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Posted 8 May 2002
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# An Introduction to Bitwise Operators

, 8 May 2002

## Introduction

I have noticed that some people seem to have problems with bitwise operators, so I decided to write this brief tutorial on how to use them.

## An Introduction to Bits

Bits, what are they, you may ask?

Well, simply put, bits are the individual ones and zeros that make up everything we do with computers. All the data you use is stored in your computer using bits. A `BYTE` is made up of eight bits, a `WORD` is two `BYTE`s, or sixteen bits. And a `DWORD` is two `WORD`S, or thirty two bits.

``` 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0
||              |               |               |              ||
|+- bit 31      |               |               |       bit 0 -+|
|               |               |               |               |
+-- BYTE 3 -----+--- BYTE 2 ----+--- BYTE 1 ----+-- BYTE 0 -----+
|                               |                               |
+----------- WORD 1 ------------+----------- WORD 0 ------------+
|                                                               |
+--------------------------- DWORD -----------------------------+```

The beauty of having bitwise operators is that you can use a `BYTE`, `WORD` or `DWORD` as a small array or structure. Using bitwise operators, you can check or set the values of individual bits or even a group of bits.

## Hexadecimal Numbers and How They Relate to Bits

When working with bits, it is kind of hard to express every number using just ones and zeros, which is known as binary notation. To get around this, we use hexadecimal (base 16) numbers.

As you may or may not know, it takes four bits to cover all the numbers from zero to fifteen, which also happens to be the range of a single digit hexadecimal number. This group of four bits, or half a `BYTE`, is called a nibble. As there are two nibbles in a `BYTE`, we can use two hexadecimal digits to show the value of one `BYTE`.

```NIBBLE   HEX VALUE
======   =========
0000        0
0001        1
0010        2
0011        3
0100        4
0101        5
0110        6
0111        7
1000        8
1001        9
1010        A
1011        B
1100        C
1101        D
1110        E
1111        F```

So if we had one `BYTE` containing the letter '`r`' (ASCII code 114), it would look like this:

```0111 0010    binary

We could write it as '`0x72`'

## Bitwise Operators

There are six bitwise operators. They are:

1.   &   The AND operator
2.    |   The OR operator
3.    ^   The XOR operator
4.    ~   The Ones Complement or Inversion operator
5.   >>   The Right Shift operator
6.   <<   The Left Shift operator.

### The & Operator

The & (AND) operator compares two values, and returns a value that has its bits set if, and only if, the two values being compared both have their corresponding bits set. The bits are compared using the following table:

```1   &   1   ==   1
1   &   0   ==   0
0   &   1   ==   0
0   &   0   ==   0
```

An ideal use for this is to set up a mask to check the values of certain bits. Say we have a BYTE that contains some bit flags, and we want to check if bit four bit is set.

```BYTE b = 50;
if ( b & 0x10 )
cout << "Bit four is set" << endl;
else
cout << "Bit four is clear" << endl;```

This would result in the following calculation:

```  00110010  - b
& 00010000  - & 0x10
----------
00010000  - result
```

So we see that bit four is set.

### The | Operator

The | (OR) operator compares two values, and returns a value that has its bits set if one or the other values, or both, have their corresponding bits set. The bits are compared using the following table:

```1   |   1   ==   1
1   |   0   ==   1
0   |   1   ==   1
0   |   0   ==   0
```

An ideal use for this is to ensure that certain bits are set. Say we want to ensure that bit three of some value is set:

```BYTE b = 50;
BYTE c = b | 0x04;
cout << "c = " << c << endl;```

This would result in the following calculation:

```  00110010  - b
| 00000100  - | 0x04
----------
00110110  - result
```

### The ^ Operator

The ^ (XOR) operator compares two values, and returns a value that has its bits set if one or the other value has its corresponding bits set, but not both. The bits are compared using the following table:

