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3 Computing System Fundamentals 3.5 Data Representation 1 3.5.1 Binary Decimal • The decimal (base 10) number 287 means ‣ ‣ ‣ 2 hundreds, 8 tens and 7 ones. • Each column leftwards represents an increase in magnitude by x10 (powers of 10, 0 1 2 i.e. 10 , 10 , 10 ... 3 Decimal • The decimal number 7451: LSD MSD Thousands 103 7 Hundreds 102 4 Tens 101 5 • MSD = most significant digit • LSD = least significant digit. 4 Ones 100 1 Binary • Counting in tens is an accident of evolution (most of us have 10 fingers). • Computers are built from millions of on/off switches, so it is more logical for them to work in base 2 (binary). • The only allowed digits are 0 and 1 (binary digits or bits). 5 Binary • MSB LSB Bit no. 7 6 5 4 3 2 1 0 Power of 2 27 26 25 24 23 22 21 20 Which is 128 64 32 16 8 4 2 1 Number in binary 0 1 0 0 1 0 0 1 Representin g 64 8 and 64 + 8 + 1 = 73 in decimal • MSB = most significant bit. 6 1 Binary • As with decimal, a leading zero is usually dropped i.e. 01001001 = 1001001. • Subscripting is used to denote the base you are working in: i.e. 7310 = 10010012 • If there is no subscript and you are not told the base, assume it is decimal. 7 Uses of binary • ASCII (American Standard Code for Information Interchange) is a binary code used to represent characters. • Each letter, digit and symbol is represented by a unique code e.g. 01000001 is the capital letter A, (represented as the decimal number 65 in a table of ASCII characters). 8 Uses of binary • In standard ASCII, only the first 7 bits are used so the LSB can be used as a parity bit when characters are transmitted across a network. • This allow 2 7 = 128 combinations and therefore 128 keyboard characters, which is enough for the common western European languages. 9 Uses of binary • To accommodate non-Latin characters, most operating systems (and Java) now use Unicode , which uses 2 bytes per character. • Using 16 bits allows 2 16 or 65 536 possibilities i.e. the more bits you use the greater the range of data you can represent. • This is enough to cover other alphabets such as Chinese and Arabic. 10 Uses of binary • Bit patterns can be used to represent data other than numbers or characters e.g. colours • Very old computers might use 4 bit colour i.e. 16 colours in all i.e. every pixel has a 4 bit colour code. • 16 bit colour (65 536 colours) is referred to as “thousands of colours”, 32 bit as “millions”. 11 Uses of binary • Locations in computer memory need their own unique address, which is a binary number. • Most PCs use a 32 bit address bus i.e. there are 232 or 4 294 967 296 possible memory addresses in the RAM - such computers cannot usually work with more than about 4GB of RAM. 12