A system to represent or express numbers.

2.3 NUMBER SYSTEM

We are familiar with decimal number system which uses ten distinct symbols from 0…9, and has base 10. In the decimal number system a number n4n3n2n1 is interpreted as n4×103 +n3×102 +n2×101 +n1×100 . Thus decimal number 5632 represents 5000+600+30+2. It is a weighted code system since numbers 5632, 2563, 3562, 6532 all represent different quantities despite the fact that all of them use the same symbols (2,3,5,6). The magnitude/ value of a number is determined both by the symbols used and the places at which they are present. Thus, symbol 3 at ten’s place represent 30, but when written at thousands’ place it represent 3000. Although we use only the decimal number system in everyday applications but there are many other number systems possible. In fact, we can have number system with any base r.

A number system with base r will have r distinct symbols, from 0 to r-1. Binary number system (r = 2), octal number system (r = 8) and hexadecimal number system (r = 16) are some of the frequently used number systems in computer science. Binary number system has two distinct symbols 0 & 1; Octal has seven distinct symbols 0,1,2,3,4,5,6,7; and Hexadecimal number system has sixteen distinct symbols namely 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. The numbers written in a particular number system can be transformed to an equivalent value in a different number system. For example a number 3F in hexadecimal is equivalent to 63 (3×161 + F × 160 ) in decimal number system. And similarly a number 302 in octal is equivalent to 194 (3× 8 2 + 0 × 8 1 +2 × 8 0 ) in decimal number system

2.3.1 Binary Number System

As stated above, the binary number system has base 2 and therefore uses only two distinct symbols 0 and 1. Any number in binary is written only by using these two symbols. Though it uses just two symbols but it has enough expressive power to represent any number. A binary number 1001 thus represents the number 1× 2 3 + 0 × 2 2 + 0× 2 1 + 1× 2 0 . It is equivalent to number 9 in decimal number system. Similarly 11001 and 10001 represent numbers 25 and 17 respectively in decimal number system. Please note that the base in binary number system is 2

A binary number can be converted to its decimal equivalent by forming the sum of powers of 2 of those coefficients whose value is 1. For example:

The conversion from decimal to binary or to any other base-r system is done by separating the number into an integer part and a fraction part and then converting each part separately. For example the decimal number 41.6875 can be converted to binary equivalent, by converting the integer and fraction parts separately, as follows:

Operation Quotient Remainder

41 / 2 20 1

20 / 2 10 0

10 / 2 5 0 ⬆

5 / 2 2 1 ⬆

2 / 2 1 0 ⬆

1 / 2 0 1 ⬆

The number is divided by 2 and the remainder part is extracted. The quotient obtained is again divided by 2 and this process is repeated until the quotient becomes 0. Every time the remainder obtained is recorded. The set of remainders obtained, read from the bottom to top form the binary equivalent of the integer part of the number. Thus (41)10 = (101001)2.

In order to convert the fraction part, it is multiplied by 2 to obtain resultant integer and fraction parts. The integer part in the resultant is extracted and the fraction part is again multiplied by 2. This multiplication process is repeated till the fraction part becomes 0 or a desired precision is achieved. The integer part of the given number (.6875) can be converted to binary as follows:

Operation Resulting Integer part Resulting Fraction part

0.6875 X 2 1 ↓ .3750

0.3750 X 2 0 ↓ .7500

0.7500 X 2 1 ↓ .5000

0.5000 X 2 1 ↓ .0000

The binary equivalent of fraction 0.6875 is 1011, obtained by reading the integer parts from top to down. Thus, the decimal number 41.6875 is equivalent to 101001.1011 binary number system.

Conversion
Decimal to Binary:

Q. Convert (13)₁₀ to an equivalent binary number.

Quotient Remainder

13/2 6 1

6/2 3 0

3/2 1 1

1/2 0 1

We take the remainder from bottom to top. Hence (13)₁₀ = (1101)₂

Binary To Decimal

Q. Convert (101011)₂ to an equivalent decimal number.

5 4 3 2 1 0

(1 0 1 0 1 1)₂

= 2⁵ + 2³ + 2 ¹ + 2 ⁰= 32 + 8 + 2 + 1 = 43

Hence, (101011)₂ = 43

2.3.2 Binary Codes

We have seen earlier that digital computers use signals that have two distinct values and there exists a direct analogy between binary signals and binary digits. Computers not only manipulate numbers but also other discrete elements of information. The distinct discrete quantities can be represented by a group of binary digits (known as bits). For example, to represent two different quantities uniquely two symbols are sufficient and hence one binary digit (either 0 or 1) will be sufficient to uniquely represent the two symbols. But one bit will not suffice if one has to represent more than two quantities. In those cases more than one bits are required, i.e. the bits have to used repeatedly. For example if we have to give unique codes for three distinct items we need at least 2 bits. With two bits we can have codes 00, 01, 10 and 11. Out of this we can use first three to assign unique codes to three distinct quantities and leave the fourth one unused. In general, an n-bit binary code can be used to represent 2n distinct quantities. Thus group of two bits can represent four distinct quantities through unique symbols 00, 01, 10 & 11. Three bits can be used to represent eight distinct quantities by unique symbols 000, 001, 010, 011, 100, 101, 110 & 111. In other words, to assign unique codes to m distinct items we need at least n bit code such that 2ⁿ>= m.