```1   ^   1   ==   0
1   ^   0   ==   1
0   ^   1   ==   1
0   ^   0   ==   0
```

An ideal use for this is to toggle certain bits. Say we want to toggle the bits three and four:

```BYTE b = 50;
cout << "b = " << b << endl;
b = b ^ 0x18;
cout << "b = " << b << endl;
b = b ^ 0x18;
cout << "b = " << b << endl;```

This would result in the following calculations:

```  00110010  - b
^ 00011000  - ^ 0x18
----------
00101010  - result

00101010  - b
^ 00011000  - ^ 0x18
----------
00110010  - result
```

### The ~ Operator

The ~ (Ones Complement or inversion) operator acts only on one value and it inverts it, turning all the ones into zeros, and all the zeros into ones. An ideal use of this would be to set certain bytes to zero, and ensuring all other bytes are set to one, regardless of the size of the data. Say we want to set all the bits to one except bits zero and one:

```BYTE b = ~0x03;
cout << "b = " << b << endl;
WORD w = ~0x03;
cout << "w = " << w << endl;```

This would result in the following calculations:

```00000011  - 0x03
11111100  - ~0x03  b

0000000000000011  - 0x03
1111111111111100  - ~0x03  w
```

Another ideal use, is to combine it with the & operator to ensure that certain bits are set to zero. Say we want to clear bit four:

```BYTE b = 50;
cout << "b = " << b << endl;
BYTE c = b & ~0x10;
cout << "c = " << c << endl;```

This would result in the following calculations:

```  00110010  - b
& 11101111  - ~0x10
----------
00100010  - result
```

### The >> and << Operators

The >> (Right shift) and << (left shift) operators move the bits the number of bit positions specified. The >> operator shifts the bits from the high bit to the low bit. The << operator shifts the bits from the low bit to the high bit. One use for these operators is to align the bits for whatever reason (check out the `MAKEWPARAM`, `HIWORD`, and `LOWORD` macros):

```BYTE b = 12;
cout << "b = " << b << endl;
BYTE c = b << 2;
cout << "c = " << c << endl;
c = b >> 2;
cout << "c = " << c << endl;```

This would result in the following calculations:

```00001100  - b
00110000  - b << 2
00000011  - b >> 2
```

## Bit Fields

Another interesting thing that can be done using bits is to have bit fields. With bit fields, you can set up minature structures within a `BYTE`, `WORD` or `DWORD`. Say, for example, we want to keep track of dates, but we want to use the least amount of memory as possible. We could declare our structure this way:

```struct date_struct {
BYTE day   : 5,   // 1 to 31
month : 4,   // 1 to 12
year  : 14;  // 0 to 9999
} date;```

In this example, the `day` field takes up the lowest 5 bits, `month` the next four, and `year` the next 14 bits. So we can store the date structure in twenty three bits, which is contained in three `BYTE`s. The twenty fourth bit is ignored. If I had declared it using an integer for each field, the structure would have taken up 12 `BYTE`s.

```|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|
|                           |       |         |
+------ year ---------------+ month +-- day --+```

Now let's pick this declaration apart to see what we are doing.

First, we will look at the data type we are using for the bit field structure. In this case, we used a `BYTE`. A `BYTE` is 8 bits, and by using it, the compiler will allocate one `BYTE` for storage. If however, we use more than 8 bits in our structure, the compiler will allocate another `BYTE`, as many `BYTE`s as it takes to hold our structure. If we had used a `WORD` or `DWORD`, the compiler would have allocated a total of 32 bits to hold our structure.

Now let's look at how the various fields are declared. First, we have the variable (`day`, `month`, and `year`), followed by a colon that separates the variable from the number of bits that it contains. Each bit field is separated by a comma, and the list is ended with a semicolon.

Now we get to the `struct` declaration. We put the bit fields into a `struct` like this so that we can use convention structure accessing notation to get at the structure members. Also, since we cannot get the addresses of bit fields, we can now use the address of the structure.

```date.day = 12;

dateptr = &date;
dateptr->year = 1852;```

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