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Digital Number System

In this number system, we are going to learn about various methods which are commonly used. The most commonly used systems are binary, octal, decimal and hexadecimal. Let us discuss them in detail.
Note: The word Digit is derived from the Latin word Digitus meaning finger or toe.

Decimal Number system

  • The most commonly used number system is the decimal number system.

  • It is composed of digits 0(zero) through 9(nine).

  • The fact that this system has ten digits is commonly attributed to the ten fingers of man.

  • The decimal system is the example of a system which employs positional notations.

  • In such a system the position of each digit in a number determines its value or weight.

  • For example

374.29

Is a shorthand notation for

 

(3 * 102) + (7 * 101) +(4 * 100) +(2 * 10-1) +(9 * 10-2)

  • Obviously, the position of any digit determines the power of 100 by which the digit is multiplied
  • In this example, 3 is in the hundreds (102) position, 7 is in the tens (101) position, 4 is the ones (100) position, 2 is in the tenths (10-1) and 9 is in the hundredths (10-2) position.
  • The radix or base of the decimal number system is 10 three important characteristics of a number systems having positional notation are
    • The number of digits is equal to the base
    • The largest number is one less than the base
    • Each digit is multiplied by the base raised to the appropriate power of digit position.
  • ​In the above example, a radix point is used to separate the integer art of the number from the fractional part of the number.
  • Representing this number in this manner is very useful in conversion between bases.​

Binary Number System

  • The most widely used number system in digital system is binary number system.
  • Bi means two
  • It uses only two representations 0 and 1 which make it useful for digital systems.
  • Where the on and off positions of switches are used to build up data.
  • Hence the radix is 2.
  • The digits in the numbers in binary are given a special name BITS,
  • The binary system is also a positional value system, where in each binary digit has its own value or weight expressed as a power of 2.
  • Here, places to the left of the binary point are positive powers of 2 and places to the right are negative powers of 2.
  • For example,

10112

Is equivalent in decimal terms to?

(1 * 23) + (0 * 22) + (1* 21) + (1 * 20) = 112

Let us consider a fractional example

1011.11012

is equivalent in decimal terms to

(1 * 23) + (0 * 22) + (1* 21) + (1 * 20) + (1 * 2-1) + (1 * 2-2) + (0* 2-3) + (1 * 2-4) = 11.812510.

  • The subscript of the above example indicates the base of the number.
  • These are often omitted when the base of the number is understood.
  • It is to be noted that the radix point separates the integer and fractional parts of the number in a manner identical to that of the decimal point of the previous section.
  • Also the conversion of binary to decimal is straight forward and easily performed using positional notation format.
  • For example

Octal Number system

  • The octal number system is an important system which is often used in microcomputers.

  • It has a base of 8 and employs the digits 0 through 7.

  • To illustrate the positional value for this system, consider the octal number

673.188

The equivalent decimal number is given by

(6 * 82) + (7 * 81) + (3* 80) + (1 * 8-1) + (2* 8-2) = 1443.156258

  • As in the binary and decimal number systems, the radix point separates the integer and fractional parts of the number.

Hexadecimal Number System

  • The hexadecimal number system is another system often used in the microcomputers.

  • It has a base of 16 which requires sixteen digits.

  • The digits used are 0 through 9 and A through F.

  • The symbol A through F represents the equivalent decimal numbers of 10 through 15.

  • A counting sequence of the system is shown in the figure

Note: Hexadecimal number system is also called as alphanumeric number system as it uses both alphabet and numbers to represent the digits.
  • The positional value of the hexadecimal system can be demonstrated using the hexadecimal system.
  • Let us consider an example

F3D.C816

The decimal equivalent of this number is given by
(15 * 162) + (3 * 161) + (13* 160) + (12 * 16-1) + (8* 16-2) = 3901.7812516
  • The radix point, as in previously described system, separates the integer and fractional parts of the number.
  • For example





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