Information Representation Computer use a binary systems Why binary? Electronic bi-stable environment on/off, high/low voltage Bit: each bit can be either 0 or 1 Reliability With only 2 values, can be widely separated, therefore clearly differentiated “drift” causes less error Example: -3 mv 0 mv 1 Digital v.s, Analog 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1

Binary Representation in Computer System All information of diverse type is represented within computers in the form of bit patterns. e.g., text, numerical data, sound, and images One important aspect of computer design is to decide how information is converted ultimately to a bit pattern Writing software also frequently requires understanding how information is represented along with accuracies

Number Systems Decimal Number System Base is 10 or ‘D’ or ‘Dec’ Ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Each place is weighted by the power of 10 Example: 123410 or 1234D = 1 x 103 + 2 x 102 + 3 x 101 + 4 x 100 = 1000 + 200 + 30 + 4 $1,000 $100 $10 $1

Binary Number System Base is 2 or ‘b’ or ‘B’ or ‘Bin’ Two symbols: 0 and 1 Each place is weighted by the power of 2 Example: 10112 or 1011 B = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 8 + 0 + 2 +1 = 1110 11 in decimal number system is 1011 in binary number system

Conversion between Decimal and Binary Conversion from decimal number system to binary system Question: represent 3410 in the binary number system Answer: using the divide-by-2 technique repeatedly If we write the remainder from right to left : 3410  1 x 25 + 0 x 24 + 0 x 23 + 0 x 22+ 1x 21 + 0 x 20  1000102 34 8 17 2 4 1 Remainder div-by-2

Converting decimal to binary Example: 162: So, 16210 = 101000102 162 / 2 = 81 rem 0 81 / 2 = 40 rem 1 40 / 2 = 20 rem 0 20 / 2 = 10 rem 0 10 / 2 = 5 rem 0 5 / 2 = 2 rem 1 2 / 2 = 1 rem 0 1 / 2 = 0 rem 1

Some Exercises 13D = (?) B 23D = (?) B 72D = (?) B 1101B

Conversion from binary number system to decimal system Example: check if 1000102 is 3410 using the :weights” appropriately 1000102  1 x 25 + 0 x 24 + 0 x 23 + 0 x 22+ 1 x 21 + 0 x 20  32 + 0 + 0 + 0 + 2 + 0  3410

Some Exercises Ex: 0101B ( ? ) D Ex: 1100B  ( ? ) D Bit 4 23 = 8 Bit 3 22 = 4 Bit 2 21 = 2 Bit 1 20 = 1 1 4 + 1 = 5D 1 0 0 8 4 2 1 = 12D 0 1 0 1 1 1 0 0 128 64 32 16 8 4 2 1 = 92D

Binary Arithmetic on Integers Addition a b a + b 0 0 0 1 1 1 1 1 1 0 Carry bit Example: find binary number of a + b If a = 13D , b = 5D If a = 15D, b = 10D 1 1 1 1 1 0 1b 13D + 5D 18D + 0 1 0 1b 1 1 0b 1 1 1 1 1b 1 0 1 0b 1 1 0 0 1b + 15D + 10D 25D

Binary Arithmetic on Integers (cont.) Multiplication a b a x b 0 0 0 1 1 0 1 1 1 Example: if a = 100001b , b = 101b , find a x b 1 0 0 0 0 1b x 1 0 1b  33D  5D 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1b  165D

Hexadecimal Number System Base = 16 or ‘H’ or ‘Hex’ 16 symbols: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(=10), B(=11), C(=12), D(=13), E(=14), F(=15)} Hexadecimal to Decimal (an-1an-2…a1a0)16 = (an-1 x 16n-1 + an-2 x 16n-2 + …+ a1 x 161 + a0 x 160 )D Example: (1C7)16 = (1 x 162 + 12 x 161 + 7 x 160 )10 = (256 + 192 + 7)10 = (455)10 Decimal to Hexadecimal Repeated division by 16 Similar in principle to generating binary codes Example: (829)10 = (? )16 Stop, since quotient = 0 Hence, (829)10 = (3313)16 Divide-by-16 Quotient Remainder 829 / 16 51 / 16 3 / 16 51 3 13 (33D)16

Hexadecimal Conversions Hexadecimal to Binary Expand each hexadecimal digit to 4 binary bits. Example: (E29)16 = (1110 | 0010 | 1001)2 Binary to Hexadecimal Combine every 4 bits into one hexadecimal digit Example: (0101 | 1111 | 1010 | 0110)2 = (5FA6)16

Octal Number System Octal Number System Base = 8 or ‘o’ or ‘Oct’ 8 symbols: { 0, 1, 2, 3, 4, 5, 6, 7} Octal to Decimal (an-1an-2…a1a0)8 = (an-1 x 8n-1 + an-2 x 8n-2 + …+ a1 x 81 + a0 x 80 )10 Example: (127)8 = (1 x 82 + 2 x 81 + 7 x 80 )10 = (64 + 16 + 7)10 = (87)10 Decimal to Octal Repeated division by 8 (similar in principle to generating binary codes) Example: (213)10 = (? )8 Stop, since quotient = 0 Hence, (213)10 = (325)8 Divide-by -8 Quotient Remainder Octal digit 213 / 8 26 / 8 3 / 8 26 3 5 2 Lower digit = 5 Second digit =2 Third digit =3

Octal to Binary Binary to Octal Expand each octal digit to 3 binary bits. Example: (725)8 = (111 | 010 | 101)2 Binary to Octal Combine every 3 bits into one octal digit Example: (110 | 010 | 011)2 = (623)8

Mini Homework 1)  Convert the following binary numbers to decimal numbers: (a)     0011 B (b)     0101 B (c)     0001 0110 B (d)     0101 0011 B 2) Convert the following decimal numbers to binary: (a)     21 D (b)     731 D (c)     1,023 D

3) Convert the following binary numbers to hexadecimal numbers: (a)     0011 B (b)     0101 B (c)     0001 0110 B (d)     0101 0011 B (a)     21 D (b)     731 D (c)     1,023 D 4.) Perform the following binary additions and subtractions. Show your work without using decimal numbers during conversion. (a)     111 B + 101 B (b)     1001 B + 11 B