Number Systems Introduction to Computer Science Common Number

Number Systems Introduction to Computer Science

Common Number Systems System Base Symbols Used by Used in humans? computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexadecimal 16 0, 1, … 9, A, B, … F No No Introduction to Computer Science

Quantities/Counting (1 of 3) Decimal 0 Binary 0 1 2 1 10 3 4 5 6 7 11 100 101 110 111 Introduction to Computer Science

Quantities/Counting (2 of 3) Decimal 8 Binary 1000 9 10 1001 1010 11 12 13 14 15 1011 1100 1101 1110 1111 Introduction to Computer Science

Quantities/Counting (3 of 3) Decimal 16 Binary 10000 17 18 10001 10010 19 20 21 22 23 10011 10100 10101 10110 10111 Introduction to Computer Science

Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal Introduction to Computer Science

Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Introduction to Computer Science

Weight 12510 => 5 x 100 2 x 101 1 x 102 = 5 = 20 = 100 125 Base Introduction to Computer Science

Binary to Decimal Octal Binary Hexadecimal Introduction to Computer Science

Binary to Decimal • Technique – Multiply each bit by 2 n, 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 Introduction to Computer Science

Example Bit “ 0” 1010112 => 1 1 0 1 x x x 20 21 22 23 24 25 = = = 1 2 0 8 0 32 4310 Introduction to Computer Science

Decimal to Binary Decimal Octal Binary Hexadecimal Introduction to Computer Science

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. Introduction to Computer Science

Example 12510 = ? 2 2 125 2 62 2 31 2 15 7 2 3 2 1 2 0 1 1 1 12510 = 11111012 Introduction to Computer Science

Common Powers (1 of 2) • Base 10 Power Preface Symbol Value 10 -12 pico p . 0000001 10 -9 nano n . 00001 10 -6 micro . 000001 10 -3 milli m . 001 103 kilo k 1000 106 mega M 1000000 109 giga G 100000 1012 tera T 1000000 Introduction to Computer Science

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 Introduction to Computer Science

Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 230 = Introduction to Computer Science

Review – multiplying powers • For common bases, add powers ab ac = ab+c 26 210 = 216 = 65, 536 or… 26 210 = 64 k Introduction to Computer Science

Binary Addition (1 of 2) • Two 1 -bit values A 0 0 1 1 B 0 1 A+B 0 1 1 10 “two” pp. 36 -38 Introduction to Computer Science

Binary Addition (2 of 2) • Two n-bit values – Add individual bits – Propagate carries – E. g. , 1 1 10101 + 11001 101110 21 + 25 46 Introduction to Computer Science

Multiplication (2 of 3) • Binary, two 1 -bit values A 0 0 1 1 B 0 1 A B 0 0 0 1 Introduction to Computer Science

Multiplication (3 of 3) • Binary, two n-bit values – As with decimal values – E. g. , 1110 x 1011 1110 0000 1110 10011010 Introduction to Computer Science

Fractions • Binary to decimal 10. 1011 => 1 1 0 1 x x x 2 -4 2 -3 2 -2 2 -1 20 21 = = = 0. 0625 0. 125 0. 0 0. 5 0. 0 2. 6875 Introduction to Computer Science

Fractions • Decimal to binary 3. 14579 11. 001001. . 14579 x 2 0. 29158 x 2 0. 58316 x 2 1. 16632 x 2 0. 33264 x 2 0. 66528 x 2 1. 33056 etc. Introduction to Computer Science