1. Binary & Hex Review

2. Bits & Bytes

3. Bits And Bit Patterns N bits gives 2N possible patterns
2 bits = 4 patterns
3 bits = 8 patterns

4. Bytes Smallest usable chunk of memory: byte : 8 bits
Special names for large collections of bytes:
Name
Number of Bytes
power of 2
byte 1 20
kilobyte (KB)
1024
210
megabyte (MB)
1,048,576
220
gigabyte (GB)
1,073,741,824
230
terabyte (TB)
1,099,511,627,776
240

5. Metric Units
Standard metric units

6. Explaining Size
2TB Drive:

7. Bi units Metric prefixes May refer to powers of 2 or 10
Roughly equivalent
GB GiB
109 = ~ = 230

8. Bi units Memory measured in powers of 2
Network / Processor in powers of 10
Disk
Manufactures powers of 10
OS powers of 2
2 TB = 2 * * 1012 / 240 = GiB

9. This Class
Unless stated otherwise, assume all measurements are powers of 2

10. Powers of 2
How many bytes is 16GB?

11. Powers of 2
How many bytes is 16GB?
= 16 * 230
= 24 * 230
= 234

12. Unsigned Binary

13. Bases Place based number representations: 1 2 5 9 1 ten thousands 104
103
hundreds
102
tens
101
ones
100 1 2 5 9
thirty-twos 25
sixteens 24
eights 23
fours 22
twos 21
ones 20 1

14. Specifying Base
Specify base as subscript: 610 = 1102

15. Binary Number Interpretation
Table Method: = = 10510
128 64 32 16 8 4 2 1

16. Leading 0's Leading 0's do not matter: 101 = 0101 = 00000101
Byte : fundamental storage unit : 8 bits
Often write binary numbers in multiples 8 bits
128 64 32 16 8 4 2 1

17. Conversion With Division/ Multiplication
Binary works in powers of 2
Multiplying/dividing by 2 shifts digits

18. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105

19. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1

20. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1

21. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2

22. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01

23. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01
26 ÷ 2 13
001

24. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01
26 ÷ 2 13
001
13 ÷ 2 6
1001
6 ÷ 2 3
01001
3 ÷ 2
101001
1 ÷ 2

25. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01
26 ÷ 2 13
001
13 ÷ 2 6
1001
6 ÷ 2 3
01001
3 ÷ 2
101001
1 ÷ 2
128 64 32 16 8 4 2 1
= 105

26. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101

27. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
0 * 2

28. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101
0 * 2 + 1

29. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1

30. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1
1 * 2

31. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01
1 * 2 + 1

32. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3

33. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3
3 * 2

34. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3
3 * 2 + 0

35. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6

36. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6
6 * 2

37. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6 _
6 * 2 + 1

38. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6 _ 13

39. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6 _ 13
128 64 32 16 8 4 2 1
= 13

40. Unsigned Binary Math

41. Adding Adding base 2: 1 1 = 1 2 = 10 = keep 0 and carry 1
1 = 1
2 = 10 = keep 0 and carry 1
3 = 11 = keep 1 carry 1 1

42. Adding Adding base 2: 1 7 5 12 1 = 1 2 = 10 = keep 0 and carry 1
1 = 1
2 = 10 = keep 0 and carry 1
3 = 11 = keep 1 carry 1 7 5 12 1

43. Overflow
What is ? 1

44. Overflow
What is ?
Overflow: number gets too big to hold in the digits we have 1

45. Signed Binary

46. Sign Magnitude?? - / + are two choices Ex: + 5 - 5 sign 4 2 1 sign 4 2
Use a bit value to represent sign
Ex:
+ 5
- 5
sign 4 2 1
sign 4 2 1

47. Sign Magnitude??
Would need new math algorithm… 1 -3 1

48. 2’s complement 2's Complement System: First bit indicates sign:
0 : Positive #: works like normal
What is:
00110
Bit Values:
4+2
Value: 6

49. 2’s complement 2's Complement System: First bit indicates sign:
1 : Negative #: defined as indicated value – 2n where n is number of bits
What is:
11010
Bit Values:
= 26
2n :
25 = 32
Value:
26 – 32 = -6

