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# Number systems

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### Number systems

1. 1. 1. Number Systems
2. 2. Common Number Systems 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
3. 3. Quantities/Counting (1 of 3) Decimal Binary HexaOctal decimal 0 1 2 0 1 10 0 1 2 0 1 2 3 4 5 6 7 11 100 101 110 111 3 4 5 6 7 3 4 5 6 7
4. 4. Quantities/Counting (2 of 3) Decimal Binary HexaOctal decimal 8 9 10 1000 1001 1010 10 11 12 8 9 A 11 12 13 14 15 1011 1100 1101 1110 1111 13 14 15 16 17 B C D E F
5. 5. Quantities/Counting (3 of 3) Decimal Binary HexaOctal decimal 16 17 18 10000 10001 10010 20 21 22 10 11 12 19 20 21 22 23 10011 10100 10101 10110 10111 23 24 25 26 27 13 14 15 16 17 Etc.
6. 6. Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal pp. 40-46
7. 7. Quick Example 2510 = 110012 = 318 = 1916 Base
8. 8. Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide…
9. 9. Weight 12510 => 5 x 100 2 x 101 1 x 102 Base = 5 = 20 = 100 125
10. 10. Binary to Decimal Decimal Octal Binary Hexadecimal
11. 11. 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
12. 12. Example Bit “0” 1010112 => 1 1 0 1 0 1 x x x x x x 20 21 22 23 24 25 = = = = = = 1 2 0 8 0 32 4310
13. 13. Octal to Decimal Decimal Octal Binary Hexadecimal
14. 14. Octal to Decimal • Technique – Multiply each bit by 8n, 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
15. 15. Example 7248 => 4 x 80 = 2 x 81 = 7 x 82 = 4 16 448 46810
17. 17. 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
18. 18. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
19. 19. Decimal to Binary Decimal Octal Binary Hexadecimal
20. 20. Decimal to Binary • There are two methods that can be used to convert decimal numbers to binary: – Repeated subtraction method – Repeated division method • Both methods produce the same result and you should use whichever one you are most comfortable with.
21. 21. The Repeated Subtraction method – Step 1: • Starting with the 1s place, write down all of the binary place values in order until you get to the first binary place value that is GREATER THAN the decimal number you are trying to convert. 1024 512 256 128 64 32 16 8 4 2 1
22. 22. The Repeated Subtraction method – Step 2: • Mark out the largest place value (it just tells us how many place values we need). 853 1024 512 256 128 64 32 16 8 4 2 1
23. 23. The Repeated Subtraction method • – Step 3: • Subtract the largest place value from the decimal number. Place a “1” under that place value. 853 - 512 = 341 512 256 128 64 32 16 8 4 2 1 1
24. 24. The Repeated Subtraction method – Step 4: • For the rest of the place values, try to subtract each one from the previous result. – If you can, place a “1” under that place value. – If you can’ t, place a “0” under that place value.
25. 25. The Repeated Subtraction method • – Step 5: • Repeat Step 4 until all of the place values have been processed. • The resulting set of 1s and 0s is the binary equivalent of the decimal number you started with.
26. 26. The Repeated Subtraction method
27. 27. Repeated division method 12510 = ?2 2 125 2 62 2 31 2 15 7 2 3 2 1 2 0 1 0 1 1 1 1 1 12510 = 11111012
28. 28. Octal to Binary Decimal Octal Binary Hexadecimal
29. 29. Octal to Binary • Technique – Convert each octal digit to a 3-bit equivalent binary representation
30. 30. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
32. 32. Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation
33. 33. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
34. 34. Decimal to Octal Decimal Octal Binary Hexadecimal
35. 35. Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder
36. 36. Example 123410 = ?8 8 8 8 8 1234 154 19 2 0 2 2 3 2 123410 = 23228
38. 38. Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder
39. 39. Example 123410 = ?16 16 16 16 1234 77 4 0 2 13 = D 4 123410 = 4D216
40. 40. Binary to Octal Decimal Octal Binary Hexadecimal
41. 41. Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits
42. 42. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278