# Math Module
# Rounding: round, floor, ceil, trunc
In addition to the built-in
round function, the
math module provides the
x = 1.55 y = -1.55 # round to the nearest integer round(x) # 2 round(y) # -2 # the second argument gives how many decimal places to round to (defaults to 0) round(x, 1) # 1.6 round(y, 1) # -1.6 # math is a module so import it first, then use it. import math # get the largest integer less than x math.floor(x) # 1 math.floor(y) # -2 # get the smallest integer greater than x math.ceil(x) # 2 math.ceil(y) # -1 # drop fractional part of x math.trunc(x) # 1, equivalent to math.floor for positive numbers math.trunc(y) # -1, equivalent to math.ceil for negative numbers
round always return a
round(1.3) # 1.0
round always breaks ties away from zero.
round(0.5) # 1.0 round(1.5) # 2.0
trunc always return an
Integral value, while
round returns an
Integral value if called with one argument.
round(1.3) # 1 round(1.33, 1) # 1.3
round breaks ties towards the nearest even number. This corrects the bias towards larger numbers when performing a large number of calculations.
round(0.5) # 0 round(1.5) # 2
As with any floating-point representation, some fractions cannot be represented exactly. This can lead to some unexpected rounding behavior.
round(2.675, 2) # 2.67, not 2.68!
# Warning about the floor, trunc, and integer division of negative numbers
Python (and C++ and Java) round away from zero for negative numbers. Consider:
>>> math.floor(-1.7) -2.0 >>> -5 // 2 -3
# Calculating the length of the hypotenuse
math.hypot(2, 4) # Just a shorthand for SquareRoot(2**2 + 4**2) # Out: 4.47213595499958
# Converting degrees to/from radians
math functions expect radians so you need to convert degrees to radians:
math.radians(45) # Convert 45 degrees to radians # Out: 0.7853981633974483
All results of the inverse trigonometic functions return the result in radians, so you may need to convert it back to degrees:
math.degrees(math.asin(1)) # Convert the result of asin to degrees # Out: 90.0
# Sine, cosine, tangent and inverse functions
# Sine and arc sine math.sin(math.pi / 2) # Out: 1.0 math.sin(math.radians(90)) # Sine of 90 degrees # Out: 1.0 math.asin(1) # Out: 1.5707963267948966 # "= pi / 2" math.asin(1) / math.pi # Out: 0.5 # Cosine and arc cosine: math.cos(math.pi / 2) # Out: 6.123233995736766e-17 # Almost zero but not exactly because "pi" is a float with limited precision! math.acos(1) # Out: 0.0 # Tangent and arc tangent: math.tan(math.pi/2) # Out: 1.633123935319537e+16 # Very large but not exactly "Inf" because "pi" is a float with limited precision
math.atan(math.inf) # Out: 1.5707963267948966 # This is just "pi / 2"
math.atan(float('inf')) # Out: 1.5707963267948966 # This is just "pi / 2"
Apart from the
math.atan there is also a two-argument
math.atan2 function, which computes the correct quadrant and avoids pitfalls of division by zero:
math.atan2(1, 2) # Equivalent to "math.atan(1/2)" # Out: 0.4636476090008061 # ≈ 26.57 degrees, 1st quadrant math.atan2(-1, -2) # Not equal to "math.atan(-1/-2)" == "math.atan(1/2)" # Out: -2.677945044588987 # ≈ -153.43 degrees (or 206.57 degrees), 3rd quadrant math.atan2(1, 0) # math.atan(1/0) would raise ZeroDivisionError # Out: 1.5707963267948966 # This is just "pi / 2"
# Hyperbolic sine, cosine and tangent
# Hyperbolic sine function math.sinh(math.pi) # = 11.548739357257746 math.asinh(1) # = 0.8813735870195429 # Hyperbolic cosine function math.cosh(math.pi) # = 11.591953275521519 math.acosh(1) # = 0.0 # Hyperbolic tangent function math.tanh(math.pi) # = 0.99627207622075 math.atanh(0.5) # = 0.5493061443340549
math.log(x) gives the natural (base
e) logarithm of
math.log(math.e) # 1.0 math.log(1) # 0.0 math.log(100) # 4.605170185988092
math.log can lose precision with numbers close to 1, due to the limitations of floating-point numbers. In order to accurately calculate logs close to 1, use
math.log1p, which evaluates the natural logarithm of 1 plus the argument:
math.log(1 + 1e-20) # 0.0 math.log1p(1e-20) # 1e-20
math.log10 can be used for logs base 10:
math.log10(10) # 1.0
When used with two arguments,
math.log(x, base) gives the logarithm of
x in the given
log(x) / log(base).
