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Data Representation in Computers Number systems The additive approach – Number earlier consisted of symbols e.g. Roman number system - I for 1, II for 2, III for 3 etc. Positional numbering – Symbols represent different values depending on the position they occupy e.g. the Decimal system Data Representation in Computers/Session 3 / 2 of 33 Decimal Number System 365 = (3 * 100) + (6*10) + (5*1) The value of each digit in the number system is determined by: The digit itself The position of the digit in the number The base/radix of the system (6*10) Base Position number Data Representation in Computers/Session 3 / 3 of 33 Binary Number System The binary number system has a base of two and symbols used are 0 and 1. In this number system, as we move to the left, the value of the digit will be two times greater than its predecessor because the base is two. Thus the value of the places are : 128 64 32 16 8 4 2 1 0001 1110 0101 0111 Most Significant bit Binary Number Least Significant bit Data Representation in Computers/Session 3 / 4 of 33 Octal number systems Uses a base of 8 Binary 000 Values increase 001 from right to left 1, 010 8, 64, 512, 4096 ... Example: 1204 = (1 * 512) + (2 * 64) + (0 * 8) + (4 * 1) = 512 + 128 + 0 + 4 = 644 011 100 101 110 111 Octal 0 1 2 3 4 5 6 7 Data Representation in Computers/Session 3 / 5 of 33 Hexadecimal Number Systems Uses a base of 16 The 16 symbols required for the hexadecimal number system obtainedby using the alphabets A, B, C, D, E and F Example: A0119 = (10 * 65,536)+(0 * 4,096)+(1 * 256)+ ( 1 * 16) + ( 9 * 1) = 6,55,360 + 0 + 256 + 16 + 9 = 6, 55, 641 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Data Representation in Computers/Session 3 / 6 of 33 Octal Binary Conversion Binary to octal : group 3 bits from right to left Otal to binary : each digit is represented by 3 bits Binary Octal 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7 Data Representation in Computers/Session 3 / 7 of 33 Binary Decimal Conversion Binary to decimal: …128<64 <32 <16 <8 <4 <2 <1 Or anan-1….a2a1a0 = an*2n + an-1*2n-1 +….+a2*22+a1*21+a0*20 The decimal equivalent of 110100 is = (1 * 32 ) + (1 * 16) + (0 * 8) + ( 1 * 4) + ( 0 * 2) + (0 * 1) = 32 + 16 + 0 + 4 + 0 + 0 = 52 Data Representation in Computers/Session 3 / 8 of 33 Binary Decimal Conversion Decimal to Binary: Divide the decimal number by 2 Note the remainder in one column and divide the quotient again with 2 Keep repeating this process until quotient is reduced to a zero Reading remainders in the reverse order gives the binary equivalent Data Representation in Computers/Session 3 / 9 of 33 Example E.g. Converting the decimal number 52 to its binary equivalent. 52 |_2_ 0 26 |_2_ 0 13 |_2_ 1 6 |_2_ 0 3 |_2_ 1 1 |_2_ 1 0 Thus the binary equivalent of the decimal number 52 is 11 01 00 Data Representation in Computers/Session 3 / 10 of 33 Binary Hexadecimal conversion Hexa to binary : Each hexadecimal digit is represented by 4 bits Example: 1A412C is represented 0001 1010 0100 0001 0010 1100 Delete digits 0 on the left 110100100000100101100 Binary to hexa: Group 4 digits from right to left Convert groups to 16 base Example: 10101011000010 0010 1010 1100 0010 2 A C 2 Data Representation in Computers/Session 3 / 11 of 33 Data Representation Digital computers use binary code to represent characters. Binary code is made up of binary digits or bits. A string of "0s" and "1s" is used to represent characters. Byte is a sequence of 8 bits. Most computers have words that consist of 8 or 16 bits. In large computers the number of bits per word could be 16 or 32 bits. Data Representation in Computers/Session 3 / 12 of 33 Binary Arithmetic Addition The following rules of binary addition are to be remembered: 0+0=0 0+1=1=1+0 1 + 1 = 0 carry 1 to the next column to the left 1 + 1 + 1 = 1 carry 1 to the next column e.g. Carry 1 1 1 1 1 10 11 + 1 11 10 00 10 Data Representation in Computers/Session 3 / 13 of 33 Complementary Subtraction Three steps to perform subtraction : Find the complement of the number you are subtracting To the complement of the number add the number we are subtracting from If there is a carry of 1 add the carry to the result of the addition Else re-complement the sum and attach a negative sign e.g. Number 10 00 11 01 00 10 10 10 Complement 01 11 00 10 11 01 01 01 Data Representation in Computers/Session 3 / 14 of 33 Complementary Subtraction (Contd.) Example of subtraction : e.g. 1010101 - 1001100 Step 1. Find the complement of 1001100 0110011 Step 2. Do the Add operation carry 11 1 011 1 1 01 01 01 + 0 11 00 11 0 00 10 00 Since there is a carry of 1, Add the carry 0 00 10 00 + 1 0 00 10 01 Data Representation in Computers/Session 3 / 15 of 33 Complementary Subtraction (Contd.) e.g.2 101100 - 11100101 Step 1. Complement of 11100101 is 00011010 Step 2. Carry 01 11 00 10 11 00 +00 01 10 10 01 00 01 10 Step 3. Since there is no carry we re-complement the result and add a negative sign Thus the answer is -10111001 Data Representation in Computers/Session 3 / 16 of 33 Multiplication Rules for Multiplication: 0x0=0 0x1=0 1x0=0 1x1=1 E.g.. 10101 * 11001 10101 x11001 ------10101 00000 00000 10101 10101 ----------1000001101 Data Representation in Computers/Session 3 / 17 of 33 Division 1. Start from the left of the dividend 2. Perform subtraction i.e. divisor should be subtracted from the dividend a) if subtraction is possible put 1 in the quotient and subtract the divisor from digits of the dividend else put 0 in the quotient b) bring down the next digit to the right of the remainder 3. Do step 2 till no more digits remain in the dividend Data Representation in Computers/Session 3 / 18 of 33 Example The complete table for binary division is: 0/1 = 0 1/1 = 1 E.g 100001 / 110 Then 0101 (Quotient) ________ (Divisor) 110 | 1000 01 (Dividend) - 110 10 0 10 01 1 10 11 Quotient 101 (Remainder) Data Representation in Computers/Session 3 / 19 of 33