of x with respect to y. Forfinite 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, thenearest 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) isx for any finite x, and remainder(x, 0) andremainder(math.inf, x) raise ValueError for any non-NaN x.If the result of the remainder operation is zero, that zero will havethe same sign as x.

On platforms using IEEE 754 binary floating-point, the result of thisoperation is always exactly representable: no rounding error is introduced.

Added in version 3.7.

math.sumprod(p, q)

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(itertools.starmap(operator.mul, zip(p, q, strict=True)))

For float and mixed int/float inputs, the intermediate productsand sums are computed with extended precision.

Added in version 3.12.

math.trunc(x)

Return x with the fractional partremoved, leaving the integer part. This rounds toward 0: trunc() isequivalent 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.

math.ulp(x)

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 positivedenormalized representable float (smaller than the minimum positivenormalized 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 firstfloat smaller than x is x - ulp(x).

  • Otherwise (x is a positive finite number), return the value of the leastsignificant bit of x, such that the first float bigger than xis x + ulp(x).

ULP stands for “Unit in the Last Place”.

See also math.nextafter() and sys.float_info.epsilon.

Added in version 3.9.

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.

Power and logarithmic functions

math.cbrt(x)

Return the cube root of x.

Added in version 3.11.

math.exp(x)

Return e raised to the power x, where e = 2.718281… is the baseof natural logarithms. This is usually more accurate than math.e ** xor pow(math.e, x).

math.exp2(x)

Return 2 raised to the power x.

Added in version 3.11.

math.expm1(x)

Return e raised to the power x, minus 1. Here e is the base of naturallogarithms. 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

Added 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).

Added 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 followthe IEEE 754 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.

Changed in version 3.11: The special cases pow(0.0, -inf) and pow(-0.0, -inf) werechanged to return inf instead of raising ValueError,for consistency with IEEE 754.

math.sqrt(x)

Return the square root of x.

Trigonometric functions

math.acos(x)

Return the arc cosine of x, in radians. The result is between 0 andpi.

math.asin(x)

Return the arc sine of x, in radians. The result is between -pi/2 andpi/2.

math.atan(x)

Return the arc tangent of x, in radians. The result is between -pi/2 andpi/2.

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.dist(p, q)

Return the Euclidean distance between two points p and q, eachgiven as a sequence (or iterable) of coordinates. The two pointsmust 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.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 pointgiven by the coordinates.

For a two dimensional point (x, y), this is equivalent to computingthe 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 twodimensional case was supported.

Changed in version 3.10: Improved the algorithm’s accuracy so that the maximum error isunder 1 ulp (unit in the last place). More typically, the resultis almost always correctly rounded to within 1/2 ulp.

math.sin(x)

Return the sine of x radians.

math.tan(x)

Return the tangent of x radians.

Angular conversion

math.degrees(x)

Convert angle x from radians to degrees.

math.radians(x)

Convert angle x from degrees to radians.

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.

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

Added 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.

Added in version 3.2.

math.gamma(x)

Return the Gamma function atx.

Added in version 3.2.

math.lgamma(x)

Return the natural logarithm of the absolute value of the Gammafunction at x.

Added in version 3.2.

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!

Added in version 3.6.

math.inf

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

A floating-point “not a number” (NaN) value. Equivalent to the output offloat('nan'). Due to the requirements of the IEEE-754 standard, math.nan and float('nan') arenot considered to equal to any other numeric value, including themselves. To checkwhether a number is a NaN, use the isnan() function to testfor NaNs instead of is or ==.Example:

>>> import math>>> math.nan == math.nanFalse>>> 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 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.

math — Mathematical functions (2024)

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