Representation of Number of Different Radix

Representation of Number of Different Radix

1. Decimal Number System

• In decimal number system we can express any decimal number in units, tens, hundreds, thousands and so on.

• When we write a decimal number say, 5678.9, we know it can be represented as

5000 + 600 + 70 + 8 + 0.9 = 5678.9

• The decimal number 5678.9 can also be written as 5678.910, where the 10 subscript indicates the radix or base.

• The position of a digit with reference to the decimal point determines its value/weight. The sum of all the digits multiplied by their weights gives the total number being represented.

• The leftmost digit, which has the greatest weight is called the most significant digit and the rightmost digit, which has the least weight, is called the least significant digit.

• Fig. 1.2.1 shows decimal digit and its weights expressed as a power of 10.

2. Binary  Number System

• Binary system with its two digits is a base-two system.

• The two binary digits (bits) are 1 and 0.

• In binary system, weight is expressed as a power of 2.

• The Fig. 1.2.2 (a) shows representation of binary number 1101.101 in power of 2.

• By adding each digit of a binary number in a power of 2 we can find the decimal equivalent of the given binary number. 

3. Octal Number System

• The octal number system uses first eight digits of decimal number system : 0, 1, 2, 3, 4, 5, 6 and 7. As it uses 8 digits, its base is 8.

• For example, the octal number 475.25 can be represented in power of 8 as shown in Fig. 1.2.2 (b).

• By adding each digit of an octal number in a power of 8 we can find the decimal equivalent of the given octal number.

4. Hexadecimal Number System

• The hexadecimal number system has a base of 16 having 16 characters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

• Table 1.2.1 shows the relationship between decimal, binary, octal and hexadecimal.

• For example, 3FD.84 can be represented in power of 16 as shown below.

• By adding each digit of a hexadecimal number in a power of 16 we can find decimal equivalent of the given hexadecimal number. 

5. Format of a Binary Number

• A single digit in the binary number is called bit.

• The following figure shows the format of binary number. Four binary digits form a nibble, eight binary digits form a byte, sixteen binary digits form a word and thirty-two binary digits form a double-word.

Nibble : 4-bits can represent 2⁴ = 16 distinct values

Byte : 8-bits can represent 2⁸ = 256 distinct values 

Word : 16-bits can represent 2¹⁶ = 65536 distinct values

Double word : 32-bits can represent 2³² = 4294967296 distinct values

6. Counting in Radix (Base) r

• Each number system has r set of characters. For example, in decimal number system r equals to 10 has 10 characters from 0 to 9, in binary number system r equals to 2 has 2 characters 0 and 1 and so on.

• In general we can say that, a number represented in radix r, has r characters in its set and r can be any value. This is illustrated in Table 1.2.2.

Ex. 1.2.1 Find the decimal equivalent of (231.23)₄

Sol. :

N = 2 × 4² + 3 × 4¹ + 1 × 4⁰ + 2x4^-1 + 3 × 4^-2

= 32+12+1 +0.5 + 0.1875

= 45.6875₁₀

Ex. 1.2.2 Count from 0 to 9 in radix 5.

Sol. :

The Table 1.2.2 indicates that radix 5 has 5 characters. A count sequence from 0 decimal to 9 decimal

00, 01, 02, 03, 04, 10, 11, 12, 13, 14

Ex. 1.2.3 What is the largest binary number that can be expressed with 12-bits? What is the equivalent decimal and hexadecimal ?

Sol. :

(1111 1111 1111)₂, (4095) ₁₀, (FFF) _H .