# math --- 數學函式¶

## 數論與表現函式¶

math.ceil(x)

math.comb(n, k)

k <= n 時其值為 n! / (k! * (n - k)!)，否則其值為 0

math.copysign(x, y)

math.fabs(x)

math.factorial(n)

math.floor(x)

math.fmod(x, y)

math.frexp(x)

(m, e) 對的格式回傳 x 的尾數 m 及指數 em 是浮點數而 e 是整數，且兩者精確地使 x == m * 2**e。若 x 為零，回傳 (0.0, 0)，否則令 0.5 <= abs(m) < 1。此函式用於以可攜的方式「分割」浮點數內部表示法。

math.fsum(iterable)

Return an accurate floating-point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums.

For further discussion and two alternative approaches, see the ASPN cookbook recipes for accurate floating-point summation.

math.gcd(*integers)

math.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)

ab 兩值足夠接近便回傳 True，否則回傳 False

rel_tol 為相對容許偏差 ── ab 兩數差的最大容許值，與 ab 兩數的絕對值中較大者相關。例如欲設置 5% 的容許偏差，則傳入 rel_tol=0.05。其預設值為 1e-09，該值可確保兩數於大約 9 個十進數位內相同。rel_tol 須大於 0

abs_tol 為最小絕對容許偏差 ── 於接近零的比較時很有用。abs_tol 須大於等於 0

PEP 485 ── 用於測試近似相等的函式

math.isfinite(x)

x 不是無限值或 NaN 便回傳 True，否則回傳 False。（注意 0.0 被視為有限數。）

math.isinf(x)

x 是正無限值或負無限值便回傳 True，否則回傳 False

math.isnan(x)

xNaN ── 即非數字（NaN, not a number）── 便回傳 True，否則回傳 False

math.isqrt(n)

math.lcm(*integers)

math.ldexp(x, i)

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, 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.

• 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.

math.perm(n, k=None)

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!.

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 is intended specifically for use with numeric values and may reject non-numeric types.

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, 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.

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 products and sums are computed with extended precision.

math.trunc(x)

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.

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 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 "Unit in the Last Place".

Note that 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).

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.

## Power and logarithmic functions¶

math.cbrt(x)

Return the cube root of x.

math.exp(x)

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

math.exp2(x)

Return 2 raised to the power x.

math.expm1(x)

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

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). 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).

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)

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.

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 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)

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)

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)))

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).

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.

Convert angle x from degrees to radians.

## Hyperbolic functions¶

Hyperbolic functions 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.

## Special functions¶

math.erf(x)

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

math.erfc(x)

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.

math.gamma(x)

Return the Gamma function at x.

math.lgamma(x)

Return the natural logarithm of the absolute value of the Gamma function at x.

## 常數¶

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 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!

math.inf

A floating-point positive infinity. (For negative infinity, use -math.inf.) Equivalent to the output of float('inf').

math.nan

A floating-point "not a number" (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

CPython 實作細節： 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.

cmath 模組

Complex number versions of many of these functions.