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1. Number System Prepared by: Risala Tasin Khan PROFESSOR IIT

2. 2 Base-N Number System Base N N Digits: 0, 1, 2, 3, 4, 5, …, N-1 Example: 1045 N Positional Number System Digit d o is the least significant digit (LSD). Digit d n -1 is the most significant digit (MSD).

3. 3 Decimal Number System Base 10 Ten Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 1045 10 Positional Number System Digit d is the least significant digit (LSD). Digit d n -1 is the most significant digit (MSD).

4. 4 Binary Number System Base 2 Two Digits: 0, 1 Example: 1010110 2 Positional Number System B inary Dig its are called Bits Bit b o is the least significant bit (LSB). Bit b n-1 is the most significant bit (MSB).

5. 5 Hexadecimal Number System Base 16 Sixteen Digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Example: EF56 16 Positional Number System 0000 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 A 1011 B 1100 C 1101 D 1110 E 1111 F

6. 6 Definitions nybble = 4 bits byte = 8 bits (short) word = 2 bytes = 16 bits (double) word = 4 bytes = 32 bits (long) word = 8 bytes = 64 bits 1K (kilo or “kibi”) = 1,024 1M (mega or “mebi”) = (1K)*(1K) = 1,048,576 1G (giga or “gibi”) = (1K)*(1M) = 1,073,741,824

7. Quantities/Counting (1 of 3) Decimal Binary Octal Hexa- decimal 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

8. Quantities/Counting (2 of 3) 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

9. Conversion Among Bases The possibilities: Hexadecimal Decimal Octal Binary

10. Binary to Decimal Hexadecimal Decimal Octal 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 = 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. 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 1 x 2 1 = 2.0 2.6875

14. Decimal to Binary Hexadecimal Decimal Octal Binary

15. 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.

16. 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

17. 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...

18. Octal to Binary Hexadecimal Decimal Octal Binary

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

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

21. Binary to Octal Hexadecimal Decimal Octal Binary

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

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

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

27. Binary to Hexadecimal Hexadecimal Decimal Octal Binary

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

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

30. Decimal to Octal Hexadecimal Decimal Octal Binary

31. Decimal to Octal Technique Divide by 8 Keep track of the remainder

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

33. Decimal to Hexadecimal Hexadecimal Decimal Octal Binary

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

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

36. Hexadecimal to Octal Hexadecimal Decimal Octal Binary

37. Hexadecimal to Octal Technique Use binary as an intermediary

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

39. Binary Addition Two 1-bit values pp. 36-38 A B A + B 1 1 1 1 1 1 10 “two”

40. Binary Addition Two n -bit values Add individual bits Propagate carries E.g., 10101 21 + 11001 + 25 101110 46 1 1

41. Multiplication Binary, two 1-bit values A B A  B 1 1 1 1 1

42. Multiplication Binary, two n -bit values As with decimal values E.g., 1110 x 1011 1110 1110 0000 1110 10011010

43. 43 Complements 1’s complement To calculate the 1’s complement of a binary number just “flip” each bit of the original binary number. E.g. 0  1 , 1  01010100100  10101011011

44. 44 Complements 2’s complement To calculate the 2’s complement just calculate the 1’s complement, then add 1. 01010100100  10101011011 + 1= 10101011100

45. 45 Complements Note the 2’s complement of the 2’s complement is just the original number N EX: let N = 01010100100 2’s comp of N = M = 10101011100 2’s comp of M = 01010100100 = N

46. 46 Arithmetic Subtraction Borrow Method This is the technique you learned in grade school For binary numbers, we have 0 - 0 = 0 1 - 0 = 1 1 - 1 = 0 0 - 1 = 1 with a “borrow” 1

47. 47 Binary Subtraction Note: A – (+B) = A + (-B) A – (-B) = A + (-(-B))= A + (+B) In other words, we can “subtract” B from A by “adding” –B to A. However, -B is just the 2’s complement of B, so to perform subtraction, we 1. Calculate the 2’s complement of B 2. Add A + (-B)

48. 48 Binary Subtraction - Example Let n=4, A=0100 2 (4 10 ), and B=0010 2 (2 10 ) Let’s find A+B, A-B and B-A 0 1 0 0 + 0 0 1 0  (4) 10  (2) 10 0 11 0 6 A+B

49. 49 Binary Subtraction - Example 0 1 0 0 - 0 0 1 0  (4) 10  (2) 10 10 0 1 0 2 A-B 0 1 0 0 + 1 1 1 0  (4) 10  (-2) 10 A+ (-B) “Throw this bit” away since n=4

50. 50 Binary Subtraction - Example 0 0 1 0 - 0 1 0 0  (2) 10  (4) 10 1 1 1 0 -2 B-A 0 0 1 0 + 1 1 0 0  (2) 10  (-4) 10 B + (-A) 1 1 1 0 2 = - 0 0 1 0 2 = -2 10