# Lecture.5. Number System:

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1 Number System: Lecturer: Dr. Laith Abdullah Mohammed Lecture.5. The number system that we use in day-to-day life is called Decimal Number System. It makes use of 10 fundamental digits i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Computer doesn't make use of decimal number system. Internally it makes use of another number system, called Binary Number system. Binary number system is the fundamental base of computers. It makes use of two digits only i.e. 0 and 1. All other numbers in binary number system are formed using these digits. A bit is the smallest unit of data that a binary computer can recognize. Therefore, the input you enter via a keyboard, the software program you use to play your music collection, the term paper stored on your computer, and the digital photos located on your mobile phone are all just groups of bits. Representing data in a form that can be understood by a digital computer is called digital data representation. a kilobyte (KB) =1,024 bytes; a megabyte (MB) =1 million bytes; a gigabyte (GB) =1 billion bytes; a terabyte (TB) is about 1 trillion bytes; a petabyte (PB) is about 1,000 terabytes (2 50 ) ; an exabyte (EB) is about 1,000 petabytes (2 60 ) ; a zettabyte (ZB) is about 1,000 exabytes (2 70 ) ; a yottabyte (YB) is about 1,000 zettabytes (2 80 ).

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3 Conversion Of Decimal Number Into Binary Number: A given decimal number can be converted into equivalent binary number by following method: 1. Since binary number system makes use of only two digits hence divide the decimal number by 2. Write the quotient below the number and remainder on the right hand side. 2. Now divide the quotient by 2 and repeat above-mentioned process till the time quotient becomes Now arrange the remainders (x1, x2, x3. etc.) in reverse order (. x3, x2, x1). The number thus obtained will be binary equivalent of given decimal number. To understand the procedure of conversion, consider following examples: Example -1 Convert decimal number (109) 10 to equivalent binary number. Thus above mentioned process yields: (109) 10 = ( ) 2

4 How do you convert fractional numbers (that have decimal point in them) into equivalent binary numbers? 1. Since binary number system consists of only two digits hence multiply fractional decimal part by 2. When you do so, integer part obtained on the left hand side of the decimal number will either be 1 or 0. Whatever you get, write it separately (say you write x1. Here x can either be 1 or 0). 2. Now take the resultant fractional part and multiply it by 2 again. Repeat the process, as many times as number of digits are required on the right hand side of the binary point ( thus obtaining x2, x3, x4 etc.). For example, if 4 digits are required on the right hand side of the binary point, repeat above-mentioned process 4 times. 3. Arrange the digits that were written separately in the sequence, in which they were obtained (e.g. x1, x2, x3, and x4). The number thus obtained (x1x2x3x4) will be binary equivalent of given fractional decimal number. Example (1): Convert decimal number (0.862) 10 to equivalent binary number = = = = = = Thus above mentioned process yields: (0.862) 10 = ( ) 2 Example (2) Convert decimal number ( ) 10 to equivalent binary number. First Step: Integer part of the number is taken: Second Step: In second step, fractional part of the number is taken: = = = Number = (0.0111) = Third Step: Joining both the parts together will result in following number: Thus (122) 10 = ( ) 2 ( ) 10 = ( ) 2

5 Converting Binary Numbers into Equivalent Decimal Numbers Positional values for different positions in binary number are illustrated below: Perform following steps to convert a binary number into equivalent decimal number: 1. Multiply each digit of the number, by its positional value. 2. Now add all the products to get the sum. The sum, thus obtained will be the equivalent decimal number of given binary number. Example: Convert ( )2 into equivalent decimal number. Solution: ( ) 2 = ( ) 10 = ( ) 10 = (102) 10 ONE'S COMPLEMENT: In computers, one's complement is used for representing negative numbers. When 0's in a binary number are replaced by 1s and 1s are replaced by 0s then the resultant number is said to be 1's complement of the number. For example: 1's complement of will be Similarly 's one's complement will be

