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
What the hell MS?

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. Unsigned Binary

10. 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

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

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

13. 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

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

15. 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

16. 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

17. 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

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 ÷ 2 52 1
52 ÷ 2

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
52 ÷ 2 26 01

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
52 ÷ 2 26 01
26 ÷ 2 13
001

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 26 01
26 ÷ 2 13
001
13 ÷ 2 6
1001
6 ÷ 2 3
01001
3 ÷ 2
101001
1 ÷ 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
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

23. 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

24. 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

25. 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

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
101 1

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
101 1
1 * 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 1 01
1 * 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 01 3

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 01 3
3 * 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 3
3 * 2 + 0

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 6

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 6
6 * 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 6 _
6 * 2 + 1

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 _ 13

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 _ 13
128 64 32 16 8 4 2 1
= 13

37. Unsigned Binary Math

38. 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

39. 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

40. Overflow
What is ? 1

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

42. Signed Binary

43. 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

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

45. 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

46. 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

47. 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

48. 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

49. More Examples
What are? 1 1 1

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

51. 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

52. 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

53. 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

54. 4 bits Biggest positive: 7 -1 : Lowest negative: -8 1 1 Flip (0)1 1
Flip (0)
+1 = 1 1
Flip (7)
+1 = 8

55. 8 bits Biggest positive: 127 -1 : Lowest negative: -128 1 1 Flip (0)1 1
Flip (0)
+1 = 1 1
Flip (127)
+1 = 128

56. 2’s complement addition
Standard addition algorithm works fine: : 1 -3 1

57. 2’s complement addition
Standard addition algorithm works fine: : 1 -3 -2 1
???
110
Reverse Bits
001
Interpret 1
Add One 2
Value -2

58. 2’s complement
2 + 4 = 1

59. 2’s complement
2 + 4 = 6 Success – answer fits in four bits Ignore extra carry 1

60. 2’s complement -3 + -2 = 1 3 0011 Reverse Bits 1100 Add one 1101 2
0010
Reverse Bits
1101
Add one
1110 = 1

61. 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

62. 2’s complement = 1

63. 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

64. 2’s complement -4 + -6 = 1 4 0100 Reverse Bits 1011 Add one 1100 6
0110
Reverse Bits
1001
Add one
1010 1

65. 2’s complement
= 6??? Overflowed the 4 bits! 1

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

67. Hexadecimal

68. 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

69. 13?
How do we represent 13???
4096’s
163
256’s
162
16’s
161
1’s
160

70. 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

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

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

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

74. 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