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Binary

A computer operates on binary digits which use a base 2 numbering system. To deter-mine the decimal equivalent of a binary number each column is represented by two raised to the power of 0, 1, 2, and so on. For example, the decimal equivalents of 1000 0001 and 0101 0011 are:

Thus 01010011 gives:

(0x128) + (1x64) + (0x32) + (1x16)+ (0x8) + (0x4) + (1x2) + (1x1) = 83

As seen the number of different representations of the binary digits is determined by the number of bits used to represent the value. With a single binary digit we can represent two values (21), with two binary digits we can represent four values (22), and so on. For example:

- 8 bits gives 0 to 2^8-1 (255) different representations.
- 16 bits gives 0 to 2^16-1 (65,535) different representations.
- 32 bits gives 0 to 2^32-1 (4,294,967,295) different representations.

Just as in the decimal system (with units, tens, hundreds, and so on), the most-significant bit (msb) is at the left-hand side of the binary number and the least-significant bit (lsb) on the right-hand side. To convert from decimal (base 10) to binary the decimal value is divided by two recursively and the remainder noted. The first remainder gives the lsb and the last gives the msb. For example:

Thus 54 in decimal is 110110 in binary. Normally computer system use groups of 4 and 8 bits, thus it is important to memorize some of the key values for groups of 4 and 8, such as:

Typical groupings of bits are:

Nibble. A group of four bits. A nibble 16 (24) different combinations of ON/OFF, from 0000 to 1111.
Byte. A group of eight bits. A byte gives 256 (28) different combinations of ON/OFF, from 00000000 to 11111111.
Word. A group of 16 bits (2 bytes). A word gives 65,536 (216) different combinations of ON/OFF, from 0000000000000000 to 1111111111111111.
Long Word. A group of 32 bits (4 bytes). A long word gives 4,294,967,296 (232) different combinations of ON/OFF.

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Digital  computers are digital systems, and operate on binary information(base-2), that is, 0’s and 1’s. Unfortunately, humans have difficulties in converting binary information into a decimal format. Some of the operations within the computer require that the user or programmer defines a binary value. Typically requirements are:

To specify a memory address. Typically memory addresses in a computer are specified with their binary  address, thus there must be a method to display this in a form which the user can easily convert to and from.

To display or set the value of a variable. Sometimes the actual binary  contents of a value needs to be interrogated or set, thus there must be a form in which the user can easily read and convert it into a binary form.

To specify network addresses. A network address of a computer (such as 146.176.151.140) often needs to be converted into a form which the computer understands, thus there must be a conversion between decimal and binary, or vice versa.

These problems are solved by either converting between binary and decimal, or between binary and hexadecimal  (base-16) or octal  (base-8). Without the aid of a calculator, the conversion between binary and decimal is relatively difficult for large binary numbers, but hexadecimal and octal conversion make it easier, as they allow the binary digits to be split into groups of four (for hexadecimal) or three (for octal), and then converted. Hexadecimal is the conversion most often used to specify a memory  address or in defining the contents of a memory address.

Often it is difficult to differentiate binary  numbers from decimal numbers (as one hundred and one can be seen as 101 in binary, and vice-versa). A typical convention is to use a proceeding b for binary numbers, for example 010101111010b and 101111101010b are binary numbers. Hexadecimal and octal  are often used to represent binary digits, as they are relatively easily to convert to and from binary. Table 2 shows the basic conversion between decimal, binary, octal and hexadecimal  numbers. A typical convention is to append a hexadecimal value with an ‘h’ at the end (and octal number with an o). For example, 43F1h is a hexadecimal value whereas 4310o is octal.

To represent a binary  digit as a hexadecimal value, the binary digits are split into groups of four bits (starting from the least-significant bit). A hexadecimal equivalent value then replaces each of the binary groups. For example, to represent 0111 0101 1100 0000b the bits are split into sections of four to give:

Thus, 75C0h represents the binary  number 0111010111000000b. To convert from decimal to hexadecimal  the decimal value is divided by 16 recursively and each remainder noted. The first remainder gives the least-significant digit and the final remainder the most-significant digit. For example, the following shows the hexadecimal equivalent of the decimal number 1103:

Thus, the decimal value 1103 is equivalent to 044Fh.

Table 2: Decimal, binary, octal and hexadecimal conversions