1. 1. Number Systems

2. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 160, 1, … 9, A, B, … F No

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

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

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. Hexadecimal to Decimal Hexadecimal DecimalOctal Binary

14. 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

15. 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

16. Decimal to Binary Hexadecimal DecimalOctal Binary

17. 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.

18. 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

19. Hexadecimal to Binary Hexadecimal DecimalOctal Binary

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

21. Decimal to Hexadecimal Hexadecimal DecimalOctal Binary

22. Decimal to Hexadecimal Technique –Divide by 16 –Keep track of the remainder

23. Example 1234 10 = ? 16 1234 10 = 4D2 16 16 1234 77 2 16 4 13 = D 16 0 4

24. Binary to Hexadecimal Hexadecimal DecimalOctal Binary

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

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

27. Exercise – Convert... Don’t use a calculator! DecimalBinary Hexa- decimal 33 1110101 1AF

28. Exercise – Convert … DecimalBinary Hexa- decimal 3310000121 117111010175 4511110000111C3 4311101011111AF Answer

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

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

31. Multiplication (1 of 3) Decimal (just for fun) pp. 39 35 x 105 175 000 35 3675

32. Multiplication (2 of 3) Binary, two 1-bit values AB A  B 000 010 100 111

33. Multiplication (3 of 3) Binary, two n-bit values –As with decimal values –E.g., 1110 x 1011 1110 1110 0000 1110 10011010

34. Thank you