Bit patterns

• Bits can be used to represent patterns • Specifically, any system or set of symbols can be translated into bit patterns – patterns of ones and zeros Binary Numbers – 10100001101 • Example: characters from any language alphabet • Require enough bits so that all symbols have a unique bit pattern to represent them – How many bits are needed to represent the English alphabet? • Require set of symbols is finite

How many bits? Decimal (base 10) representation

• A bit pattern consisting of a single bit can represent at • We commonly represent numbers in decimal (base 10) most two symbols • Numbers are represented using patterns of the digits – possible patterns are 0 and 1 { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } • A bit pattern consisting of two bits can represent at most four symbols • Position of each digit represents a power of ten – possible patterns are 00, 01, 10 and 11 • Example: Consider the decimal representation 2307 • In general, a bit pattern consisting of n bits can represent n at most 2 symbols 2307 • How many bits are needed to represent the English ↑↑↑↑ alphabet? position: 3 2 1 0 – we can represent 26 symbols using 5 bits (25=32) – 4 bits is not enough (24=16) 2307 = 2×103 + 3×102 + 0×101 + 7×100

Base n representation Binary (base 2) representation

• A base n system contains n distinct symbols, the digits 0 • The binary representation contains two symbols: { 0, 1 } through n -1 • Position of each symbol represents a power of two • Numeric values greater than n -1 are represented by a • What is the value of the binary representation 111? pattern of the n symbols • The value of any symbol in the string is found by 111 p multiplying that symbol by n , where p is the distance ↑↑↑ from the rightmost symbol in the pattern position: 2 1 0 • Computers represent information using bit patterns, or binary (base 2) representation 2 1 0 • Numbers represented in base 2 are usually called binary 111 = 1×2 + 1×2 + 1×2 numbers = 1×4+ 1×2+ 1×1 = 4 + 2 + 1 = 7

1 Binary representation Binary addition

• What is the value of the binary representation 1010? 0 1 0 1 + 0 + 0 + 1 + 1 ------1010 0 1 1 10 ↑↑↑↑ position: 3 2 1 0 • Represent sum of binary numbers as a binary number

1010 = 1×23 + 0×22 + 1×21 + 0×20 decimal addition binary addition = 1×8 + 0×4+ 1×2+ 0×1 1+1 = 2 1+1 = 10 = 8 + 0 + 2 + 0 = 10 1+1+1 = 3 1+1+1 = 10+1 = 11

Adding binary numbers Converting decimal to binary

→→conversion →→ 1 1 ← carry 1 1 ← carry Decimal Binary 101 101 111 0 + 10 + 11 + 110 0 = 0×2 = 0 ------1 = 1×20 = 1 111 1000 1101 2 = 1×21 + 0×20 = 10 3 = 1×21 + 1×20 = 11 2 1 0 1 1 1 ← carry 4 = 1×2 + 0×2 + 0×2 = 100 10101010111 5 = 1×22 + 0×21 + 1×20 = 101 + 110000110 6 = 1×22 + 1×21 + 0×20 = 110 ------7 = 1×22 + 1×21 + 1×20 = 111 11011011101 8 = 1×23 + 0×22 + 0×21 + 0×20 = 1000

Converting decimal to binary Converting decimal to binary

0 remainder 1 0 remainder 1 2 1 • Repeated division by 2 1 • Repeated division by two until the quotient two until the quotient 1 remainder 0 is zero 1 remainder 1 is zero 2 2 2 3 • What is the binary • What is the binary 2 remainder 1 representation of 30? 3 remainder 1 representation of 47? 2 5 2 7 5 remainder 1 7 remainder 1 2 11 2 15 11 remainder 1 15 remainder 0 2 23 2 30 23 remainder 1 2 47 11110 111001

2 Problems Solutions

• Convert 1011000 to decimal representation • Convert 1011000 to decimal representation

1011000 = 1×26 + 0×25 + 1×24 + 1×23 + 0×22 + 0×21 + 0×20 = 64 + 16 + 8 = 88

• Add the binary numbers 1011001 and 10101 and • Add the binary numbers 1011001 1011001 express their sum in binary representation and 10101 and express their sum + 10101 in binary representation ------1101110

• Convert 77 to binary representation • Convert 77 to binary representation: 1001101

Ternary (base 3) representation

• Numbers are represented using the symbols { 0, 1, 2 } • Convert the ternary representation 1021 to decimal 1021 = 1×33 + 0×32 + 2×31 + 1×30 = 27 + 0 + 6 + 1 = 34 • Convert the ternary representation 2001 to decimal 2001 = 2×33 + 0×32 + 0×31 + 1×30 = 54 + 0 + 0 + 1 = 55 • Add the ternary numbers 1021 and 2001 and express their sum in ternary representation • Convert 77 to ternary representation

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