Numbering Systems (Part 1) PDF

 NUMERIC REPRESENTATION Decimal Binary Hex Two-state binary system consists 00 00000000 00 of only two digits called bits. 01 00000001 01 ▪ On = 1; negat...

 NUMERIC REPRESENTATION Decimal Binary Hex Two-state binary system consists 00 00000000 00 of only two digits called bits. 01 00000001 01 ▪ On = 1; negative charge. 02 00000010 02 ▪ Off = 0; no charge. 03 00000011 03 04 00000100 04 Byte = 8 bits grouped together. 05 00000101 05 Hexadecimal system. 06 00000110 06 ▪ Uses 16 digits to represent 07 00000111 07 08 00001000 08 binary numbers. 09 00001001 09 ▪ (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10 00001010 0A C, D, E, F) 11 00001011 0B 12 00001100 0C 13 00001101 0D 14 00001110 0E 15 00001111 0F INTRO. TO CS, DR. DALIA RIZK 2  CHARACTER ENCODING ASCII. ▪ American Standard Code for Information Interchange. ▪ Used by personal computers. EBCDIC. ▪ Extended Binary coded Decimal Interchange Code. ▪ Used by mainframe computers. Unicode. ▪ New encoding due to explosion of the Internet. ▪ Can be written in U T F-16 or U T F-8. ▪ Recognized by virtually all computer systems. INTRO. TO CS, DR. DALIA RIZK 3  ASCII CODES ▪ Computers only understand numbers. ▪ ASCII code is the numerical representation of a character such as ‘b' or '@‘ ▪ The ASCII code takes each character on the keyboard and convert it to a binary number Example: ▪ The letter ‘a’ has the binary number 0110 0001 (this is the decimal number 97) ▪ The letter ‘A’ has the binary number 0100 0001 (this is the decimal number 65) INTRO. TO CS, DR. DALIA RIZK 4 ASCII CODES INTRO. TO CS, DR. DALIA RIZK 5  UNICODE ▪ It is an international coding system designed to be used with different language scripts. ▪ Each character or symbol is assigned a unique numeric value, largely within the framework of ASCII. ▪ Unicode provides a unique number for every character, no matter what the platform, no matter what the program, no matter what the language. INTRO. TO CS, DR. DALIA RIZK 6  NUMBERING SYSTEMS IN COMPUTER There are different types of number systems, in which the four main types are: ▪ Binary number system(Base -2) ▪ Octal number system(Base -8) ▪ Decimal number system(Base -10) ▪ Hexadecimal number system(Base -16) INTRO. TO CS, DR. DALIA RIZK 7  NUMBERING SYSTEMS IN COMPUTER Uses of Binary number system: ▪ To represent pixels of an image. ▪ To represent On and OFF in a circuit. ▪ To represent True and False statements. ▪ To represent ASCII codes Uses of Octal Numbers: ▪ In UNIX operating system ▪ In Computing Graphics ▪ In File Protection INTRO. TO CS, DR. DALIA RIZK 8  NUMBERING SYSTEMS IN COMPUTER Uses of the Decimal system: ▪ Financial calculations ▪ Calendar numbers ▪ Counting numbers Uses of Hexadecimal: ▪ Memory locations ▪ De-bugging ▪ To define colors on web pages (red color is represented as ff0000) INTRO. TO CS, DR. DALIA RIZK 9  BINARY NUMBERS SYSTEMS Binary consists of two units 0 and 1, known as bit. ▪ 0 bit means NO (False) while 1 bit means YES (True). ▪ These bits are combined in a group of 8 to form 1 byte, to represent multiple characters and values. ▪ One byte can represent 256 values depending on its arrangement of bit units. ▪ There is no representation for negative integers because we only have 0 and 1. INTRO. TO CS, DR. DALIA RIZK 10  DECIMAL TO BINARY CONVERSION Convert the following number 59 into its equivalent binary form: Step 1:Repeat a division of 59 by 2 and get remainders 59/2 = 29 r 1 29/2 = 14 r 1 Step 2: Arrange from top into right moving to 14/2 = 7 r 0 left, then complete the (8 bit) byte 7/2 = 3 r 1 with zeros 3/2 = 1 r 1 1/2 = 0 r 1 Result = 00111011 INTRO. TO CS, DR. DALIA RIZK 11  BINARY TO DECIMAL CONVERSION Convert the following binary number 10010101 into its equivalent decimal form: Step 1:Multiply each digit from right to left by powers of 2 Step 2: Add results together 128+0+0+16+0+4+0+1 Result = 149 INTRO. TO CS, DR. DALIA RIZK 12  DECIMAL TO OCTAL CONVERSION ▪ Octal Number System has a base of eight