1. Data Representation Bits, Bytes, Binary, Hexadecimal

2. Computer Science “Joke” There are 10 types of people in the world: Those who understand binary… And those who don’t. Get it? If not, you will by the end of today’s lecture.. 1

3. Outline Bits and Bytes Binary Hexadecimal 2

4. Bits and Bytes Recall input / output of Logic Gates 0 => FALSE 1 => TRUE 3 A C B ABC = A & B 000 010 100 111

5. Bits and Bytes In reality, 0 and 1 represent voltage states (e.g., LOW / HIGH) These voltage states give the electrical circuit “life” 4 A C B ABC = A & B 000 010 100 111

6. Bits and Bytes Think of each input / output as a “light switch” Can be either “on” (TRUE, 1) or “off” (FALSE, 0) 5

7. Bits and Bytes Each “light switch” represents a bit bit stands for Binary digIT A bit can be either 0 or 1 A bit is the most basic element in computing Basis for logic gate circuits (logic gate circuits => more complex circuits) 6

8. Bits and Bytes When you put 8 bits together, you get a byte A byte is used to encode integers from 0 to 255 Or integers from -128 to 127 (more on this later) 7

9. Bits and Bytes In computers, a byte is the smallest addressable memory unit Memory units are defined in terms of bytes (not bits) E.g., Kilobytes (KB), Megabytes (MB), Gigabytes (GB), etc. Kilobyte: 1,000 bytes (thousand) Megabyte: 1,000,000 bytes (million) Gigabyte: 1,000,000,000 bytes (billion) 8

10. Bits and Bytes How many bits in 10 GB of storage? 9

11. Bits and Bytes How many bits in 10 GB of storage? 80,000,000,000 bits 10

12. Outline Bits and Bytes Binary Hexadecimal 11

13. Binary Numbers A computer’s machine language only uses ONES and ZEROS How does a computer represent a number using only bits? 12

14. Binary Numbers A byte can represent any number between 0 and 255 A byte is a sequence of eight bits, b 7 through b 0 13 b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0

15. Binary Numbers 14 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER

16. Binary Numbers 15 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER b 7 through b 0 can only be 0 or 1

17. Binary Numbers 16 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER Note how b 7 gets multiplied by 2 7

18. Binary Numbers 17 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER … b 6 gets multiplied by 2 6

19. Binary Numbers 18 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER … b 5 gets multiplied by 2 5

20. Binary Numbers 19 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER Et. cetera

21. Binary Numbers 20 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (2 7 ) b 6 × (2 6 ) b 5 × (2 5 ) b 4 × (2 4 ) b 3 × (2 3 ) b 2 × (2 2 ) b 1 × (2 1 ) b 0 × (2 0 ) NUMBER All the multiplications get added together into the integer number

22. Binary Numbers 21 b7b6b5b4b3b2b1b0b7b6b5b4b3b2b1b0 b 7 × (128) b 6 × (64) b 5 × (32) b 4 × (16) b 3 × (8) b 2 × (4) b 1 × (2) b 0 × (1) NUMBER

23. Binary Numbers: Example 22 0010 1010 Represents what number?

24. Binary Numbers: Example 23 0010 1010 0 × (128) 0 × (64) 1 × (32) 0 × (16) 1 × (8) 0 × (4) 1 × (2) 0 × (1)

25. Binary Numbers: Example 24 0010 1010 0 × (128) 0 × (64) 1 × (32) 0 × (16) 1 × (8) 0 × (4) 1 × (2) 0 × (1) 32 + 8 + 2 = 42

26. Binary Numbers: Example 25 0010 1010 Represents what number? 42

27. Binary Numbers: Example 26 1001 0111 Represents what number? Please find a partner to work with Check each other’s work

28. Binary Numbers: Example 27 1001 0111 Represents what number? 151

29. Binary Numbers Same technique applies to represent larger numbers 2 bytes (16 bits) 4 bytes (32 bits) 8 bytes (64 bits) What’s the largest unsigned (positive only) number that can be represented using 16 bits? Hint: the largest number for 8-bits is 255… We’ll cover signed (positive and negative) numbers next time… 28

30. Binary Numbers Example of 16-bit number: 29 1000 0000 0000 0001

31. Binary Numbers Example of 16-bit number: 30 1000 0000 0000 0001 = 2 15 + 2 0 = 32,768 + 1 = 32,769

32. Binary Numbers To convert decimal to binary… “Reverse” the process… 1.Find largest power of 2 that “fits” within the decimal value 2.Record 1 in the associated log 2 placeholder (in binary sequence) 3.Subtract power of 2 value from original decimal number 4.Repeat until nothing remains 31

