1. NUMBER SYSTEMS Prepared By K.Anitha Associate Professor Dept of ECE PVPSIT

2. Common Number Systems System BaseSymbols Used Decimal100, 1, … 9 Binary20, 1 Octal80, 1, … 7 Hexa-decimal160, 1, … 9, A, B, … F

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

6. Conversion Among Bases The possibilities: DecimalOctal Binary Hexadecimal

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

8. Decimal to Decimal 125 10 =>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = 100 125 Base Weight

9. 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 the right most position – Add the results

10. 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

11. Octal to Decimal Technique – Multiply each bit by 8 n, where n is the “weight” of the bit – The weight is the position of the bit, starting from right most position – Add the results

12. Example 724 8 => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10

13. 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 right most position – Add the results

14. 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

15. Decimal to Binary Technique – Continuously divide the number by 2, keep track of the remainders generated – Write the remainders from bottom to top

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

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 Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation

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

21. Decimal to Octal Technique – Continuously Divide by 8 – Keep track of the remainders

22. Example 8 1234 8 154 2 8 19 2 8 2 3 0 2 1234 10 = 2322 8 1234 10 = ? 8

23. Decimal to Hexadecimal Technique – Divide by 16 – Keep track of the remainders

24. Example 1234 10 = ? 16 16 1234 16 77 2 164 13 0 4 1234 10 = 4D2 16

25. Binary to Octal Technique – Make three bits as a Group, starting from right – Convert to octal digits

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

27. Binary to Hexadecimal Technique – Make four bits as a Group, starting from right – Convert to hexadecimal digits

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

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

30. Example 0010/0011/1110 2 3 E 1076 8 = ? 16 1 0 7 6 001 000 111 110 1076 8 = 23E 16

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

32. Example 0/001/111/100/001/100 0 1 7 4 1 4 1F0C 16 = ? 8 1 F 0 C 0001 1111 0000 1100 1F0C 16 = 17414 8

33. Exercise – Convert DecimalBinaryOctal Hexa- decimal 33 1110101 703 1AF

34. Exercise – Convert DecimalBinaryOctal Hexa- decimal 331000014121 117111010116575 4511110000117031C3 4311101011116571AF

35. Thank you