and uses the set of values = {0,1,2,3,4,5,6,7} Convert the decimal number (560)10 to its equivalent octal form: Step 1: Repeat a division of decimal 560 by 8 and get remainder (remainder should be between 0 and 7) 560/8 = 70 r 0 70/8 = 8 r 6 Step 2: Arrange digits from top into right 8/8 = 1 r 0 moving to left 1/8 = 0 r 1 Result = (1060)8 INTRO. TO CS, DR. DALIA RIZK 13  OCTAL TO DECIMAL CONVERSION Convert the octal number (215)8 to its equivalent decimal form: Step 1:Multiply each digit from right to left by powers of 8 Step 2: Add results together Result = 128 + 8 +5 = (141)10 INTRO. TO CS, DR. DALIA RIZK 14  OCTAL TO BINARY CONVERSION Convert the octal number (540)8 to its equivalent binary form: Step 1: Put each digit in its equivalent three binary bits from right to left Step 2: Arrange binary numbers from right to left Result = (101100000)2 INTRO. TO CS, DR. DALIA RIZK 15  BINARY TO OCTAL CONVERSION Example: Convert the following binary number (0100010)2 to its equivalent octal form: Step 1: Divide the binary bits into groups of three starting from right to left (complete with zeros for most significant bits) Step 2: Get the octal number of each three bits Step 3: Arrange digits in their octal form from right to left Result = (42)8 INTRO. TO CS, DR. DALIA RIZK 16  HEXADECIMAL NUMBER SYSTEM ▪ Hexadecimal number system has the base of sixteen and has a set of values = {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} ▪ Equivalent hexadecimal and binary numbers for decimal numbers are shown in table below: INTRO. TO CS, DR. DALIA RIZK 17  DECIMAL TO HEXADECIMAL CONVERSION Convert the decimal number (960)10 to its equivalent hexadecimal form: Step 1: Repeat a division of 960 by 16 and get remainder 960 / 16 = 60 r 0 60 / 16 = 3 r 12 ------(12 is equivalent to C) 3 / 16 = 0 r 3 Step 2: Arrange remainder digits from top into right moving to left, and substitute the equivalent hexadecimal values Result = (3C0)16 INTRO. TO CS, DR. DALIA RIZK 18  HEXADECIMAL TO DECIMAL CONVERSION Convert the hexadecimal number (7CF)16 to its equivalent decimal form: Step 1: Multiply each digit from right to left (after hexadecimal substitution) by powers of 16 7= 7 , C= 12 , F=15 Step 2: Add results together = (7 * 256) + (12 * 16) + (15 * 1) = 1792 + 192 + 15 = 1999 Result = (1999)10 INTRO. TO CS, DR. DALIA RIZK 19  BINARY TO HEXADECIMAL CONVERSION Convert the binary number (11100011)2 to its equivalent hexadecimal form: Step 1: Divide the binary number into groups of four bits starting from right to left = 1110 0011 Step 2: Get the equivalent hexadecimal for each four bits (convert binary to decimal then get its equivalent hexadecimal) Step 3: Arrange hexadecimal digits from right to left Result = (E3)16 INTRO. TO CS, DR. DALIA RIZK 20  HEXADECIMAL TO BINARY CONVERSION Convert the hexadecimal number (A2B)16 to its equivalent binary form: Step 1: Put each hexadecimal digit in its binary four bits starting from right to left (after substitution: A = 10 , 2 = 2 , B =11) Step 2: Arrange the binary numbers from right to left = (101000101011)2 Result = (101000101011)2 INTRO. TO CS, DR. DALIA RIZK 21  HEXADECIMAL TO OCTAL CONVERSION Convert the hexadecimal number (1BC)16 to its equivalent octal form: Step 1: Put each hexadecimal digit in its binary four bits starting from right to left, and after substitution: 1= 1, B = 11, C = 12 (1 1011 1100)2 Step 2: Divide the resulted binary bits into groups of three bits and get their octal number = 110 111 100 Step 3: Arrange digits in their octal form from right to left Result = (674)8 INTRO. TO CS, DR. DALIA RIZK 22  OCTAL TO HEXADECIMAL CONVERSION Convert the octal number (56)8 to its equivalent hexadecimal form: Step 1: Put each octal digit in its binary three bits form from right to left= 101 110 Step 2: Divide the resulted binary bits into groups of four bits and get its hexadecimal number = 10 1110 Step 3: Arrange digits in their octal form from right to left Result = (2E)16 INTRO. TO CS, DR. DALIA RIZK 23