6 BIT: Computer stores all its data in the form of 0s and 1s. Digit 0 and 1 and called Bits. That means 0 is one bit and 1 is another bit. The data 1100 comprises of 4 bits. You can say that bit is the smallest storage unit of computer. BYTE: 8 bit put together form 1 byte. For example, will occupy one byte space. WORD: Number of bits that are used for representing a character within the computer are called word. Note that in most of the computers 8 bit are used to represent a character. Thus in those computers words comprises of one byte. Coding Systems for Text-Based Data While numeric data is represented by the binary numbering system, text-based data is represented by binary coding systems specifically developed for text-based data namely, ASCII, EBCDIC, and Unicode. These codes are used to represent all characters that can appear in text data such as numbers, letters, and special characters and symbols like the dollar sign, comma, percent symbol and many mathematical characters.

7 ASCII and EBCDIC ASCII (American Standard Code for Information Interchange) is the coding system traditionally used with personal computers. ASCII is a 7-digit (7-bit) code, although there are several different 8-bit extended versions of ASCII that contain additional symbols not included in the 7-bit ASCII code. EBCDIC (Extended Binary-Coded Decimal Interchange Code) was developed by IBM, primarily for use with mainframes. EBCDIC represent each character as a unique combination of 8 bits. One group of 8 bits (1 byte) allows 256 (28) unique combinations. Therefore, an 8-bit code can represent up to 256 characters (twice as many as a 7-bit code) enough to include the characters used in the English alphabet, as well as some non-english characters, the 10 decimal digits, the other characters usually found on a keyboard, and many special characters not included on a keyboard such as mathematical symbols, graphic symbols, additional punctuation marks, and other symbols.

8 ASCII and EBCDIC binary codes for typical keyboard symbols.

9 Unicode Unlike ASCII and EBCDIC, which are limited to only the Latin alphabet used with the English language, Unicode is a universal international coding standard designed to represent text-based data written in any ancient or modern language, including those with different alphabets, such as Chinese, Greek, Arabic, Tibetan, and Russian. Unicode uniquely identifies each character using 0s and 1s, no matter which language, program, or computer platform is being used. It is a longer code, consisting of 1 to 4 bytes (8 to 32 bits) per character, and can represent over one million characters, which is more than enough unique combinations to represent the standard characters in all the world s written languages, as well as thousands of mathematical and technical symbols, punctuation marks, and other symbols and signs. The biggest advantage of Unicode is that it can be used worldwide with consistent and unambiguous results. Unicode is quickly replacing ASCII as the primary text-coding system. In fact, Unicode includes the ASCII character set so ASCII data can be converted easily to Unicode when needed. Unicode is used by most Web browsers and is widely used for Web pages and Web applications (Google data, for instance, is stored exclusively in Unicode). Most recent software programs, including the latest versions of Microsoft Windows, Mac OS, and Microsoft Office, also use Unicode, as do modern programming languages, such as Java and Python. Unicode is updated regularly to add new characters and new languages not originally encoded the most recent version is Unicode 5.2.

10 Selected Unicode codes.

11 Graphics Data Graphics data consists of still images, such as photographs or drawings. One of the most common methods for storing graphics data is in the form of a bitmap image an image comprised of a grid of small dots, called pixels (short for picture elements), that are colored appropriately to represent an image. The color to be displayed at each pixel is represented by some combination of 0s and 1s, and the number of bits required to store the color for each pixel ranges from 1 to 24 bits. For example, Each pixel in a monochrome graphic can be only one of two possible colors (such as black or white). These monochrome images require only one bit of storage space per pixel (for instance, the bit would contain a 1 when representing a pixel that should display as white, and the bit would contain a 0 for a pixel that should display as black). Images with more than two colors can use 4, 8, or 24 bits to store the color data for each pixel this allows for 16 (24), 256 (28), or 16,777,216 (224) colors respectively The number of bits used per pixel depends on the type of image being stored; for instance, The JPEG images taken by most digital cameras today use 24-bit true color images.

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