Hexadecimal Number System
The hexadecimal system uses a radix of 16. Therefore, it has 16 possible digit symbols. The first ten digits in the hexadecimal system are represented by the numbers 0 through 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), and the letters A through F are used to represent the numbers 10, 11, 12, 13, 14, and 15, respectively. The adjoining table illustrates the relationships among hexadecimal, decimal, and binary representations. Note that each hexadecimal digit represents a group of four binary digits.
| Hexadecimal | Decimal | Binary |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
As is true for binary and decimal numbers, each digit in the hexadecimal system has a positional value or weight. For the right-most digit of a hex (abbreviation for hexadecimal) number, the positional weight is \(16^{0}\) (= 1), the next digit to the left has a positional weight of \(16^{1}\) (= 16), and so on. The positional weight distribution of a hex number system is given below: etc.
| 3 | 2 | 1 | 0 |
|---|---|---|---|
| \(16^{3}\) | \(16^{2}\) | \(16^{1}\) | \(16^{0}\) |
| 4096 | 256 | 16 | 1 |
Decimal-to-Hex Conversion.
To convert a decimal number to a hex number, the technique is the same as used for decimal-to-binary conversion or decimal-to-octal conversion. Recall that we did decimal-to-binary conversion using repeated division by 2 and decimal-to-octal conversion using repeated division by 8. Likewise, decimal-to-hex conversion is done using repeated division by 16. The decimal-to-hex conversion procedure is given below.
Suppose we are to convert the decimal number 423 to hex number.
| Division | Remainder |
|---|---|
| 423 ÷ 16 = 26 | 7 (LSB) |
| 26 ÷ 16 = 1 | 10 |
| 1 ÷ 16 = 0 | 1 (MSB) |
| ∴ \((423)_{10}\) = \((1A7)_{16}\) |
The 10 is represented by the letter A.
Hex-to-Decimal Conversion.
In order to convert a hex number to its decimal equivalent, simply add up the position weight of each digit in the hex number. The following example illustrates this conversion.
\[(356)_{16} = (3 × 16^{2}) + (5 × 16^{1}) + (6 × 16^{0})\] \[= 768 + 80 + 6 = 854\] \[∴ (356)_{16} = (854)_{10}\]
Hex-to-Binary Conversion.
The conversion from hex to binary is performed by converting each hex digit to its 4-bit binary equivalent. The following example illustrates this point. Here, we will convert the hexadecimal number \((9F2)_{16}\) to its binary equivalent.
| 9 | F | 2 |
|---|---|---|
| ↓ | ↓ | ↓ |
| 1001 | 1111 | 0010 |
\[∴ (9F2)_{16} = (100111110010)_{2}\]
Binary-to-Hex Conversion.
The conversion from binary to hex is just the reverse of the above process. The binary number is grouped into groups of four bits and each group is converted to its equivalent hex digit. The following example illustrates this point. Here, we shall convert the binary number (1110100110)2 to its *equivalent hex number.
| 0011 | 1010 | 0110 |
|---|---|---|
| ↓ | ↓ | ↓ |
| 3 | 6 | A |
\[(1110100110)_{2} = (3A6)_{16}\]