Python Programming/Operators

Basics
Python math works as expected:

Note that Python adheres to the PEMDAS order of operations.

Powers
There is a built in exponentiation operator **, which can take either integers, floating point or complex numbers. This occupies its proper place in the order of operations.

Floor Division and True Division
In Python 3.x, slash operator ("/") does true division for all types including integers, and therefore, e.g.  . The result is of a floating-point type even if both inputs are integers: 4 / 2 yields 2.0.

In Python 3.x and latest 2.x, floor division for both integer arguments and floating-point arguments is achieved by using the double slash ("//") operator. For negative results, this is unlike the integer division in the C language since -3 // 2 == -2 in Python while -3 / 2 == -1 in C: C rounds the negative result toward zero while Python toward negative infinity.

Beware that due to the limitations of floating point arithmetic, rounding errors can cause unexpected results. For example:

>>> print(0.6/0.2) 3.0 >>> print(0.6//0.2) 2.0

For Python 2.x, dividing two integers or longs using the slash operator ("/") uses floor division (applying the floor function after division) and results in an integer or long. Thus, 5 / 2 == 2 and -3 / 2 == -2. Using "/" to do division this way is deprecated; if you want floor division, use "//" (available in Python 2.2 and later). Dividing by or into a floating point number will cause Python to use true division. Thus, to ensure true division in Python 2.x:.

Links:
 * 6.7. Binary arithmetic operations, in The Python Language Reference, docs.python.org
 * Python-style integer division & modulus in C, stackoverflow.com
 * Integer division rounding with negatives in C++, stackoverflow.com
 * In Python, what is a good way to round towards zero in integer division?, stackoverflow.com
 * Why Python's Integer Division Floors, python-history.blogspot.com

Modulus
The modulus (remainder of the division of the two operands, rather than the quotient) can be found using the % operator, or by the divmod builtin function. The divmod function returns a tuple containing the quotient and remainder. Note that -10 % 7 is equal to +4 while in the C language it is equal to -3. That is because Python floors towards negative infinity not zero. As a result, remainders add towards positive infinity. Consequently, since -10 / 7 = -1.4286 becomes floored to -2.0 the remainder becomes x such that -14 + x = -10.

Links:
 * 6.7. Binary arithmetic operations, in The Python Language Reference, docs.python.org

Negation
Unlike some other languages, variables can be negated directly:

Comparison
Numbers, strings and other types can be compared for equality/inequality and ordering:

Identity
The operators  and   test for object identity and stand in contrast to == (equals):   is true if and only if x and y are references to the same object in memory. yields the inverse truth value. Note that an identity test is more stringent than an equality test since two distinct objects may have the same value.

For the built-in immutable data types (like int, str and tuple) Python uses caching mechanisms to improve performance, i.e., the interpreter may decide to reuse an existing immutable object instead of generating a new one with the same value. The details of object caching are subject to changes between different Python versions and are not guaranteed to be system-independent, so identity checks on immutable objects like,  ,   may give different results on different machines.

In some Python implementations, the following results are applicable:

Links:
 * 3. Data model, python.org
 * 2. Built-in Functions # id, python.org
 * 5. Expressions # is, python.org

Augmented Assignment
There is shorthand for assigning the output of an operation to one of the inputs:

Logical Operators
Logical operators are operators that act on booleans.

or
The or operator returns true if any one of the booleans involved are true. If none of them are true (in other words, they are all false), the or operator returns false.

and
The and operator only returns true if all of the booleans are true. If any one of them is false, the and operator returns false.

not
The not operator only acts on one boolean and simply returns its opposite. So, true turns into false and false into true. The order of operations here is: not first, and second, or third. In particular, "True or True and False or False" becomes "True or False or False" which is True.

Warning, logical operators can act on things other than booleans. For instance "1 and 6" will return 6. Specifically, "and" returns either the first value considered to be false, or the last value if all are considered true. "or" returns the first true value, or the last value if all are considered false. In Python the number zero and empty strings, lists, sets, etc. are considered false. You may use  to check whether a thing is considered to be true or false in Python. For instance,  and   both return.

Bitwise Operators
Python operators for bitwise arithmetic are like those in the C language. They include & (bitwise and), | (bitwise or), ^ (exclusive or AKA xor), << (shift left), >> (shift right), and ~ (complement). Augmented assignment operators (AKA compound assignment operators) for the bitwise operations include &=, |=, ^=, <<=, and >>=. Bitwise operators apply to integers, even negative ones and very large ones; for the shift operators, the second operand must be non-negative. In the Python internal help, this is covered under the topics of EXPRESSIONS and BITWISE.

