Math Module: Rounding: round, floor, ceil, trunc
In addition to the built-in round function, the math module provides the floor, ceil, and trunc functions.
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
Python 2.x Version ≤ 2.7
floor, ceil, trunc, and round always return a float.
round(1.3) # 1.0
round always breaks ties away from zero.
round(0.5) # 1.0
round(1.5) # 2.0
Python 3.x Version ≥ 3.0
floor, ceil, and 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
Warning!
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
Math Module: Trigonometry
Calculating the length of the hypotenuse
math.hypot(2, 4) # Just a shorthand for SquareRoot(22 + 42)
Out: 4.47213595499958
Converting degrees to/from radians
All 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 trigonometric 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 Python 3.x Version ≥ 3.5 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
ath.acosh(1) # = 0.0
Hyperbolic tangent function
math.tanh(math.pi) # = 0.99627207622075
math.atanh(0.5) # = 0.5493061443340549
Math Module: 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
The built-in ** 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
Math Module: 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 math.inf and math.nan:
Python 3.x Version ≥ 3.5
pos_inf = math.inf
neg_inf = -math.inf
not_a_num = math.nan
The string representations display as inf and -inf and nan:
pos_inf, neg_inf, not_a_num
Out: (inf, -inf, nan)
We can test for either positive or negative infinity with the isinf method:
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:
Python 3.x Version ≥ 3.2
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 ZeroDivisionError exception.
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 isnan method:
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
Python 3.x Version ≥ 3.5
-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:
Python 3.x Version ≥ 3.5
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)
Math Module: Logarithms
math.log(x) gives the natural (base e) logarithm of x.
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
Python 2.x Version ≥ 2.3.0
When used with two arguments, math.log(x, base) gives the logarithm of x in the given base (i.e. log(x) / log(base).
math.log(100, 10) # 2.0
math.log(27, 3) # 3.0
math.log(1, 10) # 0.0
Math Module: Constants
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.
Python 3.x Version ≥ 3.5
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
Math Module: 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)
Math Module: 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 float.
Python 2.x Version ≥ 2.6
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
Math Module: Complex numbers and the cmath module
The 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 (or J) and not i.
z = 1 + 3j
We must use 1j since 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 conjugate:
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 abs and 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)
The 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 sqrt:
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
The cmath module also provides many functions with direct counterparts from the math module.
In addition to sqrt, there are complex versions of exp, log, log10, the trigonometric functions and their inverses (sin, cos, tan, asin, acos, atan), and the hyperbolic functions and their inverses (sinh, cosh, tanh, asinh, acosh, 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
The constants pi and e are provided. Note these are float and not complex.
type(cmath.pi)
Out:
The 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
Likewise, the 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.inf and math.nan constants (from Python 3.5 and higher)
Python 3.x Version ≥ 3.5
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 cmath and math modules.
Python 3.x Version ≥ 3.5
z = cmath.rect(*cmath.polar(1+1j))
z
Out: (1.0000000000000002+1.0000000000000002j)
cmath.isclose(z, 1+1j)
True
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