Data Representation& Programming

Here,

r = radix or base of the number system

n = number of digits in the integer part

m = number of digits in fractional part

d_n–1 = most significant digit (MSD)

d_-m = least significant digit (LSD)

On the basis of number of different symbols used, number systems are classified as:

∙   Decimal number system

∙   Binary number system

∙   Octal number system

∙   Hexadecimal number system

1.1.   Decimal number system:

The decimal number system is a radix-10 number system as it uses 10 digits from 0 to 9. These are 0,1,2,3,4,5,6,7,8, and 9. All the higher numbers after ‘9’ are represented in terms of these 10 digits only. The radix point is known as the decimal point.

The weights of different digits in a mixed decimal number, starting from the decimal point are 10⁰, 10¹, 10², and so on for the integer part and 10^-1, 10^-2, 10^-3, and so on for the fractional part. The value of a decimal number can be expressed as a sum of various digits multiplied by their place values or weights.

Example :

Let’s we have (453)₁₀ is a decimal number then, 

We can say that ‘3’ is the least significant digit (LSD) and ‘4’ is the most significant digit (MSD).

1.2.   Binary number system:

It has base or radix ‘2’ i.e. it has two numbers 0 and 1 and these base numbers are called “Bits”. In this number system, group of four bits is known as “Nibble” and group of eight bits is known as “Byte”.

4 bits = 1 Nibble, 8 bits = 1 Byte

All larger binary numbers are represented in term of ‘0’ and ‘1’.  The radix point is known as the binary point. These symbols are known as bits (binary digits). It is a positional number system; the weight of bit is defined by its position with base 2.

Starting from the binary point, the weights of different digits in a mixed binary number are 2⁰, 2¹, 2², and so on for the integer part and 2^-1, 2^-2, 2^-3, and so on for the fractional part.

Example 1: 

The decimal number representation of (101101.10101) is?

Solution:  

(101101.10101)₂ = 

1 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰ + 1 × 2^–1 + 0 × 2^–2 + 1 × 2^–3 + 0 × 2^–4 + 1 ×2^–5 

= 32 + 0 + 8 + 4 + 0 +

= (45.65625)₁₀

1.3.   Octal number system:

It has a base of ‘8’ and it possess 8 distinct symbols (0, 1, 2...,7). It is a method of grouping binary numbers in a group of three bits. The independent digits are 0,1,2,3,4,5,6,7. All the higher order number are expressed as a combination of these on the same pattern as the one followed in case of binary and decimal number system described in previous sections. In the octal number system, the radix point is known as the octal point.

The weights for different digits in the octal number system are 8⁰, 8¹, 8², and so on for the integer part and 8^-1, 8^-2, 8^-3, and so on for the fractional part.

Example : 

Convert (367.721)₈ into decimal system.

Solution: 

(367.721)₈ = 3 x 8² + 6 x 8¹ + 7 x 8⁰ + 7 x 8^–1 + 2 x 8^–2 + 1 x 8^–3

                                 = 192 + 48 + 7 + 0.875 + 0.03125 + 0.00195

                                = (247.908)₁₀

1.4.   Hexadecimal number system:

The base for this system is “16”, which requires 16 distinct symbols to represent the numbers. Its 16 basic digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. It is a method of grouping 4 bits. The decimal equivalent of A, B, C, D, E, and F are 10, 11, 12, 13, 14, and 15, respectively.

The weights of different digits in a mixed hexadecimal number are 16⁰, 16¹, 16², and so on for the integer part and 16^-1, 16^-2, 16^-3, and so on for the fractional part.

Example 5:    

Convert (3A.2F)₁₆ into decimal system.

Solution: 

(3A ⋅ 2F)₁₆ = 3 × 16¹ + 10 × 16⁰ +2 × 16^–1 + 15 × 16^–2   

                            = (58.1836)₁₀

1. CONVERSION BETWEEN DIFFERENT NUMBER SYSTEM

        Computer system process binary data, but the information given by the user may be in the form of decimal number, hexadecimal number, or octal number. So, it is required to study the conversion of numbers from one number system to another.

Table 1: Counting in different number system

2.1. OTHER BASE SYSTEM TO DECIMAL 

2.1.1. Binary to Decimal Conversion

A binary number can be converted to decimal equivalent by multiplying each binary digit by its positional weightage.

Example : 

Convert (11101.1011)₂ into decimal.

Solution:

2.1.2. Octal to Decimal Conversion

An octal number can be converted to decimal equivalent by multiplying each octal digit by its positional weightage.

Example :  Convert (6327.4051)₈ into equivalent decimal number.

                   Solution:

(6327.4051)₈ 

= 6 ×8³ +3 × 8² + 2 × 8¹ +7 × 8⁰ + 4 × 8^–1 + 0 x 8^–2 + 5 × 8^–3 + 1 ×8^–4

= (3287.5100098)₁₀

Thus,

(6327.4051)₈ = (3287.5100098)₁₀

          2.1.3. Hexadecimal to Decimal Conversion

A hexadecimal number can be converted to decimal equivalent by multiplying each hexadecimal digit by its positional weightage.

           Example 8:

           Convert (5C7)₁₆ into decimal

           Solution:

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