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Data Representation  Data Representation The term data representation means the code or technique in which the data can be represented.  Number Systems Number systems are very important to understand because the design and organization of a computer depends on the number systems. T...

Data Representation  Data Representation The term data representation means the code or technique in which the data can be represented.  Number Systems Number systems are very important to understand because the design and organization of a computer depends on the number systems. The four kind of number system used by the digital computer – 1.Decimal number system 2.Binary number system 3.Octal number system 4.Hexadecimal number system  Binary Number System Computers work only on two states– On– Off ◼ Basic memory elements hold only two states– Zero / One ◼ Thus a number system with two elements {0,1} The binary number system consists of 2 digits namely 0 and 1. Since the binary number system consists of 2 digits, the base of this system is 2. e.g (101)2 , (1001.11)2  Decimal Number System The decimal number system consists of 10 digits namely 0 to 9. Since the decimal number system consists of 10 digits, the base of this system is 10. e.g (405)10 , (145.25)10  Octal Number System The octal number system consists of 8 digits namely 0 to 7. Since the Octal number system consists of 8 digits, the base of this system is 8. e.g (76)8 , (55.25)8  Hexadecimal Number System The Hexadecimal number system, popularly known as Hex system has 16 symbols, therefore its base in 16. The 16 symbols used in Hexadecimal system are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F e.g (45)16, (11A)16 Number System  Conversion between Number Systems Decimal into Binary Step 1. Divide the decimal number by the base of binary using the repeated-division method. Step 2. Note the remainder separately. Step 3. Arrange the remainder in an order where the first remainder noted is LSD and the last remainder is MSD.  Conversion between Number Systems Decimal into Binary (Contd…)  Conversion between Number Systems Decimal into Binary (Contd…)  Conversion between Number Systems Binary into Decimal (Contd…)  Conversion between Number Systems Decimal into Octal Step 1. Divide the decimal number by the base of octal using the repeated-division method. Step 2. Note the remainder separately. Step 3. Arrange the remainder in an order where the first remainder noted is LSD and the last remainder is MSB.  Conversion between Number Systems Decimal into Octal (Contd…)  Conversion between Number Systems Octal into Decimal (Contd…)  Conversion between Number Systems Decimal into Hexadecimal Step 1. Divide the decimal number by the base of Hexadecimal using the repeated- division method. Step 2. Note the remainder separately. Step 3. Arrange the remainder in an order where the first remainder noted is LSD and the last remainder is MSB.  Conversion between Number Systems Decimal into Hexadecimal (Contd…) 16 16  Conversion between Number Systems Decimal real number into Binary, Octal and Hexadecimal Step 1. Multiply the fractional part by the base of the numbers system (2, 8 or 16). Step 2. Remove the whole number from the product (the result of the multiplication) and collect it separately. Step 3. Repeat the step 1 and 2 with the new fractional part till the fractional part becomes zero.  Conversion between Number Systems Binary to Decimal Any binary number can be converted into decimal number using the weights assigned to each bit. e.g. (11011)2 Its decimal equivalent is 1x24+1x23+0x22+1x21+1x20 = (27)10  Conversion between Number Systems Binary to Octal 1.Indirect Method: Binary → Decimal → Octal e.g. (11011)2 Its decimal equivalent is 1x24+1x23+0x22+1x21+1x20 = (27)10  Conversion between Number Systems Binary to Octal (Contd…) 2. Direct Method Binary → Octal Step 1: Make the group of 3-bits from right to left for integer from left to right for fraction. Step 2: Find decimal equivalent of each group. Note: if the left most group (in integer) and the right most group (in fraction) present with less than 3-bits make that group by adding one or two zeros.  Conversion between Number Systems Binary to Octal (Contd…) Direct Method: e.g. Binary → Octal e.g. (101111)2 = (? )8 (101111)2 = (57)8  Conversion between Number Systems Binary to Hexa 1.Indirect Method: Binary → Decimal → Hexa e.g. (11011)2 Its decimal equivalent is 1x24+1x23+0x22+1x21+1x20 = (27)10 And its Hexa equivalent is (1B)16 (Division method)  Conversion between Number Systems Binary to Hexa 2. Direct Method: Binary → Hexa Step 1: Make the group of 4-bits from right to left for integer from left to right for fraction. Step 2: Find decimal equivalent of each group. Note: if the left most group (in integer) and the right most group (in fraction) present with less than 4-bits make that group by adding one, two or three zeros.  Conversion between Number Systems Binary to Hexa Direct Method: e.g. Binary → Hexa e.g. (101111)2 = (? )16 (0010 1111)2 = (215)16 = (2F)16  Conversion between Number Systems Octal to Decimal Any octal number can be converted into decimal number using the weights assigned to each bit. e.g. (75)8 Its decimal equivalent is 7x81+5x80 = (61)10  Conversion between Number Systems Octal to Binary Any octal number can be converted into binary number by converting each bit of octal into its equivalent 3-bit binary number. e.g. (75)8 Its binary equivalent is (111101)2  Conversion between Number Systems Octal to Hexa Octal → Binary → Hexa Step1. Convert each digit of the octal into its 3 bit binary equivalent. Step2. Combine all the 3-bit binary equivalents to form the entire binary sequence. Step3. Make group of 4 bits staring from LSD. The extra zeros for the completion of a group are placed at the leftmost end of the number. Step 4. Convert each of the 4-bit groups into their hexadecimal equivalents.  Conversion between Number Systems Octal to Hexa (Contd…) Octal → Binary → Hexa  Conversion between Number Systems Hexa to decimal  Conversion between Number Systems Hexa to binary  Binary representation of integers There are three ways for representing the positive and negative integers into its binary equivalent. 1.Sign magnitude representation 2.One’s Complement 3.Two’s Complement  Binary representation of integers 1.Sign magnitude representation In the sign magnitude representation, positive number have a additional bit (sign bit) 0, while the negative number has a sign bit 1, while the magnitude is a simple binary equivalent of the number. E.g. +5 and -5 can be representing in 6 bit register as: +5 = 0 00101 and -5 = 1 00101  Binary representation of integers Note: In every representation technique , the representation of positive number is identical to that used in the sign magnitude system i.e simple binary form including sign bit 0.  Binary representation of integers 2. One’s Complement representation In one's complement, positive numbers are represented as usual in signed magnitude. However, negative numbers are represented differently. To negate a number, replace all zeros with ones, and ones with zeros - flip the bits. +12 = 0 0001100, and -12 = 1 1110011.  Binary representation of integers 3. Two’s Complement representation In two's complement, positive numbers are represented as usual in signed magnitude. However, negative numbers are represented by adding 1 in magnitude part of one’s complement. +12= 0 0001100 -12 = 1 1110011 (1’s complement) -12 = 1 1110100 (2’s complement)  Binary Addition Rule for Binary Addition: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 (Carry 1)  Binary Addition Example: Add 110101 and 101111 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0  Binary Addition Example: Add 10110 and 1101 1 0 1 1 0 0 1 1 0 1 1 0 0 0 1 1 Any Question…