Binary System

1. Binary System

2. Common terms might describe file size or memory size: Bit: smallest unit of information Byte: a grouping of eight bits of information K: (kilobyte); 1024 bytes equals 1K of storage. Bits, Bytes

3. MB: (megabyte); about 1 million bytes of information GB: (gigabyte); about 1 billion bytes of information TB: (terabyte); about 1 million megabytes of information Bits, Bytes

4. Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... Use 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 We know that 5304 =5*103+3*102+0*101+4*100 Base 10 Decimal System

5. BINARY numbers: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, .... Written with only 2 digits: "0" and "1" In the same way as for decimal,1011 (binary) = 1*23+0*22+1*21+1*20= 11 (decimal) Base 2 Converting from binary to decimal is simple, just as for 1011 above. Binary System

6. How to know if 1011 is in binary or in decimal? Subscripts are used to show the base: 10112(binary number), 101110(decimal number) Binary System

7. Positive Binary Integers Decimal      Binary      0               0000             1               0001                            2               0010                            3               0011                           4               0100                            5               0101                            6               0110                            7               0111                            8              1000                           9              1001                          10             1010                          11             1011                    

8. Decimal ÷ 2 * 2 Binary

9. Example 1 Convert Decimal to binary (35)10 = ( … )2 (110001)2 35 ÷2 1 17 ÷2 1 ÷2 8 0 ÷2 4 0 2 ÷2 0 1 ÷2 1 NO

10. Example 2 Convert Decimal to binary (45)10 = ( … )2 (101101)2 45 ÷2 1 22 ÷2 0 ÷2 11 1 ÷2 5 1 2 ÷2 0 1 ÷2 1 NO

11. Example 3 Convert Decimal to binary (55)10 = ( … )2 (110111)2 55 ÷2 1 27 ÷2 1 ÷2 13 1 ÷2 6 0 3 ÷2 1 1 ÷2 1 NO

12. Example 4 Convert Decimal to binary (72)10 = ( … )2 (1001000)2 72 ÷2 0 36 ÷2 0 ÷2 18 0 ÷2 9 1 4 ÷2 0 2 ÷2 0 1 ÷2 1 NO

13. Example 5 Convert Decimal to binary (85)10 = ( … )2 (1010101)2 85 ÷2 1 42 ÷2 0 ÷2 21 1 ÷2 10 0 5 ÷2 1 2 ÷2 0 1 ÷2 1 NO

14. Example 6 convert binary to decimal ( 100101)2 = (….)10 1*20+0*21+1*22+0*23+0*24+1*25 = 37

15. Example 7 convert binary to decimal ( 111101)2 = (….)10 1*20+0*21+1*22+1*23+1*24+1*25 = 61

16. Example 8 convert binary to decimal ( 111111)2 = (….)10 1*20+1*21+1*22+1*23+1*24+1*25 = 63

17. Example 9 convert binary to decimal ( 1100000)2 = (….)10 0*20+0*21+0*22+0*23+0*24+1*25+1*22 = 96

18. Example 10 convert binary to decimal ( 1101001)2 = (….)10 1*20+0*21+0*22+1*23+0*24+1*25+1*22 = 105

19. Example 11Convert decimal fraction into binary (0.125)10 = ( … )2 .001 integer * 0.125 2 0.250 0 2 0 0.250 0.50 * 2 1 0.50 * 1

20. Example 12Convert decimal fraction into binary (0.0625)10 = ( … )2 .0001 integer * 0.0625 2 0.1250 0 2 0 0.1250 0.2500 * 2 0 0.2500 * 0.5000 0.5000 1 * 2 1

21. Example 13Convert decimal fraction into binary (0.03125)10 = ( … )2 .00001 integer * 0.03125 2 0.06250 0 2 0 0.06250 0.12500 * 2 0 0.12500 * 0.25000 0.25000 * 2 0.5000 0 0.5000 * 2 1 1

22. Example 14convert fraction binary to decimal ( .0001)2 =(….)10 0.0625 0*2-1 + 0*2-2 +0*2-3+ 1*2-4 = 1/16 =0.0625

23. Example 15 convert fraction binary to decimal ( .0011)2 =(….)10 0.0625 0*2-1 + 0*2-2 +1*2-3+ 1*2-4

24. Example 16Convert decimal into binary (35.125)10 = ( … )2 100011.001 integer 1 35 2 0.125 *2 0.250 0 1 17 2 0.50 0 0.250 *2 0 2 8 1 1 0.50 *2 0 2 4 0 2 2 1 1 2 .001 NO 100011

25. Example 17Convert decimal into binary (42.125)10 = ( … )2 101010.001 integer 0 42 2 0.125 *2 0.250 0 1 21 2 0.50 0 0.250 *2 0 2 10 1 1 0.50 *2 1 2 5 0 2 2 1 1 2 .001 NO 101010

26. Example 18Convert decimal into binary (39.125)10 = ( … )2 100111.001 integer 1 39 2 0.125 *2 0.250 0 1 19 2 0.50 0 0.250 *2 1 2 9 1 1 0.50 *2 0 2 4 0 2 2 1 1 2 .001 NO 101010

27. Addition 1 + 0 = 1 0 + 1 = 1 0 + 0 = 0 1 + 1 = 0 carry 1

28. Example 19 1 11000 +1100 100100

29. Example 20 1 11000 +1 1100 110100

30. Example 21 1 1 1 1011000 + 111100 10010100

31. Example 22 1 1 1 100100.000 + 1111101.001 10100001.001

32. Example 23 1 1 1 1 100101.000 + 1111101.001 11100010.001

33. Subtraction 1 - 1 = 0 1 - 0 = 1 0 - 0 = 0 0 - 1 = 1 borrow 1

34. Ex 24 borrow1 11000 - 1100 1100

35. Ex 25 borrow1 1111101.000 - 100100.001 1011000.111

36. Ex 26 borrow1 11110 - 1101 10001

37. Ex 27 borrow1 11111.000 - 110.001 11000.111

38. Multiplication 1 * 0 = 0 0 * 1 = 0 0 * 0 = 0 1 * 1 = 1

39. Example 28 11000 *110 00000 11000 11000 10010000

40. Example 29 11111 *11 11111 11111 1011101

41. Example 30 1101.1 *110.01 11011 00000 00000 11011 11011 1010000.011

42. Example 31 11111 *11101 11111 00000 11111 11111 11111 1110000011

43. Example 32 11.1 *1.11 111 111 111 101.001

44. Exercises 1- Convert Decimal to binary (25)10 = ( … )2 (80)10 = ( … )2 (59.125)10 = ( … )2 (31.0625)10 = ( … )2 (125)10 = ( … )2

45. Exercises 2- Convert binary to Decimal (…..)10 = ( 0.01)2 (…..)10 = ( 0.001)2 (…..)10 = ( 0.0001)2 (….)10 = ( 0.1)2

46. Exercises 3- Convert Decimal to binary (0.125)10 = ( … )2 (0.0625)10 = ( … )2 (0.03125)10 = ( … )2 (0.5)10 = ( … )2

47. Exercises 4- binary addition 11011 + 111 1111001 + 111111 10001 + 100001 0.1101 + 0.101 0.010001+ 0.0101 + 0.010111

48. Exercises 5- binary multiplication 11011 * 111 11001 * 11101 1001 * 1001 0.101 * 0.01 0.0111 * 0.0101