Computing 6
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This document discusses different number systems including decimal, binary, octal, and hexadecimal. It provides examples of how to convert between these number systems. The key techniques described are using place value and grouping bits to convert between binary, octal, decimal, and hexadecimal representations of numbers. Conversions can be performed by using binary as an intermediary between other bases.
Computing 6
- 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 Hexa- decimal 16 0, 1, … 9, A, B, … F No No
- 3. QUANTITIES/COUNTING (1 OF 3) Decimal Binary Octal Hexa- 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
- 4. QUANTITIES/COUNTING Decimal Binary Octal Hexa- 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
- 5. CONVERSION AMONG BASES The possibilities: Hexadecimal Decimal Octal Binary
- 6. QUICK EXAMPLE 2510 = 110012 = 318 = 1916 Base
- 7. DECIMAL TO DECIMAL (JUST FOR FUN) Hexadecimal Decimal Octal Binary Next slide…
- 8. 12510 => 5 x 100= 5 2 x 101= 20 1 x 102= 100 125 Base Weight
- 9. BINARY TO DECIMAL Hexadecimal Decimal Octal Binary
- 10. 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
- 11. EXAMPLE 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 Bit “0”
- 12. OCTAL TO DECIMAL Hexadecimal Decimal Octal Binary
- 13. 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
- 14. EXAMPLE 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810
- 15. HEXADECIMAL TO DECIMAL Hexadecimal Decimal Octal Binary
- 16. 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
- 17. EXAMPLE ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 18. DECIMAL TO BINARY Hexadecimal Decimal Octal Binary
- 19. 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.
- 20. EXAMPLE 12510 = ?2 2 125 62 12 31 02 15 1 2 7 1 2 3 12 1 12 0 1 12510 = 11111012
- 21. OCTAL TO BINARY Hexadecimal Decimal Octal Binary
- 22. OCTAL TO BINARY Technique Convert each octal digit to a 3-bit equivalent binary representation
- 23. EXAMPLE 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 24. HEXADECIMAL TO BINARY Hexadecimal Decimal Octal Binary
- 25. HEXADECIMAL TO BINARY Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation
- 26. EXAMPLE 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 27. DECIMAL TO OCTAL Hexadecimal Decimal Octal Binary
- 28. DECIMAL TO OCTAL Technique Divide by 8 Keep track of the remainder
- 29. EXAMPLE 123410 = ?8 8 1234 154 2 8 19 28 2 3 8 0 2 123410 = 23228
- 30. DECIMAL TO HEXADECIMAL Hexadecimal Decimal Octal Binary
- 31. DECIMAL TO HEXADECIMAL Technique Divide by 16 Keep track of the remainder
- 32. EXAMPLE 123410 = ?16 123410 = 4D216 16 1234 77 216 4 13 = D16 0 4
- 33. BINARY TO OCTAL Hexadecimal Decimal Octal Binary
- 34. BINARY TO OCTAL Technique Group bits in threes, starting on right Convert to octal digits
- 35. EXAMPLE 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 36. BINARY TO HEXADECIMAL Hexadecimal Decimal Octal Binary
- 37. BINARY TO HEXADECIMAL Technique Group bits in fours, starting on right Convert to hexadecimal digits
- 38. EXAMPLE 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 39. OCTAL TO HEXADECIMAL Hexadecimal Decimal Octal Binary
- 40. OCTAL TO HEXADECIMAL Technique Use binary as an intermediary
- 41. EXAMPLE 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 42. HEXADECIMAL TO OCTAL Hexadecimal Decimal Octal Binary
- 43. HEXADECIMAL TO OCTAL Technique Use binary as an intermediary
- 44. EXAMPLE 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148