1. Binary System

2. Bits, Bytes 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.

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

4. Decimal System
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

5. Binary System 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.

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

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) = ( … )2
(110001)2 35 ÷2 1 17 ÷2 1 8 ÷2 4 ÷2 2 ÷2 1 ÷2 1 NO

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

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

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

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

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

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

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

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

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

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

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

21. Example 13 Convert decimal fraction into binary
( ) = ( … )2
.00001
integer * 2 2 * 2 * * 2
0.5000
0.5000 * 2 1 1

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

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

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

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

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

28. Example 19 1
11000
100100

29. Example 20 1
11000
110100

30. Example 21 1 1 1

31. Example 22 1 1 1

32. Example 23 1 1 1

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

34. Ex 24
borrow 1
11000
1100

35. Ex 25
borrow 1

36. Ex 26
borrow 1
11110
10001

37. Ex 27
borrow 1

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

39. Example 28
11000 *
00000

40. Example 29
11111 *

41. Example 30
1101.1 *
11011
00000

42. Example 31
11111 *
00000

43. Example 32
11.1 *
111

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

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

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

47. Exercises 4- binary addition 11011 + 111 1111001 + 111111

48. 5- binary multiplication
Exercises
5- binary multiplication
11011 * 111 *
1001 * 1001
0.101 * 0.01 *