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1.1 Information Representation 1.1.1 Number Representation Learning Objectives (a) show understanding of the basis of different number systems and use the binary, denary and hexadecimal number system (b) convert a number from one number system to another (c) express a positive or negative integer in two’s complement form (d) show understanding of, and be able to represent, character data in its internal binary form depending on the character set used (Candidates will not be expected to memorise any particular character codes but must be familiar with ASCII and Unicode. (e) express a denary number in Binary Coded Decimal (BCD) and vice versa Numbers • Each number system is associated with a base ▫ The decimal number system is said to be of base 10 • A number in base r contains r digits 0,1,2,...,r-1 ▫ Decimal (Base 10): 0,1,2,3,4,5,6,7,8,9 In addition to decimal, two other number systems are also important: Binary and Hexadecimal Number Systems WHAT, more number systems? Why do we need more number systems? • Humans understand decimal Check out my ten digits ! • Digital electronics (computers) understand binary Number Systems Why binary system? • Computers are made of a series of switches • Each switch has two states: ON or OFF • Each state can be represented by a number – 1 for “ON” and 0 for “OFF” Number Systems Hexadecimal • Hexadecimal (base 16) number system are used to represent binary data in a more compact form. Number Systems Common Number Systems • Denary aka Decimal System (Base 10) • (i.e. 0…9) • Binary System (base 2) • (i.e 1,0) • Hexadecimal System (base 16) • (i.e 0…9,A,B,C,D,E,F) Denary to Binary Conversion – Example 1 Divide by two and use the remainder (Method 1) 1. Divide the starting number by 2. If it divides evenly, the binary digit is 0. If it does not - if there is a remainder - the binary digit is 1. 2. Repeat the process with the integer quotient until it becomes 0 3. Record the bits from bottom to top. Binary to Denary Conversion – Answer Check 1. Multiply each bit by 2 with incrementing exponent from 0 (from the left side) until the last bit in place (from the right side). 2. Add up all the products 26 25 24 23 22 21 20 Denary to Binary Conversion – Example 2 • Convert 122 (denary) to binary Binary to Denary Conversion– Example 2 • Convert 122 (denary) to binary Denary to Binary Conversion – Example 3 Take off the biggest 2n value you can (Method 2) 1. Remove the 2n numbers from the main number and mark up the equivalent 2n column with a 1. 2. Work through the remainders until you reach zero. Stop when you reach zero 3. Complete the final columns with 0s. Denary to Binary Conversion – Example 3 Activity • Online Game Activity • Decimal to Binary Worksheet Denary to Hexadecimal Conversion – Example 1 Denary to Hexadecimal Conversion – Example 2 • Convert 921 (denary to hexadecimal) Denary to Hexadecimal Conversion – Example 3 • Convert 590 (denary to hexadecimal) • Remember that we represent 10 . . 15 as A . . F Hexadecimal to Denary Conversion – Example 1 1. Multiply each digit by 16 with incrementing exponent from 0 (from the left side) until the last digit in place (from the right side). 2. Add up all the products This is similar to binary – denary conversion but we use base 16 instead of base 2. • Convert 1128(base 16) to denary. • Solution: 1 x 163 +1 x 162 + 2 x 161 + 8 x 160 4096 + 256 + 32 +8 = 4392 Hexadecimal to Denary Conversion – Example 1 • Convert 1531 (base 16) to Decimal Hexadecimal to Denary Conversion – Example 2 • Convert FA8(base 16) to Decimal Hexadecimal to Binary • Technique • Convert each hexadecimal digit to a 4bit equivalent binary representation Number Systems Example 10AF16 = ?2 1 0 0001 0000 A F 1010 10AF16 = 00010000101011112 Number Systems 1111 Binary system complements As the binary system has base r = 2. So the two types of complements for the binary system are 2's complement and 1's complement. 1's complement The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is called as taking complement or 1's complement. Example of 1's Complement is as follows. 2's complement The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number. 2's complement = 1's complement + 1 Example of 2's Complement is as follows. Two’s Complement • Convert -28 in two’s complement • Get the binary value of +28 in 8 bits: 00011100 • Flip to get the one’s complement 11100011 • Add 1 to get the two’s complement 11100011 + 1 11100100 -28 = 11100100 Binary Coded Decimal (BCD) • BCD is a binary representation which can be used for a positive denary integer (0..9 only) • Each digit of the denary is represented in sequence with a group of four binary digits • Example: represent 859 in BCD 8 5 9 1000 0101 1001 859 denary = 100001011001 as BCD Binary Coded Decimal (BCD) • Convert 372 denary into BCD • Convert A4 • Example: represent 372 in BCD 3 7 2 0011 0111 0010 372 denary = 001101110010 as BCD • Example: represent A4 in BCD CANT BE DIRECTLY REPRESENTED Binary Coded Decimal (BCD) Application • Early computers stored date and time values in the BIOS of the OS using BCD • Later game consoles like Atari and Sony Playstation had used BCD • BCD is used in electronics too like pocket calculators, digital clocks Written Activity