Unit-5 :Data Representation (Notes of TU BSc. 4th Year Computer Science)

Data Representation

Syllabus: Number system (Binary, Octal, Decimal, and Hexadecimal); Conversion from One Number System to Another; Binary Arithmetic (Addition and Subtraction); Logic gates

Number System

A number system defines a set of values to represent quantity. It is a set of rules to represent a number. There are 2 types of number systems,

1. Non-Positional Number System
2. Positional Number System

Binary, Octal, Decimal, and Hexadecimal number systems are positional number systems. So in this chapter, we’ll learn about the different positional number systems.

In each number system, the base is the total number of digits or symbols in that number system. The maximum value of single-digit in each number system is one less than that of the base. ( That also means that the base of any number system is one more than that of the maximum value of single-digit. )

Binary Number System

The binary number system is a positional number system. 

There are only two symbols or digits (i.e., 0 and 1) in Binary Number System; hence its base is  2.

The maximum value of single-digit is 1. Each position of a digit represents a specific power of the base (2).

This number system is used in computers. 

Eg. (10111001)2

Octal Number System

In the octal number system, there are 8 symbols or digits (i.e., 0, 1, 2, 3, 4, 5, 6, and 7); hence its base is 8. 

The maximum value of single-digit is 7. Each position of a digit represents a specific power of the base (8).

Eg. (62340)8 

Decimal Number System

In the decimal number system, there are 10 symbols or digits (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9); hence its base is 10. 

The maximum value of single-digit is 9. Each position of a digit represents a specific power of the base (10).

We use this number system in our day-to-day life.

Eg. (28940583)10

Hexadecimal Number System

Hexadecimal number system has 16 symbols or digits (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F) hence its base is 16. 

The symbols A, B, C, D, E, and F represent 10, 11, 12, 13, 14, and 15, respectively.

The maximum value of single-digit is 15. Each position of a digit represents a specific power of the base (16).

Eg. (A456FC)16

Conversion from One Number System to Another

Converting a number of another base to decimal number:

Step-1: Determine the column (positional) value of each digit

Step-2: Multiply the obtained column values by the digits in the corresponding columns

Step-3: Calculate the sum of these products

Converting decimal number to a number of another base:

Step-1: Divide the decimal number to be converted by the value of the new base

Step-2: Record the remainder from step-1 as the rightmost digit (least significant digit) of the new base system

Step-3: Divide the quotient of the previous by the new base

Step-4: Record the remainder from step-3 as the next digit (to the left) of the new base number

Step-5: Repeat step-3 and step-4, recording the remainders from right to left, until the quotient becomes zero in step-3.

Note: The last remainder obtained will be the Most Significant Digit (MSD) of the new base number. 

Converting a number of some base to a number of another base:

Step-1: Convert the original number to the Decimal number

Step-2: Convert the Decimal number obtained to the new base number

Shortcut Methods

Conversion from binary to octal:

Step-1: Divide the binary digits into the group of three (starting from the right),

Step-2: Convert each group of three binary digits to one octal digit.

Example,

Step no.	Binary number	Octal number
1	11001112	12 1002 1112
2	11001112	184878
3	11001112	1478

Therefore, (1100111)₂ = (147)₈

Conversion from octal to binary:

Step-1: Convert each octal number to binary equivalent (make the digit count 3 like 1₈ = 001₂ )

Step-2: Combine converted binary numbers according to the position of the octal number

Step no.	Octal number	Binary number
1	1478	12 42 72
2	1478	001210021112
3	1478	11001112

Therefore, (147)₈ = (1100111)₂

Conversion from binary to hexadecimal:

Step-1: Divide the binary digits into a group of four (starting from the right),

Step-2: Convert each group of four binary digits to one hexadecimal digit.

Example,

Step no.	Binary number	Hexadecimal number
1	11001112	1102 01112
2	11001112	616716
3	11001112	6716

Therefore, (1100111)₂ = (67)₁₆

Conversion from hexadecimal to binary:

Step-1: Convert each Hexadecimal number to Binary equivalent (make the digit count 3 like 6₁₆ = 0110₂ )

Step-2: Combine converted Binary numbers according to the position of Hexadecimal number

Step no.	Hexadecimal number	Hexadecimal number
1	6716	6716
2	6716	616716
3	6716	01102 01112

Therefore, (67)₁₆ = (1100111)₂ 

Binary Arithmetic

Binary Number System consists of 2 digits. They are 0 and 1. As like regular Decimal Numbers, we can perform addition, subtraction, multiplication as well as division. But in this chapter, we’ll only perform addition and subtraction. 

Addition

The addition is combining two digits to produce another higher value digit. 

The rule for Binary Addition is:

0	+	0	0
0	+	1	1
1	+	0	1
1	+	1	0 ( plus a carry of 1 to next higher column )

You can learn Addition in YouTube by clicking here.

Subtraction

Subtraction is the process of taking away some value from a digit resulting in the production of lower value digit.

The rule for Binary Subtraction is:

0	-	0	0
0	-	1	1 ( with borrow from the next column )
1	-	0	1
1	-	1	0

You can learn Subtraction in YouTube by clicking here.

Decimal Number System	Binary Number System
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001

Logic Gates

Logic Gates are the electronic circuits that operate on one or more input signals. They are building blocks of all the circuits in a computer. Some of the basic and most useful logic gates are AND, OR, NOT, NAND, and NOR gates.

AND gate

The physical realization of logical multiplication (AND) operation. 

Generates an output signal 1 only if all input signals are also 1.

OR gate

The physical realization of logical addition (OR) operation.

Generates an output signal of 1 if at least one of the input signal is also 1.

NOT gate

The physical realization of complementation operation.

Generates an output signal, which is the reverse of the input signal.

NAND gate

Complemented AND gate. 

Generates an output signal of 1 if any one of the inputs is a 0 and generates 0 when all the inputs are 1.

NOR gate

Complemented OR gate.

Generates an output signal of 1 only when all inputs are 0 and generates 0 if any one of the inputs is a 1.