Number System

Number System

A number system relates quantities and symbols.In digital system how information is represented is key and there are different radices, i.e. number bases, that a numbering system can use.

Digital computer

Any class of devices capable of solving problems by processing information in discrete form.It operates on data, including letters and symbols,that are expressed in binary form i.e using only two digits 0 and 1.

The block diagram of digital computer is given below:

The memory unit stores programs as well as input, output and intermediate data. The processor unit performs arithmetic and other data processing tasks as specified by the program.The control unit supervises the flow of information between various units. The program and data prepared by the user are transferred into the memory unit by means of an input device such as punch card reader (or) tele typewriter. An output device, such as printer, receives the result of the computations and the printed results are presented to the user.

Number Representation

It can have different base values like: binary (base-2), octal (base-8), decimal (base 10) and hexadecimal (base 16),here the base number represents the number of digits used in that numbering system.

As an example, in decimal numbering system the digits used are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Therefore the digits for binary are: 0 and 1, the digits for octal are: 0, 1, 2, 3, 4, 5, 6 and 7. For the hexadecimal numbering system, base 16, the digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Binary numbers

Numbers that contain only two digit 0 and 1 are called Binary Numbers. Each 0 or 1 is called a Bit, from binary digit. A binary number of 4 bits is called a Nibble. A binary number of 8 bits is called a Byte. A binary number of 16 bits is called a Word on some systems, on others a 32-bit number is called a Word while a 16-bit number is called a Halfword.

• Using 2 bit 0 and 1 to form
• a binary number of 1 bit, numbers are 0 and 1
• a binary number of 2 bit, numbers are 00, 01, 10, 11
• a binary number of 3 bit, such numbers are 000, 001, 010, 011, 100, 101, 110, 111
• a binary number of 4 bit, such numbers are 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100,1101,1110,1111

Therefore , using n bits there are 2n binary numbers of n bits

Each digit in a binary number has a value or weight. The LSB has a value of 1. The second from the right has a value of 2, the next 4 , etc.,

The binary equivalent for some decimal numbers are given below.

Octal Number

Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0, 1, 2, 3, 4, 5, 6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits.

Decimal Numbers

Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.

Hexa-decimal Numbers

A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 14, E is 15 and F is 16.

Number Base Conversions

1. Conversion of decimal number to any number system

1. convert the integer part by doing successive division using the radix of asked number systems.
2. convert the fractional part by doing successive multiplication using radix of asked number system

2. Conversion of decimal to binary number system

• The radix of asked number system is 2
• Convert (87)₁₀ to (   )₂