.. highlight:: cpp .. _arduino-bitwisemath: Bitwise AND (&), OR (\|), XOR (^), NOT (~) ========================================== The bitwise operators perform their calculations at the bit level of variables. They help solve a wide range of common programming problems. Much of the material here is adapted for Maple from an (Arduino) `tutorial on bitwise math `_\ . Another great resource is the Wikipedia article on `bitwise operations `_\ . Below are descriptions and syntax for all of the operators. .. contents:: Contents :local: .. _arduino-bitwisemath-and: Bitwise AND (&) --------------- The bitwise AND operator in C++ is a single ampersand, ``&``, used between two other integer expressions. Bitwise AND operates on each bit position of the surrounding expressions independently, according to this rule: if both input bits are 1, the resulting output is 1, otherwise the output is 0. Another way of expressing this is:: 0 0 1 1 operand1 0 1 0 1 operand2 ---------- 0 0 0 1 (operand1 & operand2) = result On the Maple, the type ``int`` is a 32-bit value, so using ``&`` between two ``int`` expressions causes 32 simultaneous AND operations to occur. In a code fragment like:: int a = 92; // in binary: 00000000000000000000000001011100 int b = 101; // in binary: 00000000000000000000000001100101 int c = a & b; // result: 00000000000000000000000001000100, // (or 68 in decimal). Each of the 32 bits in ``a`` and ``b`` are processed using bitwise AND, and all 32 resulting bits are stored in ``c``, resulting in the value 1000100 in binary, which is 68 in decimal. .. _arduino-bitwisemath-or: Bitwise OR (\|) --------------- The bitwise OR operator in C++ is the vertical bar symbol, ``|``. Like the ``&`` operator, ``|`` operates independently on each bit in its two surrounding integer expressions, but what it does is different. The bitwise OR of two bits is 1 if either or both of the input bits is 1, otherwise it is 0. For example:: 0 0 1 1 operand1 0 1 0 1 operand2 ---------- 0 1 1 1 (operand1 | operand2) = result Here is an example of bitwise OR used in a snippet of C++ code (using ``char``, which takes up 8 bits of memory, instead of ``int``, which uses 32):: char a = 92; // in binary: 01011100 char b = 101; // in binary: 01100101 char c = a | b; // result: 01111101, or 125 in decimal. .. _arduino-bitwisemath-xor: Bitwise XOR (^) --------------- There is a somewhat unusual operator in C++ called bitwise EXCLUSIVE OR, also known as bitwise XOR. (In English, this is usually pronounced "zor" or "ex-or"). The bitwise XOR operator is written using the caret symbol, ``^``. This operator is very similar to the bitwise OR operator ``|``, except it evaluates to 0 for a given bit position when both of the input bits for that position are 1:: 0 0 1 1 operand1 0 1 0 1 operand2 ---------- 0 1 1 0 (operand1 ^ operand2) = result Another way to look at bitwise XOR is that each bit in the result is a 1 if the input bits are different, or 0 if they are the same. Here is a simple example:: int x = 12; // binary (ignoring extra bits): 1100 int y = 10; // binary: 1010 int z = x ^ y; // binary: 0110, or decimal 6 The ^ operator is often used to toggle (i.e. change from 0 to 1, or 1 to 0) some of the bits in an integer expression. In a bitwise OR operation if there is a 1 in the mask bit, that bit is inverted; if there is a 0, the bit is not inverted and stays the same. Below is a program to blink digital pin 13 (the LED pin on Maple):: // Blink Maple LED pin int led_pin = 13; int toggle = 0; // demo for Exclusive OR void setup(){ pinMode(led_pin, OUTPUT); } void loop(){ toggle = toggle ^ 1; digitalWrite(led_pin, toggle); delay(100); } .. _arduino-bitwisemath-not: Bitwise NOT (~) --------------- The bitwise NOT operator in C++ is the tilde character ``~``. Unlike ``&`` and ``|``, the bitwise NOT operator is applied to a single operand to its right. Bitwise NOT changes each bit to its opposite: 0 becomes 1, and 1 becomes 0. For example:: 0 1 operand1 ---- 1 0 ~operand1 = result Another example:: char a = 103; // binary: 01100111 char b = ~a; // binary: 10011000 = -104 You might be surprised to see a negative number like -104 as the result of this operation. This is because the highest bit in an int variable is the so-called "sign bit". If the highest bit is 1, the number is interpreted as negative. This encoding of positive and negative numbers is referred to as *two's complement*. For more information, see the Wikipedia article on `two's complement. `_ As an aside, it is interesting to note that (under two's complement arithmetic) for any integer ``x``, ``~x`` is the same as ``-x-1``. At times, the sign bit in a signed integer expression can cause some unwanted surprises. Uses ---- One of the most common uses of bitwise operations is to select or manipulate a particular bit (or bits) from an integer value, often called `bit masking `_\ . See the linked Wikipedia article for more information and examples. If you really want to see bit-twiddling techniques in their full glory, you could do much worse than to get yourself a copy of `Hacker's Delight `_\ . See Also -------- - :ref:`Boolean operations ` (``&&``, ``||``) - :ref:`Compound bitwise operations ` (``&=``, ``|=``, ``^=``).