# Bitwise Operators (&, |, ^, ~)¶

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.

## 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.

## 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.
```

## 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 toggle the built-in LED pin (you can also accomplish this with toggleLED()):

```// Toggle built-in LED pin

int toggle = 0;

// demo for Exclusive OR
void setup(){
pinMode(BOARD_LED_PIN, OUTPUT);
}

void loop(){
toggle = toggle ^ 1;
digitalWrite(BOARD_LED_PIN, toggle);
delay(100);
}
```

## 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.