This Slide show illustrates the relationship between the binary and decimal numbering system.
by Don Mendonsa
Professor of IT and CS
Tidewater Community College
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
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
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