math.log(100, 10) # 2.0 math.log(27, 3) # 3.0 math.log(1, 10) # 0.0
math modules includes two commonly used mathematical constants.
math.pi- The mathematical constant pi
math.e- The mathematical constant e (base of natural logarithm)
>>> from math import pi, e >>> pi 3.141592653589793 >>> e 2.718281828459045 >>>
Python 3.5 and higher have constants for infinity and NaN ("not a number"). The older syntax of passing a string to
float() still works.
math.inf == float('inf') # Out: True -math.inf == float('-inf') # Out: True # NaN never compares equal to anything, even itself math.nan == float('nan') # Out: False
# Infinity and NaN ("not a number")
In all versions of Python, we can represent infinity and NaN ("not a number") as follows:
pos_inf = float('inf') # positive infinity neg_inf = float('-inf') # negative infinity not_a_num = float('nan') # NaN ("not a number")
In Python 3.5 and higher, we can also use the defined constants
pos_inf = math.inf neg_inf = -math.inf not_a_num = math.nan
The string representations display as
pos_inf, neg_inf, not_a_num # Out: (inf, -inf, nan)
We can test for either positive or negative infinity with the
math.isinf(pos_inf) # Out: True math.isinf(neg_inf) # Out: True
We can test specifically for positive infinity or for negative infinity by direct comparison:
pos_inf == float('inf') # or == math.inf in Python 3.5+ # Out: True neg_inf == float('-inf') # or == -math.inf in Python 3.5+ # Out: True neg_inf == pos_inf # Out: False
Python 3.2 and higher also allows checking for finiteness:
math.isfinite(pos_inf) # Out: False math.isfinite(0.0) # Out: True
Comparison operators work as expected for positive and negative infinity:
import sys sys.float_info.max # Out: 1.7976931348623157e+308 (this is system-dependent) pos_inf > sys.float_info.max # Out: True neg_inf < -sys.float_info.max # Out: True
But if an arithmetic expression produces a value larger than the maximum that can be represented as a
float, it will become infinity:
pos_inf == sys.float_info.max * 1.0000001 # Out: True neg_inf == -sys.float_info.max * 1.0000001 # Out: True
However division by zero does not give a result of infinity (or negative infinity where appropriate), rather it raises a
try: x = 1.0 / 0.0 print(x) except ZeroDivisionError: print("Division by zero") # Out: Division by zero
Arithmetic operations on infinity just give infinite results, or sometimes NaN:
-5.0 * pos_inf == neg_inf # Out: True -5.0 * neg_inf == pos_inf # Out: True pos_inf * neg_inf == neg_inf # Out: True 0.0 * pos_inf # Out: nan 0.0 * neg_inf # Out: nan pos_inf / pos_inf # Out: nan
NaN is never equal to anything, not even itself. We can test for it is with the
not_a_num == not_a_num # Out: False math.isnan(not_a_num) Out: True
NaN always compares as "not equal", but never less than or greater than:
not_a_num != 5.0 # or any random value # Out: True not_a_num > 5.0 or not_a_num < 5.0 or not_a_num == 5.0 # Out: False
Arithmetic operations on NaN always give NaN. This includes multiplication by -1: there is no "negative NaN".
5.0 * not_a_num # Out: nan float('-nan') # Out: nan
-math.nan # Out: nan
There is one subtle difference between the old
float versions of NaN and infinity and the Python 3.5+
math library constants:
math.inf is math.inf, math.nan is math.nan # Out: (True, True) float('inf') is float('inf'), float('nan') is float('nan') # Out: (False, False)
# Pow for faster exponentiation
Using the timeit module from the command line:
> python -m timeit 'for x in xrange(50000): b = x**3' 10 loops, best of 3: 51.2 msec per loop > python -m timeit 'from math import pow' 'for x in xrange(50000): b = pow(x,3)' 100 loops, best of 3: 9.15 msec per loop
** operator often comes in handy, but if performance is of the essence, use math.pow. Be sure to note, however, that pow returns floats, even if the arguments are integers:
> from math import pow > pow(5,5) 3125.0
# Copying signs
In Python 2.6 and higher,
math.copysign(x, y) returns
x with the sign of
y. The returned value is always a
math.copysign(-2, 3) # 2.0 math.copysign(3, -3) # -3.0 math.copysign(4, 14.2) # 4.0 math.copysign(1, -0.0) # -1.0, on a platform which supports signed zero
# Imaginary Numbers
Imaginary numbers in Python are represented by a "j" or "J" trailing the target number.
