Python 3.6.5 Documentation >  "math" — Mathematical functions

"math" — Mathematical functions
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This module is always available. It provides access to the
mathematical functions defined by the C standard.

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.

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.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)" 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
or is 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 an "Integral" value.

math.fmod(x, y)

Return "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(a, b)

Return the greatest common divisor of the integers *a* and *b*. If
either *a* or *b* is nonzero, then the value of "gcd(a, b)" is the
largest positive integer that divides both *a* and *b*. "gcd(0,
0)" returns "0".

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 and
"False" otherwise.

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.

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 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.

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, and "False"
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, and
"False" 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 function
"frexp()".

math.modf(x)

Return the fractional and integer parts of *x*. Both results carry
the sign of *x* and are floats.

math.trunc(x)

Return the "Real" value *x* truncated to an "Integral" (usually an
integer). Delegates to "x.__trunc__()".

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.exp(x)

Return "e**x".

math.expm1(x)

Return "e**x - 1". 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

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*). 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)".

New in version 3.3.

See also: "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 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". 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.

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 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.hypot(x, y)

Return the Euclidean norm, "sqrt(x*x + y*y)". This is the length of
the vector from the origin to point "(x, 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.

math.radians(x)

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

New in version 3.2.

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.

New in version 3.2.

math.gamma(x)

Return the Gamma function at *x*.

New in version 3.2.

math.lgamma(x)

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

New 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 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!

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 of "float('nan')".

New in version 3.5.

**CPython implementation detail:** 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.

See also:

Module "cmath"
Complex number versions of many of these functions.