Chapter 1 Number Systems Conversions Location in course

Chapter 1 Number Systems & Conversions Location in course textbook Chapter. 1 CNET 315 Microprocessor & Assembly Language 1

Common Number Systems Number System Base Symbols Used by humans? Used in 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 CNET 315 Microprocessor & Assembly Language 2

Quantities/Counting (1 of 3) Decimal 0 Hexa. Binary Octal decimal 0 0 0 1 2 1 10 1 2 3 4 5 6 7 11 100 101 110 111 3 4 5 6 7 CNET 315 Microprocessor & Assembly Language p. 33 3

Quantities/Counting (2 of 3) Decimal 8 Hexa. Binary Octal decimal 1000 10 8 9 10 1001 1010 11 12 9 A 11 12 13 14 15 1011 1100 1101 1110 1111 13 14 15 16 17 B C D E F CNET 315 Microprocessor & Assembly Language 4

Quantities/Counting (3 of 3) Decimal 16 Hexa. Binary Octal decimal 10000 20 10 17 18 10001 10010 21 22 11 12 19 20 21 22 23 10011 10100 10101 10110 10111 23 24 25 26 27 13 14 15 16 17 CNET 315 Microprocessor & Assembly Language Etc. 5

Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 6

Quick Example 2510 = 110012 = 318 = 1916 Base CNET 315 Microprocessor & Assembly Language 7

Weight 12510 => 5 x 100 2 x 101 1 x 102 = 5 = 20 = 100 125 Base CNET 315 Microprocessor & Assembly Language 8

Binary to Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 9

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 CNET 315 Microprocessor & Assembly Language 10

Example Bit “ 0” 1010112 => 1 1 0 1 x x x 20 21 22 23 24 25 = = = 1 2 0 8 0 32 4310 CNET 315 Microprocessor & Assembly Language 11

Octal to Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 12

Octal to Decimal • Technique – Multiply each bit by 8 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 CNET 315 Microprocessor & Assembly Language 13

Example 7248 => 4 x 80 = 2 x 81 = 7 x 82 = 4 16 448 46810 CNET 315 Microprocessor & Assembly Language 14

Hexadecimal to Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 15

Hexadecimal to Decimal • Technique – Multiply each bit by 16 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 CNET 315 Microprocessor & Assembly Language 16

Example ABC 16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810 CNET 315 Microprocessor & Assembly Language 17

Decimal to Binary Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 18

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. CNET 315 Microprocessor & Assembly Language 19

Example 12510 = ? 2 2 125 2 62 2 31 2 15 7 2 3 2 1 2 0 1 1 1 12510 = 11111012 CNET 315 Microprocessor & Assembly Language 20

Octal to Binary Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 21

Octal to Binary • Technique – Convert each octal digit to a 3 -bit equivalent binary representation CNET 315 Microprocessor & Assembly Language 22

Example 7058 = ? 2 7 0 5 111 000 101 7058 = 1110001012 CNET 315 Microprocessor & Assembly Language 23

Hexadecimal to Binary Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 24

Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4 -bit equivalent binary representation CNET 315 Microprocessor & Assembly Language 25

Example 10 AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10 AF 16 = 000101011112 CNET 315 Microprocessor & Assembly Language 26

Decimal to Octal Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 27

Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder CNET 315 Microprocessor & Assembly Language 28

Example 123410 = ? 8 8 8 1234 154 19 2 0 2 2 3 2 123410 = 23228 CNET 315 Microprocessor & Assembly Language 29

Decimal to Hexadecimal Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 30

Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder CNET 315 Microprocessor & Assembly Language 31

Example 123410 = ? 16 16 1234 77 4 0 2 13 = D 4 123410 = 4 D 216 CNET 315 Microprocessor & Assembly Language 32

Binary to Octal Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 33

Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits CNET 315 Microprocessor & Assembly Language 34

Example 10110101112 = ? 8 1 010 111 1 3 2 7 10110101112 = 13278 CNET 315 Microprocessor & Assembly Language 35

Binary to Hexadecimal Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 36

Binary to Hexadecimal • Technique – Group bits in fours, starting on right – Convert to hexadecimal digits CNET 315 Microprocessor & Assembly Language 37

Example 1010112 = ? 16 10 1011 2 B B 1010112 = 2 BB 16 CNET 315 Microprocessor & Assembly Language 38

Octal to Hexadecimal Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 39

Octal to Hexadecimal • Technique – Use binary as an intermediary CNET 315 Microprocessor & Assembly Language 40

Example 10768 = ? 16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23 E 16 CNET 315 Microprocessor & Assembly Language 41

Hexadecimal to Octal Decimal Octal Binary Hexadecimal CNET 315 Microprocessor & Assembly Language 42

Hexadecimal to Octal • Technique – Use binary as an intermediary CNET 315 Microprocessor & Assembly Language 43

Example 1 F 0 C 16 = ? 8 1 0001 1 F 0 1111 7 C 0000 4 1100 1 4 1 F 0 C 16 = 174148 CNET 315 Microprocessor & Assembly Language 44

Exercise – Convert. . . Decimal 33 Binary Octal Hexadecimal 1110101 703 1 AF Don’t use a calculator! CNET 315 Microprocessor & Assembly Language 45

Exercise – Convert … Answer Decimal 33 117 Binary 100001 1110101 Octal 41 165 451 431 111000011 110101111 703 657 CNET 315 Microprocessor & Assembly Language Hexadecimal 21 75 1 C 3 1 AF 46

Binary Addition (1 of 2) • Two 1 -bit values A 0 0 1 1 B 0 1 A+B 0 1 1 10 CNET 315 Microprocessor & Assembly Language “two” 47

Binary Addition (2 of 2) • Two n-bit values – Add individual bits – Propagate carries – E. g. , 1 1 10101 + 11001 101110 21 + 25 46 CNET 315 Microprocessor & Assembly Language 48

Binary Subtraction • Have previously looked at the subtraction operation. A quick review. • Just like subtraction in any other base – Minuend – Subtrahand – Difference 10110 - 100100 • And when a borrow is needed. Note that the borrow gives us 2 in the current bit position. • . 49

Thank you CNET 315 Microprocessor & Assembly Language 50