33. Binary Numbers Example: Decimal to Binary What is the binary sequence for decimal number 250? 1.Find largest power of two that fits in 250 128 2.Record 1 in associated placeholder (128 = 2 7, 7 th position) => 1000 0000 3.Subtract power of two value from original decimal (250 – 128 = 122) 4.Repeat until there’s nothing left. 32

34. Binary Numbers Example: Decimal to Binary 1.Find largest power of two that fits in 122 64 2.Record 1 in associated placeholder (64 = 2 6, 6 th position) => 1100 0000 3.Subtract power of two value from original decimal ( 122 – 64 = 58 ) 4.Repeat until there’s nothing left. 33

35. Binary Numbers Example: Decimal to Binary 1.Find largest power of two that fits in 58 32 2.Record 1 in associated placeholder (32 = 2 5, 5 th position) => 1110 0000 3.Subtract power of two value from original decimal ( 58 - 32 = 26) 4.Repeat until there’s nothing left. 34

36. Binary Numbers Example: Decimal to Binary 1.Find largest power of two that fits in 26 16 2.Record 1 in associated placeholder (16 = 2 4, 4 th position) => 1111 0000 3.Subtract power of two value from original decimal ( 26 – 16 = 10) 4.Repeat until there’s nothing left. 35

37. Binary Numbers Example: Decimal to Binary 1.Find largest power of two that fits in 10 8 2.Record 1 in associated placeholder (8 = 2 3, 3 th position) => 1111 1000 3.Subtract power of two value from original decimal ( 10– 8 = 2 ) 4.Repeat until there’s nothing left. 36

38. Binary Numbers Example: Decimal to Binary 1.Find largest power of two that fits in 2 2 2.Record 1 in associated placeholder (2 = 2 1, 1 st position) => 1111 1010 3.Subtract power of two value from original decimal ( 2 – 2 = 0 ) 4.Repeat until there’s nothing left. DONE!! 37

39. Binary Numbers What is the binary sequence for decimal number 250? 38 250 = 128 + 64 + 32 + 16 + 8 + 2 = 2 7 + 2 6 + 2 5 + 2 4 + 2 3 + 2 1 = 1111 1010

40. Break Time… 39

41. Outline Bits and Bytes Binary Hexadecimal 40

42. Hexadecimal Writing and representing binary numbers can be cumbersome Hard to read / write large sequences of 0s and 1s Hexadecimal offers succinct way to represent binary numbers E.g., 41 1011 0101 = 0xB5

43. Hexadecimal Decimal numbers are base 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Hexadecimal numbers are base 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Note: we prefix hexadecimal numbers with 0x Why use hex? Easier to represent binary numbers… 42

44. Hexadecimal 4-bit numbers go from 0 to 15 0000 => 0 1111 => 15 Hexadecimal is base 16 Can easily represent numbers between 0 and 15 0 => 0x0 15 => 0xF Thus, we can represent 0000 as 0x0 1111 as 0xF 43

45. Hexadecimal This can be abstracted to 8-bit numbers… One hex number represents the first 4-bits One hex number represents the second 4-bits 44 1011 0101 1011 = 11 = 0xB 0xB5 0101 = 5 = 0x5

46. Hexadecimal This works with 16, 32, and 64-bit numbers One hex number per 4-bits 45 1111 0110 1100 0001 0x6 0xF6C1 0xC0xF0x1

47. Hexadecimal 46 1111 0110 1100 0001 (binary) = 0xF6C1 (hex) = 63,169 (decimal)

48. Hexadecimal (practice) Find a partner to work with Convert decimal number 77 A) to binary B) to hexadecimal 47

49. Hexadecimal (practice) Find a partner to work with Convert decimal number 77 A) to binary 01001101 B) to hexadecimal 0x4D 48

50. Hexadecimal (practice) Convert binary number 1110 0111 A) to decimal B) to hexadecimal 49

51. Hexadecimal (practice) Convert binary number 1110 0111 A) to decimal 231 B) to hexadecimal 0xE7 50

52. Hexadecimal (practice) Convert hexadecimal number 0xDF A) to binary B) to decimal 51

53. Hexadecimal (practice) Convert hexadecimal number 0xDF A) to binary 11011111 B) to decimal 223 52

54. Next steps… Next lecture: Binary Arithmetic (addition / subtraction) 2’s complement (signed numbers) Simple logic gate circuits to do addition… 53