1j # Equivalent to the square root of -1. 1j * 1j # = (-1+0j)
# Complex numbers and the cmath module
cmath module is similar to the
math module, but defines functions appropriately for the complex plane.
First of all, complex numbers are a numeric type that is part of the Python language itself rather than being provided by a library class. Thus we don't need to
import cmath for ordinary arithmetic expressions.
Note that we use
J) and not
z = 1 + 3j
We must use
j would be the name of a variable rather than a numeric literal.
1j * 1j Out: (-1+0j) 1j ** 1j # Out: (0.20787957635076193+0j) # "i to the i" == math.e ** -(math.pi/2)
We have the
real part and the
imag (imaginary) part, as well as the complex
# real part and imaginary part are both float type z.real, z.imag # Out: (1.0, 3.0) z.conjugate() # Out: (1-3j) # z.conjugate() == z.real - z.imag * 1j
The built-in functions
complex are also part of the language itself and don't require any import:
abs(1 + 1j) # Out: 1.4142135623730951 # square root of 2 complex(1) # Out: (1+0j) complex(imag=1) # Out: (1j) complex(1, 1) # Out: (1+1j)
complex function can take a string, but it can't have spaces:
complex('1+1j') # Out: (1+1j) complex('1 + 1j') # Exception: ValueError: complex() arg is a malformed string
But for most functions we do need the module, for instance
import cmath cmath.sqrt(-1) # Out: 1j
Naturally the behavior of
sqrt is different for complex numbers and real numbers. In non-complex
math the square root of a negative number raises an exception:
import math math.sqrt(-1) # Exception: ValueError: math domain error
Functions are provided to convert to and from polar coordinates:
cmath.polar(1 + 1j) # Out: (1.4142135623730951, 0.7853981633974483) # == (sqrt(1 + 1), atan2(1, 1)) abs(1 + 1j), cmath.phase(1 + 1j) # Out: (1.4142135623730951, 0.7853981633974483) # same as previous calculation cmath.rect(math.sqrt(2), math.atan(1)) # Out: (1.0000000000000002+1.0000000000000002j)
The mathematical field of complex analysis is beyond the scope of this example, but many functions in the complex plane have a "branch cut", usually along the real axis or the imaginary axis. Most modern platforms support "signed zero" as specified in IEEE 754, which provides continuity of those functions on both sides of the branch cut. The following example is from the Python documentation:
cmath.phase(complex(-1.0, 0.0)) # Out: 3.141592653589793 cmath.phase(complex(-1.0, -0.0)) # Out: -3.141592653589793
cmath module also provides many functions with direct counterparts from the
In addition to
sqrt, there are complex versions of
log10, the trigonometric functions and their inverses (
atan), and the hyperbolic functions and their inverses (
atanh). Note however there is no complex counterpart of
math.atan2, the two-argument form of arctangent.
cmath.log(1+1j) # Out: (0.34657359027997264+0.7853981633974483j) cmath.exp(1j * cmath.pi) # Out: (-1+1.2246467991473532e-16j) # e to the i pi == -1, within rounding error
e are provided. Note these are
float and not
type(cmath.pi) # Out: <class 'float'>
cmath module also provides complex versions of
isinf, and (for Python 3.2+)
isfinite. See "Infinity and NaN". A complex number is considered infinite if either its real part or its imaginary part is infinite.
cmath.isinf(complex(float('inf'), 0.0)) # Out: True
cmath module provides a complex version of
isnan. See "Infinity and NaN". A complex number is considered "not a number" if either its real part or its imaginary part is "not a number".
cmath.isnan(0.0, float('nan')) # Out: True
Note there is no
cmath counterpart of the
math.nan constants (from Python 3.5 and higher)
cmath.isinf(complex(0.0, math.inf)) # Out: True cmath.isnan(complex(math.nan, 0.0)) # Out: True cmath.inf # Exception: AttributeError: module 'cmath' has no attribute 'inf'
In Python 3.5 and higher, there is an
isclose method in both
z = cmath.rect(*cmath.polar(1+1j)) z # Out: (1.0000000000000002+1.0000000000000002j) cmath.isclose(z, 1+1j) # True