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Octal and Hexadecimal Numbers

The conversion from and to binary, octal, and hexadecimal representation plays an important part in digital computers. Since 23 = 8 and 24 = 16, each octal digit corresponds to three binary digits and each hexadecimal digit corresponds to four binary digits. The conversion from binary to octal is easily accomplished by partitioning the binary number into groups of three bits each. The corresponding octal digit is then assigned to each group of bits and the string of digits so obtained gives the octal equivalent of the binary number. Consider, for example, a 16-bit register. Physically, one may think of the register as composed of 16 binary storage cells, with each cell capable of holding either a 1 or a 0. Suppose that the bit configuration stored in the register is as shown in Fig. 3-2. Since a binary number consists of a string of 1's and 0's, the 16-bit register can be used to store any binary number from 0 to 216 - 1. For the particular example shown, the binary number stored in the register is the equivalent of decimal 44899. Starting from the low-order bit, we partition the register into groups of three bits each (the sixteenth bit remains in a group by itself). Each group of three bits is assigned its octal equivalent and placed on top of the register. The string of octal digits so obtained represents the octal equivalent of the binary number.

Table 3-1

Conversion from binary to hexadecimal is similar except that the bits are divided into groups of four. The corresponding hexadecimal digit for each group of four bits is written as shown below the register of Fig. 3-2. The string of hexadecimal digits so obtained represents the hexadecimal equivalent of the binary number. The corresponding octal digit for each group of three bits is easily remembered after studying the first eight entries listed in Table 3-1. The correspondence between a hexadecimal digit and its equivalent 4-bit code can be found in the first 16 entries of Table 3-2.

Table 3-1 lists a few octal numbers and their representation in registers in binary-coded form. The binary code is obtained by the procedure explained above. Each octal digit is assigned a 3-bit code as specified by the entries of the first eight digits in the table. Similarly, Table 3-2 lists a few hexadecimal numbers and their representation in registers in binary-coded form. Here the binary code is obtained by assigning to each hexadecimal digit the 4-bit code listed in the first 16 entries of the table.

Figure 3-2

Comparing the binary-coded octal and hexadecimal numbers with their binary number equivalent we find that the bit combination in all three representations is exactly the same. For example, decimal 99, when converted to binary, becomes 1100011. The binary-coded octal equivalent of decimal 99 is 001 100 011 and the binary-coded hexadecimal of decimal 99 is 0110 0011. If we neglect the leading zeros in these three binary representations, we find that their bit combination is identical. This should be so because of the straightforward conversion that exists between binary numbers and octal or hexadecimal. The point of all this is that a string of 1's and 0's stored in a register could represent a binary number, but this same string of bits may be interpreted as holding an octal number in binary-coded form (if we divide the bits in groups of three) or as holding a hexadecimal number in binary-coded form (if we divide the bits in groups of four). The registers in a digital computer contain many bits. Specifying the content of registers by their binary values will require a long string of binar-, digits. It is more convenient to specify content of registers by their octal or hexadecimal equivalent. The number of digits is reduced by one-third in the octal designation and by one-fourth in the hexadecimal designation. For example, the binary number 1111 1111 1111 has 12 digits. It can be expressed in octals as 7777 (four digits) or in hexadecimal as FFF (three digits). Computer manuals invariably choose either the octal or the hexadecimal designation for specifying contents of registers.