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# 01.Number Systems

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### 01.Number Systems

1. 1. 1. Number Systems
2. 2. Common Number Systems No No 0, 1, … 9, A, B, … F 16 Hexa- decimal No No 0, 1, … 7 8 Octal Yes No 0, 1 2 Binary No Yes 0, 1, … 9 10 Decimal Used in computers? Used by humans? Symbols Base System
3. 3. Quantities/Counting (1 of 3) p. 33 7 7 111 7 6 6 110 6 5 5 101 5 4 4 100 4 3 3 11 3 2 2 10 2 1 1 1 1 0 0 0 0 Hexa- decimal Octal Binary Decimal
4. 4. Quantities/Counting (2 of 3) F 17 1111 15 E 16 1110 14 D 15 1101 13 C 14 1100 12 B 13 1011 11 A 12 1010 10 9 11 1001 9 8 10 1000 8 Hexa- decimal Octal Binary Decimal
5. 5. Quantities/Counting (3 of 3) Etc. 17 27 10111 23 16 26 10110 22 15 25 10101 21 14 24 10100 20 13 23 10011 19 12 22 10010 18 11 21 10001 17 10 20 10000 16 Hexa- decimal Octal Binary Decimal
6. 6. Conversion Among Bases <ul><li>The possibilities: </li></ul>Hexadecimal Decimal Octal Binary pp. 40-46
7. 7. Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base
8. 8. Decimal to Decimal (just for fun) Hexadecimal Decimal Octal Binary Next slide…
9. 9. 125 10 => 5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = 100 125 Base Weight
10. 10. Binary to Decimal Hexadecimal Decimal Octal Binary
11. 11. Binary to Decimal <ul><li>Technique </li></ul><ul><ul><li>Multiply each bit by 2 n , where n is the “weight” of the bit </li></ul></ul><ul><ul><li>The weight is the position of the bit, starting from 0 on the right </li></ul></ul><ul><ul><li>Add the results </li></ul></ul>
12. 12. Example 101011 2 => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = 32 43 10 Bit “0”
13. 13. Octal to Decimal Hexadecimal Decimal Octal Binary
14. 14. Octal to Decimal <ul><li>Technique </li></ul><ul><ul><li>Multiply each bit by 8 n , where n is the “weight” of the bit </li></ul></ul><ul><ul><li>The weight is the position of the bit, starting from 0 on the right </li></ul></ul><ul><ul><li>Add the results </li></ul></ul>
15. 15. Example 724 8 => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10
17. 17. Hexadecimal to Decimal <ul><li>Technique </li></ul><ul><ul><li>Multiply each bit by 16 n , where n is the “weight” of the bit </li></ul></ul><ul><ul><li>The weight is the position of the bit, starting from 0 on the right </li></ul></ul><ul><ul><li>Add the results </li></ul></ul>
18. 18. Example ABC 16 => C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 = 2560 2748 10
19. 19. Decimal to Binary Hexadecimal Decimal Octal Binary
20. 20. Decimal to Binary <ul><li>Technique </li></ul><ul><ul><li>Divide by two, keep track of the remainder </li></ul></ul><ul><ul><li>First remainder is bit 0 (LSB, least-significant bit) </li></ul></ul><ul><ul><li>Second remainder is bit 1 </li></ul></ul><ul><ul><li>Etc. </li></ul></ul>
21. 21. Example 125 10 = ? 2 125 10 = 1111101 2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1
22. 22. Octal to Binary Hexadecimal Decimal Octal Binary
23. 23. Octal to Binary <ul><li>Technique </li></ul><ul><ul><li>Convert each octal digit to a 3-bit equivalent binary representation </li></ul></ul>
24. 24. Example 705 8 = ? 2 705 8 = 111000101 2 7 0 5 111 000 101
26. 26. Hexadecimal to Binary <ul><li>Technique </li></ul><ul><ul><li>Convert each hexadecimal digit to a 4-bit equivalent binary representation </li></ul></ul>
27. 27. Example 10AF 16 = ? 2 10AF 16 = 0001000010101111 2 1 0 A F 0001 0000 1010 1111
28. 28. Decimal to Octal Hexadecimal Decimal Octal Binary
29. 29. Decimal to Octal <ul><li>Technique </li></ul><ul><ul><li>Divide by 8 </li></ul></ul><ul><ul><li>Keep track of the remainder </li></ul></ul>
30. 30. Example 1234 10 = ? 8 8 1234 154 2 1234 10 = 2322 8 8 19 2 8 2 3 8 0 2
32. 32. Decimal to Hexadecimal <ul><li>Technique </li></ul><ul><ul><li>Divide by 16 </li></ul></ul><ul><ul><li>Keep track of the remainder </li></ul></ul>
