1. 1. Number Systems Chapt. 2

2. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7NoYes Hexa- decimal 160, 1, … 9, A, B, … F NoYes

3. Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal 0000 1111 21022 31133 410044 510155 611066 711177 p. 33

4. Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal 81000108 91001119 10101012A 11101113B 12110014C 13110115D 14111016E 15111117F

5. Quantities/Counting (3 of 3) DecimalBinaryOctal Hexa- decimal 16100002010 17100012111 18100102212 19100112313 20101002414 21101012515 22101102616 23101112717 Etc.

6. Conversion Among Bases The possibilities: Hexadecimal DecimalOctal Binary pp. 40-46

7. Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base

8. Decimal to Decimal (just for fun) Hexadecimal DecimalOctal Binary Next slide…

9. 125 10 =>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = 100 125 Base Weight

10. Binary to Decimal Hexadecimal DecimalOctal Binary

11. 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

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. Decimal to Binary Hexadecimal DecimalOctal Binary

14. 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.

15. Example 125 10 = ? 2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 125 10 = 1111101 2

16. Octal to Binary Hexadecimal DecimalOctal Binary

17. Octal to Binary Technique –Convert each octal digit to a 3-bit equivalent binary representation

18. Example 705 8 = ? 2 7 0 5 111 000 101 705 8 = 111000101 2

19. Hexadecimal to Binary Hexadecimal DecimalOctal Binary

20. Hexadecimal to Binary Technique –Convert each hexadecimal digit to a 4-bit equivalent binary representation

21. Example 10AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10AF 16 = 0001000010101111 2

22. Binary to Octal Hexadecimal DecimalOctal Binary

23. Binary to Octal Technique –Group bits in threes, starting on right –Convert to octal digits

24. Example 1011010111 2 = ? 8 1 011 010 111 1 3 2 7 1011010111 2 = 1327 8

25. Binary to Hexadecimal Hexadecimal DecimalOctal Binary

26. Binary to Hexadecimal Technique –Group bits in fours, starting on right –Convert to hexadecimal digits

27. Example 1010111011 2 = ? 16 10 1011 1011 2 B B 1010111011 2 = 2BB 16

28. Octal to Hexadecimal Hexadecimal DecimalOctal Binary

29. Octal to Hexadecimal Technique –Use binary as an intermediary

30. Example 1076 8 = ? 16 1 0 7 6 001 000 111 110 2 3 E 1076 8 = 23E 16

31. Hexadecimal to Octal Hexadecimal DecimalOctal Binary

32. Hexadecimal to Octal Technique –Use binary as an intermediary

33. Example 1F0C 16 = ? 8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C 16 = 17414 8

34. Exercise – Convert... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal 33 1110101 703 1AF Skip answer Answer

35. Exercise – Convert … DecimalBinaryOctal Hexa- decimal 331000014121 117111010116575 4511110000117031C3 4311101011116571AF Answer

36. Common Powers (1 of 2) Base 10 PowerPrefaceSymbol 10 -12 picop 10 -9 nanon 10 -6 micro  10 -3 millim 10 3 kilok 10 6 megaM 10 9 gigaG 10 12 teraT Value.000000000001.000000001.000001.001 1000 1000000 1000000000 1000000000000

37. Common Powers (2 of 2) Base 2 PowerPrefaceSymbol 2 10 kilok 2 20 megaM 2 30 GigaG Value 1024 1048576 1073741824 What is the value of “k”, “M”, and “G”? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

38. Binary Addition (1 of 2) Two 1-bit values pp. 36-38 ABA + B 000 011 101 1110 “two”

39. Binary Addition (2 of 2) Two n-bit values –Add individual bits –Propagate carries –E.g., 10101 21 + 11001 + 25 101110 46 11

40. Fractions Decimal to decimal (just for fun) pp. 46-50 3.14 =>4 x 10 -2 = 0.04 1 x 10 -1 = 0.1 3 x 10 0 = 3 3.14

41. Fractions Binary to decimal 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

42. Fractions Decimal to binary 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...

43. Exercise – Convert... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal 29.8 101.1101 3.07 C.82 Skip answer Answer

44. Exercise – Convert … DecimalBinaryOctal Hexa- decimal 29.811101.110011…35.63…1D.CC… 5.8125101.11015.645.D 3.10937511.0001113.073.1C 12.50781251100.1000001014.404C.82 Answer

45. Thank you Next topic