50. 2’s complement negation
To negate a number, reverse bits and add 1
What is -3 as 4-bit 2's complement number?:
0011 start with reverse in two's complement
Check: = = 13
2n = 24 = 16
13 – 16 = -3

51. 2’s complement negation
To negate a number, reverse bits and add 1
What is 1101 as a 4-bit 2’s complement number?
1101 start with negative pattern reverse interpret add one
negated is 3, so it must be -3

52. More Examples
What are? 1 1 1

53. More Examples
What are? 1 1 1
Starts with 0 = normal:

54. More Examples What are? 4 + 2 + 1 7 1 -1 1 1 1 11 1 1
Starts with 0 = normal:
Starts with 1 = negative:
Flip Bits
Add one:
Read number, make negative:
1 -1 1

55. More Examples What are? 4 + 2 + 1 7 1 -1 8 -8 1 1 1 1 1 11 1 1
Starts with 0 = normal:
Starts with 1 = negative:
Flip Bits
Add one:
Read number, make negative:
1 -1
Starts with 1 = negative:
Flip Bits
Add one:
Read number, make negative: 1 1 1

56. Range
2's complement uses half of range for negative, half for 0/positive
4 bits:
24 = 16 values
Unsigned 0 to 15
2's Complement -8 to 7

57. 4 bits Biggest positive: 7 -1 : Lowest negative: -8 1 1 Flip (0) 11 1
Flip (0)
+1 = 1 1
Flip (7)
+1 = 8

58. 8 bits Biggest positive: 127 -1 : Lowest negative: -128 1 1 Flip (0) 11 1
Flip (0)
+1 = 1 1
Flip (127)
+1 = 128

59. 2’s complement addition
Standard addition algorithm works fine: : 1 -3 1

60. 2’s complement addition
Standard addition algorithm works fine: : 1 -3 -2 1
???
110
Reverse Bits
001
Interpret 1
Add One 2
Value -2

61. 2’s complement
2 + 4 = 1

62. 2’s complement
2 + 4 = 6 Success – answer fits in four bits Ignore extra carry 1

63. 2’s complement -3 + -2 = 1 3 0011 Reverse Bits 1100 Add one 1101 2
0010
Reverse Bits
1101
Add one
1110 = 1

64. 2’s complement
???
1011
Reverse Bits
0100
Add one
0101
Bit Values
1 + 4
Value -5
= -5 Success – answer fits in four bits Ignore extra carry 1

65. 2’s complement = 1

66. 2’s complement 4 + 6 = -6!?!?! Overflowed the 4 bits! 1 ??? 1010
Reverse Bits
0101
Add one
0110
Bit Values
2 + 4
Value -6
= -6!?!?! Overflowed the 4 bits! 1

67. 2’s complement -4 + -6 = 1 4 0100 Reverse Bits 1011 Add one 1100 6
0110
Reverse Bits
1001
Add one
1010 1

68. 2’s complement
= 6??? Overflowed the 4 bits! 1

69. Overflow
Big + numbers and Low – numbers can wrap!

70. 2’s complement Final column carry out and in are == : OK
Final column carry out and in are != : Overflow 1

71. Hexadecimal

72. Hexdecimal Base 16 Each column is a power of 16:
= 4096 * * * * 9 = =
4096’s
163
256’s
162
16’s
161
1’s
160 2 5 9

73. 13?
How do we represent 13???
4096’s
163
256’s
162
16’s
161
1’s
160

74. 13? How do we represent 13??? A-F = 10-15 4096’s 256’s 16’s 1’s 163
162
16’s
161
1’s
160

75. Hex Numbers 1A316 = 1 * 256 + 10 * 16 + 3 * 1 = 256 + 160 + 3 = 41910
4096
256 16 1 A 3

76. Decimal   Hex Division Methods: ÷ 16 7 12 (C) C7 1C7 Current Value
Quotient
Remainder
455
÷ 16 28 7 1
12 (C) C7
1C7

77. Hex
Hexdecimal = base 16
4 bits = 1 hex digit

78. Summary Hex is a bridge: Easier for us Still just binary for computers
System
Base
Symbols
Used by humans?
Used in computers?
Decimal 10
0, 1, … 9
Yes No
Binary 2
0, 1
Hex 16
0-9ABCDEF
No, but better than binary
No, but easily translates to binary