33. 33. Example 1234 10 = ? 16 1234 10 = 4D2 16 16 1234 77 2 16 4 13 = D 16 0 4
34. 34. Binary to Octal Hexadecimal Decimal Octal Binary
35. 35. Binary to Octal <ul><li>Technique </li></ul><ul><ul><li>Group bits in threes, starting on right </li></ul></ul><ul><ul><li>Convert to octal digits </li></ul></ul>
36. 36. Example 1011010111 2 = ? 8 1011010111 2 = 1327 8 1 011 010 111 1 3 2 7
38. 38. Binary to Hexadecimal <ul><li>Technique </li></ul><ul><ul><li>Group bits in fours, starting on right </li></ul></ul><ul><ul><li>Convert to hexadecimal digits </li></ul></ul>
39. 39. Example 1010111011 2 = ? 16 <ul><li>10 1011 1011 </li></ul><ul><li>B B </li></ul>1010111011 2 = 2BB 16
41. 41. Octal to Hexadecimal <ul><li>Technique </li></ul><ul><ul><li>Use binary as an intermediary </li></ul></ul>
42. 42. Example 1076 8 = ? 16 1076 8 = 23E 16 1 0 7 6 001 000 111 110 2 3 E
44. 44. Hexadecimal to Octal <ul><li>Technique </li></ul><ul><ul><li>Use binary as an intermediary </li></ul></ul>
45. 45. Example 1F0C 16 = ? 8 1F0C 16 = 17414 8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4
46. 46. Exercise – Convert ... Skip answer Answer Don’t use a calculator! 1AF 703 1110101 33 Hexa- decimal Octal Binary Decimal
47. 47. Exercise – Convert … Answer 1AF 657 110101111 431 1C3 703 111000011 451 75 165 1110101 117 21 41 100001 33 Hexa- decimal Octal Binary Decimal
48. 48. Common Powers (1 of 2) <ul><li>Base 10 </li></ul>T tera 10 12 G giga 10 9 M mega 10 6 k kilo 10 3 m milli 10 -3  micro 10 -6 n nano 10 -9 p pico 10 -12 Symbol Preface Power 1000000000000 1000000000 1000000 1000 .001 .000001 .000000001 .000000000001 Value
49. 49. Common Powers (2 of 2) <ul><li>Base 2 </li></ul><ul><li>What is the value of “k”, “M”, and “G”? </li></ul><ul><li>In computing, particularly w.r.t. memory , the base-2 interpretation generally applies </li></ul>G Giga 2 30 M mega 2 20 k kilo 2 10 Symbol Preface Power 1073741824 1048576 1024 Value
50. 50. Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 2 30 =
51. 51. Exercise – Free Space <ul><li>Determine the “free space” on all drives on a machine in the lab </li></ul>etc. E: D: C: A: GB Bytes Drive Free space
52. 52. Review – multiplying powers <ul><li>For common bases, add powers </li></ul>2 6  2 10 = 2 16 = 65,536 or… 2 6  2 10 = 64  2 10 = 64k a b  a c = a b+c
53. 53. Binary Addition (1 of 2) <ul><li>Two 1-bit values </li></ul>pp. 36-38 “two” 10 1 1 1 0 1 1 1 0 0 0 0 A + B B A
54. 54. Binary Addition (2 of 2) <ul><li>Two n -bit values </li></ul><ul><ul><li>Add individual bits </li></ul></ul><ul><ul><li>Propagate carries </li></ul></ul><ul><ul><li>E.g., </li></ul></ul>10101 21 + 11001 + 25 101110 46 1 1
55. 55. Multiplication (1 of 3) <ul><li>Decimal (just for fun) </li></ul>pp. 39 35 x 105 175 000 35 3675
56. 56. Multiplication (2 of 3) <ul><li>Binary, two 1-bit values </li></ul>1 1 1 0 0 1 0 1 0 0 0 0 A  B B A
57. 57. Multiplication (3 of 3) <ul><li>Binary, two n -bit values </li></ul><ul><ul><li>As with decimal values </li></ul></ul><ul><ul><li>E.g., </li></ul></ul>1110 x 1011 1110 1110 0000 1110 10011010
58. 58. Fractions <ul><li>Decimal to decimal (just for fun) </li></ul>pp. 46-50 3.14 => 4 x 10 -2 = 0.04 1 x 10 -1 = 0.1 3 x 10 0 = 3 3.14
59. 59. Fractions <ul><li>Binary to decimal </li></ul>pp. 46-50 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 2 0 = 0.0 1 x 2 1 = 2.0 2.6875
60. 60. Fractions <ul><li>Decimal to binary </li></ul>p. 50 3.14579 .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. 11.001001...
61. 61. Exercise – Convert ... Skip answer Answer Don’t use a calculator! C.82 3.07 101.1101 29.8 Hexa- decimal Octal Binary Decimal
62. 62. Exercise – Convert … Answer C.82 14.404 1100.10000010 12.5078125 3.1C 3.07 11.000111 3.109375 5.D 5.64 101.1101 5.8125 1D.CC… 35.63… 11101.110011… 29.8 Hexa- decimal Octal Binary Decimal
63. 63. Thank you