1. Dale Roberts Department of Computer and Information Science, School of Science, IUPUI CSCI 230 Information Representation: Positive Integers Dale Roberts, Lecturer IUPUIdroberts@cs.iupui.edu

2. Dale Roberts 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 000111 Digital v.s, Analog 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1

3. Dale Roberts 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

4. Dale Roberts 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: 1234.21 10 or 1234.21 D = 1 x 10 3 + 2 x 10 2 + 3 x 10 1 + 4 x 10 0 + 2 x 10 -1 + 1 x 10 -2 = 1000 + 200 + 30 + 4 + 0.2 + 0.01 $1,000 $100 $10 $1 10¢ 1¢1¢

5. Dale Roberts 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: 1011 2 or 1011 B = 1 x 2 3 + 0 x 2 2 + 1 x 2 1 + 1 x 2 0 = 8 + 0 + 2 +1 = 11 10 11 in decimal number system is 1011 in binary number system

6. Dale Roberts Conversion between Decimal and Binary Conversion from decimal number system to binary system Question: represent 34 10 in the binary number system Answer: using the divide-by-2 technique repeatedly If we write the remainder from right to left : 34 10  1 x 2 5 + 0 x 2 4 + 0 x 2 3 + 0 x 2 2 + 1x 2 1 + 0 x 2 0  100010 2 34817 2 4 1 010001 Remainder div-by-2

7. Dale Roberts Practice Exercises 13 D = (?) B 23 D = (?) B 72 D = (?) B 8 4 2 1 16 8 4 1 1101 B 8 4 2 116 8 4 2 1 32 64 32 16 8 4 2 1 Blocks: 512 256 128 64 32 16 8 4 2 1

8. Dale Roberts Conversion between Binary and Decimal Conversion from binary number system to decimal system Example: check if 100010 2 is 34 10 using the :weights” appropriately 100010 2  1 x 2 5 + 0 x 2 4 + 0 x 2 3 + 0 x 2 2 + 1 x 2 1 + 0 x 2 0  32 + 0 + 0 + 0 + 2 + 0  34 10

9. Dale Roberts Practice Exercises Ex: 0101 B  ( ? ) D Ex: 1100 B  ( ? ) D Ex: 0101 1100 B  ( ? ) D Bit 4 2 3 = 8 Bit 3 2 2 = 4 Bit 2 2 1 = 2 Bit 1 2 0 = 1 0101 41+= 5 D 11 0 0 8 4 2 1 = 12 D 0 1 0 1 1 1 0 0 128 64 32 16 8 4 2 1 = 92 D

10. Dale Roberts Binary Arithmetic on Integers Addition a ba + b 0 01 1 01 13 D + 5 D 18 D 15 D + 10 D 25 D 0 1 1 0 1 Example: find binary number of a + b If a = 13 D, b = 5 D If a = 15 D, b = 10 D 111 1 1 0 1 b 0 1 0 1 b + 111 1 1 1 1 b 1 0 1 0 b 1 1 0 0 1 b + 10b0b 001 Carry bit

11. Dale Roberts Multiplication Binary Arithmetic on Integers a ba x b 0 01 101 1 0 0 0 0 1 b x 1 0 1 b 0 1 0 0  33 D  5 D  165 D 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 1 b Example: if a = 100001 b, b = 101 b, find a x b

12. Dale Roberts 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 (a n-1 a n-2 …a 1 a 0 ) 16 = (a n-1 x 16 n-1 + a n-2 x 16 n-2 + …+ a 1 x 16 1 + a 0 x 16 0 ) D Example: (1C7) 16 = (1 x 16 2 + 12 x 16 1 + 7 x 16 0 ) 10 = (256 + 192 + 7) 10 = (455) 10 Decimal to Hexadecimal Decimal to Hexadecimal Repeated division by 16 binary codes Similar in principle to generating binary codes Example: (829) 10 = (? ) 16 Stop, since quotient = 0 Hence, (829) 10 = (33D) 16 Hence, (829) 10 = (33D) 16 Divide-by-16QuotientRemainderHexadecimal digit 829 / 16 51 / 16 3 / 16 51 3 0 13 3 Lower digit = D Second digit =3 Third digit =3

13. Dale Roberts 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

14. Dale Roberts Octal Number System Base = 8 or ‘o’ or ‘Oct’ 8 symbols: { 0, 1, 2, 3, 4, 5, 6, 7} Octal to Decimal (a n-1 a n-2 …a 1 a 0 ) 8 = (a n-1 x 8 n-1 + a n-2 x 8 n-2 + …+ a 1 x 8 1 + a 0 x 8 0 ) 10 Example: (127) 8 = (1 x 8 2 + 2 x 8 1 + 7 x 8 0 ) 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 Stop, since quotient = 0 Hence, (213) 10 = (325) 8 Hence, (213) 10 = (325) 8 Divide-by -8QuotientRemainderOctal digit 213 / 8 26 / 8 3 / 8 26 3 0 523523 Lower digit = 5 Second digit =2 Third digit =3

15. Dale Roberts Octal Conversions Octal to Binary 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

16. Dale Roberts Practice Exercises 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

17. Dale Roberts Practice Exercises 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

18. Dale Roberts Acknowledgements These slides where originally prepared by Dr. Jeffrey Huang, updated by Dale Roberts.