Lecture 1 Number System PDF

1. Number Systems  Common Number Systems Used by Used in System Base Symbols humans? computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- 16 0,...

1. Number Systems  Common Number Systems Used by Used in System Base Symbols humans? computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- 16 0, 1, … 9, No No decimal A, B, … F Quantities/Counting (1 of 3) Hexa- Decimal Binary Octal decimal 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 Quantities/Counting (2 of 3) Hexa- Decimal Binary Octal decimal 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F Quantities/Counting (3 of 3) Hexa- Decimal Binary Octal decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 20 10100 24 14 21 10101 25 15 22 10110 26 16 23 10111 27 17 Etc.  Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal  Quick Example 2510 = 110012 = 1916 Base Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal  Weight 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base  Binary to Decimal Decimal Octal Binary Hexadecimal  Binary to Decimal Technique – Multiply each bit by 2n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results  Example Bit “0” 1010112 => 1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 = 32 4310 Hexadecimal to Decimal Decimal Octal Binary Hexadecimal  Hexadecimal to Decimal Technique – Multiply each bit by 16n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results  Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810  Decimal to Binary Decimal Octal Binary Hexadecimal  Decimal to Binary Technique – Divide by two, keep track of the remainder – First remainder is bit 0 (LSB, least-significant bit) – Second remainder is bit 1 – Etc.  Example 12510 = ?2 2 125 2 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 0 1 12510 = 11111012 Hexadecimal to Binary Decimal Octal Binary Hexadecimal  Hexadecimal to Binary Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation  Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112 Decimal to Hexadecimal Decimal Octal Binary Hexadecimal  Decimal to Hexadecimal Technique – Divide by 16 – Keep track of the remainder  Example 123410 = ?16 16 1234 16 77 2 16 4 13 = D 0 4 123410 = 4D216 Binary to Hexadecimal Decimal Octal Binary Hexadecimal  Binary to Hexadecimal Technique – Group bits in fours, starting on right – Convert to hexadecimal digits  Example 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16  Common Powers (1 of 2) Base 10 Power Preface Symbol Value 10-12 pico p.000000000001 10-9 nano n.000000001 10-6 micro .000001 10-3 milli m.001 103 kilo k 1000 106 mega M 1000000 109 giga G 1000000000 1012 tera T 1000000000000  Common Powers (2 of 2) Base 2 Power Preface Symbol Value 210 kilo k 1024 220 mega M 1048576 230 Giga G 1073741824 What is the value of “k”, “M”, and “G”? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 230 =  Exercise – Free Space Determine the “free space” on all drives on a machine in the lab Free space Drive Bytes GB A: C: D: E: etc.  Review – multiplying powers For common bases, add powers ab  ac = ab+c 26  210 = 216 = 65,536 or… 26  210 = 64  210 = 64k  Binary Addition (1 of 2) Two 1-bit values A B A+B 0 0 0 0 1 1 1 0 1 1 1 10 “two”  Binary Addition (2 of 2) Two n-bit values – Add individual bits – Propagate carries – E.g., 1 1 10101 21 + 11001 + 25 101110 46  Multiplication (1 of 3) Decimal (just for fun) 35 x 105 175 000 35 3675  Multiplication (2 of 3) Binary, two 1-bit values A B AB 0 0 0 0 1 0 1 0 0 1 1 1  Multiplication (3 of 3) Binary, two n-bit values – As with decimal values – E.g., 1110 x 1011 1110 1110 0000 1110 10011010  Fractions Decimal to decimal (just for fun) 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14  Fractions Binary to decimal 10.1011 => 1 x 2-4 = 0.0625 1 x 2-3 = 0.125 0 x 2-2 = 0.0 1 x 2-1 = 0.5 0 x 20 = 0.0 1 x 21 = 2.0 2.6875  Fractions Decimal to binary.14579 x 2 3.14579 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 11.001001... etc.