Examples:
 * 0b1101 & 0b111 == 0b101
 * Note: 0b starts a binary literal, just like 0x starts a hexadecimal literal.
 * 0b1 | 0b100 == 0b101
 * 0b111 ^ 0b101 == 0b10
 * 1 << 4 == 16
 * 7 >> 1 == 3
 * 1 << 100 == 0x10000000000000000000000000
 * Large results are supported.
 * 1 << -1
 * An error: the 2nd operand must be non-negative.
 * -2 & 15 == 14
 * For bitwise operations, negative integers are treated as if represented using two's complement with an infinite sequence of leading ones. Thus -2 is as if 0x...FFFFFFFFFE, which ANDed with 15 (0xF) yields 0xE, which is 14.
 * format(-2 % (1 << 32), "032b")
 * Determines a string showing the last 32 bits of the implied two's complement representation of -2.
 * ~-2 == 1
 * The above note about treatment of negative integers applies. For the complement (~), this treatment yields ~x == -1 * (x + 1). This can be verified: min((~x == -1 * (x + 1) for x in range(-10 ** 6, 10 ** 6))) == True.
 * ~1 == -2
 * The formula ~x == -1 * (x + 1) mentioned above applies. In the bitwise complement interpretation, all the imaginary leading leading zeros of 1 are toggled to leading ones, which is then interpreted as two's complement representation of -2, which is as if 0x...FFFFFFFFFE.
 * x = 0b11110101; x &= ~(0xF << 1); x == 0b11100001
 * A common idiom clears the least significant bits 5 to 2 using complement, showing the usefulness of the infinite-leading-ones two's complement implied representation of negative numbers for bitwise operations. Works for arbitrarily large x.
 * ~2 & 0xFFFFFFFF == 0xFFFFFFFD
 * We can emulate bitwise complement on 32-bit integers by ANDing the complement with the maximum unsigned 32-bit integer, 0xFFFFFFFF. We can proceed similarly for 8-bit complement, 16-bit complement, etc., by ANDing the complement with 0xFF, 0xFFFF, etc.
 * 2 ^ 0xFFFFFFFF == 0xFFFFFFFD
 * Another way to emulate fixed-size bitwise complement is by XORing with all Fs for the size.
 * v = 2.0 & 2
 * Yields an error: no automatic conversion of floats to ints, and no operation on the underlying representation of the float.
 * int.bit_length(0x8000) == 16
 * Determines how many bits are needed to represent the integer. Therefore, min((int.bit_length(1 << x) == x + 1 for x in range(100000))) == True.

Examples of augmented assignment operators:
 * a = 0b1101; a &= 0b111; a == 0b101
 * a = 0b1; a |= 0b100; a == 0b101
 * a = 0b111; a ^= 0b101; a == 0b10
 * a = 1; a <<= 4; a == 16
 * a = 7; a >>= 1; a == 3

Class definitions can overload the operators for the instances of the class; thus, for instance, sets overload the pipe (|) operator to mean set union: {1,2} | {3,4} == {1,2,3,4}. The names of the override methods are __and__ for &, __or__ for |, __xor__ for ^, __invert__ for ~, __lshift__ for <<, __rshift__ for >>, __iand__ for &=, __ior_ for |=, __ixor__ for ^=, __ilshift__ for <<=, and __irshift__ for >>=.

Examples of use of bitwise operations include calculation of CRC and MD5. Admittedly, these would usually be implemented in C rather than Python for maximum speed; indeed, Python has libraries for these written in C. Nonetheless, implementations in Python are possible and are shown in the links to Rosetta Code below.

Links:
 * BitwiseOperators, wiki.python.org
 * BitManipulation, wiki.python.org
 * Bitwise Operations on Integer Types in Library Reference, docs.python.org
 * 2.5. Operators in The Python Language Reference, docs.python.org
 * 6. Expressions in The Python Language Reference, docs.python.org
 * 3. Data model in in The Python Language Reference, docs.python.org
 * "Integers (int) [...] For the purpose of shift and mask operations, a binary representation is assumed, and negative numbers are represented in a variant of 2’s complement which gives the illusion of an infinite string of sign bits extending to the left."
 * , wikipedia.org
 * , wikipedia.org
 * PEP 203 – Augmented Assignments, python.org
 * Bitwise operation and usage, stackoverflow.com
 * CRC-32 # Python, rosettacode.org
 * MD5/Implementation # Python, rosettacode.org
 * longobject.c at python/cpython, github.com - functions long_bitwise, long_rshift, long_lshift, and long_invert