Department of Computer and Information Science, School of Science, IUPUI

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

2. 1 0 1 0 1 0 -3 mv 0 mv 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: Digital v.s, Analog 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1

3. 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. 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: 1234.2110 or 1234.21D = 1 x 103 + 2 x 102 + 3 x 101 + 4 x 100 + 2 x 10-1 + 1 x 10-2 = 1000 + 200 + 30 + 4 + 0.2 + 0.01 $1,000 $100 1¢ $10 10¢ $1

5. 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: 10112or1011B = 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

6. 34 17 8 4 2 1 div-by-2 div-by-2 div-by-2 div-by-2 div-by-2 div-by-2 Remainder 0 1 0 0 0 1 Conversion between Decimal and Binary • Conversion from decimal number system to binary system • Question: represent 3410in 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

7. Some Exercises Blocks: 512 256 128 64 32 168 4 2 1 • 13D = (?) B • 23D = (?) B • 72D = (?) B 32 16 8 4 2 1 2 8 8 1 1101B 4 1 4 16 2 8 16 4 1 2 8 32 4 64 1 16

8. Conversion from binary number system to decimal system Example: check if 1000102is 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

9. Some Exercises • Ex: 0101B ( ? ) D • Ex: 1100B  ( ? ) D • Ex: 0101 1100B ( ? ) D 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

10. a b a + b • 0 0 • 0 1 • 0 • 1 1 1 1 1 13D + 5D 18D + + 1 1 1 15D + 10D 25D 1 1 1 1b 1 0 1 0b 1 1 0 0 1b Binary Arithmetic on Integers • Addition 0 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 0 1b 0 1 0 1b 1 0 0 1 0b

11. a b a x b 0 0 0 1 1 0 1 1 1 0 0 0 0 1b x 1 0 1b Binary Arithmetic on Integers (cont.) • Multiplication 0 0 0 1 • Example: if a = 100001b , b = 101b ,find a x b  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

12. Hexadecimal Number System • 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+a0x 160)D Example: (1C7)16 = (1 x 162+ 12x 161+7x 160)10 = (256 + 192 + 7)10 = (455)10 • Decimal to HexadecimalRepeated division by 16 • Similar in principle to generatingbinary codes Example: (829)10=(?)16 Stop, since quotient = 0 Hence, (829)10 = (33D)16

13. 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. 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+ 2x 81+7x 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

15. 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. Mini Homework #01 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. 3)Convert the following binary numbers to hexadecimal numbers: (a)     0011 B (b)0101B (c)0001 0110B (d)0101 0011B (a)21D (b)731D (c)1,023D 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. Acknowledgements • These slides where originally prepared by Dr. Jeffrey Huang, updated by Dale Roberts.