Number Systems

Number Systems

• Number system is a basis for counting various items.

• The decimal number system has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

• Modern computers communicate and operate with binary numbers which use only the digits 0 and 1.

• When decimal quantities are represented in the binary form, they take more digits.

• For large decimal numbers people have to deal with very large binary strings and therefore, they do not like working with binary numbers. This fact gave rise to three new number systems : Octal, Hexadecimal and Binary Coded Decimal (BCD).

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 ad 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.

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

N = 1×2³ + 1×2² + 0×2¹ + 1×2⁰    +    1×2^–1  + 0×2 ^{– 2} + 1×2^–3    = (13.625)₁₀

• 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 5632.471 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.

N = 5×8³ + 6×8² + 3×8¹ + 2×8⁰      +     4×8^–1 + 7×8^–2 + 1×8^–3 = (2970.611328)₁₀

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.

N = 3 × 16² + F × 16¹ + D × 16⁰       +   8 × 16^–1 + 4 ×16^–2

     = 3 × 16² + 15 × 16¹ + 13 × 16⁰    +   8 × 16^–1 + 4 ×16^–2   = (1021.515625)₁₀

• 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.

Example: 1

Find the decimal equivalent of (231.23)₄

Solution :

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

    = 32 + 12 + 1 + 0.5 + 0.1875 = 45.6875₁₀

Example: 2

Count from 0 to 9 in radix 5.

Solution :

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

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

Example: 3

What is the largest binary number that can be expressed with 12–bits? What is the equivalent decimal and hexadecimal ?

Solution :

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

Review Questions

1. Explain various number systems.

2. Name the number system used in computers.