Digital computers use binary codes to represent all kinds of information ranging from numbers to alphabets. Whether we have to input an alphabet, a number or a punctuation symbol; we need to convey it to machine through a unique code for each item. Thus, the instruction to be performed by the CPU and the input data which form the operands of the instruction are represented using a binary code system. A typical machine instruction in a digital computer system could therefore look like a set of 0s and 1s. Many binary codes are used in digital systems. BCD code for representing decimal numbers, ASCII code for information interchange between computer and keyboard, Unicode for use over Internet and Reflected (Gray) code are some commonly studied binary code systems.

2.3.3 ASCII & Unicode (ASCII is an alphanumeric code used for representing numbers, alphabets, punctuation symbols and other control characters)

An alphanumeric code has to represent 10 decimal digits, 26 alphabets and certain other symbols such as punctuation marks and special characters. Therefore, a minimum of six bits is required to code alphanumeric characters (26 = 64, but 25 = 32 is insufficient). With a few variations this 6 bit code is used to represent alphanumeric characters internally. However, the need to represent more than 64 characters (to incorporate lowercase and uppercase letters and special characters), have given rise to seven- and eight- bit alphanumeric codes. ASCII code is one such seven bit code that is used to identify key press on the keyboard. ASCII stands for American Standard Code for Information Interchange. It's an alphanumeric code used for representing numbers, alphabets, punctuation symbols and other control characters. It’s a seven bit code, but for all practical purposes it’s an eight bit code, where eighth bit is added for parity. Table 2.1 below presents the ASCII code chart.

ASCII codes represent text in computers, communications equipment, and other devices that use text. Most modern character-encoding schemes are based on ASCII, though they support many more characters than did ASCII. Historically, ASCII developed from telegraphic codes. Its first commercial use was as a seven-bit teleprinter code promoted by Bell data services. Work on ASCII formally began on October 6, 1960, with the first meeting of the American Standards Association's (ASA) X3.2 subcommittee. The first edition of the standard was published during 1963 a major revision during 1967, and the most recent update during 1986. ASCII includes definitions for 128 characters: 33 are non-printing control characters (now mostly obsolete) that affect how text and space is processed; 94 are printable characters, and the space is considered an invisible graphic. The most commonly used character encoding on the World Wide Web was US-ASCII until December 2007, when it was surpassed by UTF-8.

Unicode (is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems)

is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems. Developed in conjunction with the Universal Character Set standard and published in book form as The Unicode Standard, the latest version of Unicode consists of a repertoire of more than 107,000 characters covering 90 scripts, a set of code charts for visual reference, an encoding methodology and set of standard character encodings, an enumeration of character properties such as upper and lower case, a set of reference data computer files, and a number of related items, such as character properties, rules for normalization, decomposition, collation, rendering, and bidirectional display order (for the correct display of text containing both right-to-left scripts, such as Arabic and Hebrew, and leftto-right scripts. Unicode can be implemented by different character encodings. The most commonly used encodings are UTF-8 (which uses one byte for any ASCII characters, which have the same code values in both UTF-8 and ASCII encoding, and up to four bytes for other characters), the now-obsolete UCS-2 (which uses two bytes for each character but cannot encode every character in the current Unicode standard), and UTF16 (which extends UCS-2 to handle code points beyond the scope of UCS-2).

The Unicode Consortium, the nonprofit organization that coordinates Unicode's development, has the ambitious goal of eventually replacing existing character encoding schemes with Unicode and its standard Unicode Transformation Format (UTF) schemes, as many of the existing schemes are limited in size and scope and are incompatible with multilingual environments. Unicode's success at unifying character sets has led to its widespread and predominant use in the internationalization and localization of computer software. The standard has been implemented in many recent technologies, including XML, the Java programming language, the Microsoft .NET Framework, and modern operating systems.

👉Check Your Progress 2

1) Choose the best alternative:

(a) ASCII is practically a ………………. bit code:

(i) six (ii) seven (iii) eight (iv) nine

(b) A number system having base r will have ……….. distinct symbols:

(i) r/2 (ii) 2r (iii) r (iv) r2

(i) processing (ii) Internet technologies (iii) logical operations

(d) The binary equivalent of (25.25)10 is:

(i) 11001.001 (ii) 11101.010 (iii) 10001.110

2) State True or False:

(a) A decimal number can be converted into its binary equivalent integer and fraction parts separately.

True False

(b) An n-bit binary code can represent n2 distinct codes.

True False

3) Briefly state the reason why ASCII can not be less than 7 bit code?

4) What was the main motivation behind the development of Unicode?