The document explains the concept of number systems based on different bases, specifically base 10 and base 2. It details how values are determined using place-value systems where the value of digits is influenced by their position. Additionally, the document illustrates how to calculate the weighted sum of digits to derive their respective numerical values.

1. Making sense out of 0’s and 1’s

2. Numbers
4710

4. • Based on 10

5. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

6. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)

7. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)
• Place-Value Based System

8. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)
• Place-Value Based System
• Value of symbol depends partially on place in number

9. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1

10. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1
• Place values increase by a factor of 10 as you move to left
Place values decrease by a factor of 10 as you move to right
Place values are 10 raised to power

11. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1
• Place values increase by a factor of 10 as you move to left
Place values decrease by a factor of 10 as you move to right
Place values are 10 raised to power
• Value of a number is the weighted sum of the digits where
each digit is multiplied by its place value before being added to the sum

13. • Based on 2

14. • Based on 2
• Uses 2 symbols (0, 1)

15. • Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)

16. • Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)
• Place-Value Based System

17. • Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)
• Place-Value Based System
• Value of symbol depends partially on place in number

18. • Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1

19. • Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1
• Place values increase by a factor of 2 as you move to left
Place values decrease by a factor of 2 as you move to right
Place values are 2 raised to power

20. • Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1
• Place values increase by a factor of 2 as you move to left
Place values decrease by a factor of 2 as you move to right
Place values are 2 raised to power
• Value of a number is the weighted sum of the digits where
each digit is multiplied by its place value before being added to the sum

21. • Based on 10
• Uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Largest symbol has value 1 less than 10 (9)
• Place-Value Based System
• Value of symbol depends partially on place in number
• Place values start with 1
• Place values increase by a factor of 10 as you move to
left
Place values decrease by a factor of 10 as you move to
right
Place values are 10 raised to power
• Value of a number is the weighted sum of the digits
where
each digit is multiplied by its place value before being
added
to the sum
• Based on 2
• Uses 2 symbols (0, 1)
• Largest symbol has value 1 less than 2 (1)
• Place-Value Based System
• Value of symbol depends partially on place in
number
• Place values start with 1
• Place values increase by a factor of 2 as you move to
left
Place values decrease by a factor of 2 as you move to
right
Place values are 2 raised to power
• Value of a number is the weighted sum of the digits
where
each digit is multiplied by its place value before
being
added to the sum

22. 1239
1 2 3 9
103 102 101 100
1000 100 10 1
9 x 1 = 9
3 x 10 = 30
2 x 100 = 200
1 x 1000 = 1000
----------------------
Total = 1239

23. 10011010111
1 0 0 1 1 0 1 0 1 1 1
210 29 28 27 26 25 24 23 22 21 20
1024 512 256 128 64 32 16 8 4 2 1
1 x 1 = 1
1 x 2 = 2
1 x 4 = 4
0 x 8 = 0
1 x 16 = 16
0 x 32 = 0
1 x 64 = 64
1 x 128 = 128
0 x 256 = 0
0 x 512 = 0
1 x 1024 = 1024
----------------------
Total = 1239

24. 10011010111
1 0 0 1 1 0 1 0 1 1 1
210 29 28 27 26 25 24 23 22 21 20
1024 512 256 128 64 32 16 8 4 2 1
1 x 1 = 1
1 x 2 = 2
1 x 4 = 4
1 x 16 = 16
1 x 64 = 64
1 x 128 = 128
1 x 1024 = 1024
----------------------
Total = 1239
We find the value
of the number by
simply adding all
the place values
that contain 1’s

25. What is the value of the Binary number 1 0 1 1 0 0 1 ?
Remember: Place values start at the right at 1 and double as you move to the lef
Place values: 64 32 16 8 4 2 1

26. What is the value of the Binary number 1 0 1 1 0 0 1 ?
Place values : 64 32 16 8 4 2 1
Answer: 64 + 16 + 8 + 1 => 89

27. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0

28. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0

29. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
128 64 32 16 8 4 2 1

30. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
128 64 32 16 8 4 2 1

31. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13 = 8 + 4 + 1
128 64 32 16 8 4 2 1

32. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 8 + 4 + 1
128 64 32 16 8 4 2 1

33. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 8 + 4 + 1
= 16 + 1
128 64 32 16 8 4 2 1

34. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 8 + 4 + 1
= 16 + 1
128 64 32 16 8 4 2 1

35. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
128 64 32 16 8 4 2 1

36. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1 = 32 -1
128 64 32 16 8 4 2 1

37. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1 = 32 -1
128 64 32 16 8 4 2 1

38. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 32 -1
128 64 32 16 8 4 2 1

39. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 32 -1
= 31 – 8
128 64 32 16 8 4 2 1

40. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 32 -1
= 31 – 8
128 64 32 16 8 4 2 1

41. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 16 + 8 + 4 + 1
= 32 -1
= 31 – 8
128 64 32 16 8 4 2 1

42. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 16 + 8 + 4 + 1
= 32 -1
= 31 – 8
= 31 – 2
128 64 32 16 8 4 2 1

43. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 198
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 16 + 8 + 4 + 1
= 32 -1
= 31 – 8
= 31 – 2
128 64 32 16 8 4 2 1

44. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 198
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 16 + 8 + 4 + 1
= 128 + 64 + 4 + 2
= 32 -1
= 31 – 8
= 31 – 2
128 64 32 16 8 4 2 1

45. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 198
= 170
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 16 + 8 + 4 + 1
= 128 + 64 + 4 + 2
= 32 -1
= 31 – 8
= 31 – 2
128 64 32 16 8 4 2 1

46. 1 1 0 1
1 0 0 0 1
1 1 1 1 1
1 0 1 1 1
1 1 1 0 1
1 1 0 0 0 1 1 0
1 0 1 0 1 0 1 0
= 13
= 17
= 31
= 23
= 29
= 198
= 170
= 8 + 4 + 1
= 16 + 1
= 16 + 8 + 4 + 2 + 1
= 16 + 4 + 2 + 1
= 16 + 8 + 4 + 1
= 128 + 64 + 4 + 2
= 128 + 32 + 8 + 2
= 32 -1
= 31 – 8
= 31 – 2
128 64 32 16 8 4 2 1

47. The binary representation of a number is the collection of
powers of two that add up to the number we want to represent.
The key will be to discover what powers of two will add up to the target
Base 10 number.

48. What is the binary representation of 27?
(What powers of 2 will add up to 27?)
We can find these by subtracting different powers of two (no more than once)
until we reach zero.
These Powers of two must then be the ones we need to represent the number
in binary.
Start with highest that is less than or equal to the present value and work our
way down to zero.

49. What is the binary representation of 27?
(What powers of 2 will add up to 27?)
Start with highest that is less than or equal to the present value and work our
way down to zero.
The highest power that is less than or equal to 27 is 16:
27 – 16 = 11
Repeat for 11
11 – 8 = 3
Repeat for 3
3 – 2 = 1
Repeat for 1
1 – 1 = 0
Fill in the remaining places
with 0 0
Answer: 2710 = 110112
The binary number
must have a 1 in the
following positions:
16, 8, 2 and 1:
1 1 1 1
--- --- --- --- ---
16 8 4 2 1

50. 5
18
25
53
78
113
143
179
220

51. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0

52. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0 = 101

53. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 101

54. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 101
= 10010

55. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 101
= 10010

56. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 101
= 10010
= 11001

57. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 101
= 10010
= 11001

58. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 101
= 10010
= 11001
= 110101

59. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 101
= 10010
= 11001
= 110101

60. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 101
= 10010
= 11001
= 110101
= 1001100

61. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 101
= 10010
= 11001
= 110101
= 1001100

62. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001

63. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 143-128 = 15-8 = 7-4 = 3-2 = 1-1 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001

64. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 143-128 = 15-8 = 7-4 = 3-2 = 1-1 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001
= 10001111

65. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 143-128 = 15-8 = 7-4 = 3-2 = 1-1 = 0
= 179-128 = 51-32 = 19-16 = 3-2 = 1-1 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001
= 10001111

66. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 143-128 = 15-8 = 7-4 = 3-2 = 1-1 = 0
= 179-128 = 51-32 = 19-16 = 3-2 = 1-1 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001
= 10001111
= 10110011

67. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 143-128 = 15-8 = 7-4 = 3-2 = 1-1 = 0
= 179-128 = 51-32 = 19-16 = 3-2 = 1-1 = 0
= 220-128 = 92-64 = 28-16 = 12-8 = 4-4 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001
= 10001111
= 10110011

68. 5
18
25
53
78
113
143
179
220
= 5-4 = 1-1 = 0
= 18-16 = 2-2 = 0
= 25-16 = 9-8 = 1-1 = 0
= 53-32 = 21-16 = 5-4 = 1-1 = 0
= 78-64 = 12-8 = 4-4 = 0
= 113-64 = 49-32 = 17-16 = 1-1 = 0
= 143-128 = 15-8 = 7-4 = 3-2 = 1-1 = 0
= 179-128 = 51-32 = 19-16 = 3-2 = 1-1 = 0
= 220-128 = 92-64 = 28-16 = 12-8 = 4-4 = 0
= 101
= 10010
= 11001
= 110101
= 1001100
= 1110001
= 10001111
= 10110011
= 11011100