1. Number Systems
Common Number Systems

System

Base Symbols

Used by
humans?

Used in
computers?

Decimal

10

0, 1, … 9

Yes

No

Binary
...
Quantities/Counting (1 of 3)
Decimal

Binary

HexaOctal decimal

0
1
2

0
1
10

0
1
2

0
1
2

3
4
5
6
7

11
100
101
110
11...
Quantities/Counting (2 of 3)
Decimal

Binary

HexaOctal decimal

8
9
10

1000
1001
1010

10
11
12

8
9
A

11
12
13
14
15

...
Quantities/Counting (3 of 3)
Decimal

Binary

HexaOctal decimal

16
17
18

10000
10001
10010

20
21
22

10
11
12

19
20
21...
Conversion Among Bases
• The possibilities:
Decimal

Octal

Binary

Hexadecimal
pp. 40-46
Quick Example

2510 = 110012 = 318 = 1916
Base
Decimal to Decimal (just for fun)

Decimal

Octal

Binary

Hexadecimal
Next slide…
Weight
12510 =>

5 x 100
2 x 101
1 x 102

Base

=
5
= 20
= 100
125
Binary to Decimal
Decimal

Octal

Binary

Hexadecimal
Binary to Decimal
• Technique
– Multiply each bit by 2n, where n is the “weight”
of the bit
– The weight is the position o...
Example
Bit “0”
1010112 =>

1
1
0
1
0
1

x
x
x
x
x
x

20
21
22
23
24
25

=
=
=
=
=
=

1
2
0
8
0
32
4310
Octal to Decimal
Decimal

Octal

Binary

Hexadecimal
Octal to Decimal
• Technique
– Multiply each bit by 8n, where n is the “weight”
of the bit
– The weight is the position of...
Example

7248 =>

4 x 80 =
2 x 81 =
7 x 82 =

4
16
448
46810
Hexadecimal to Decimal
Decimal

Octal

Binary

Hexadecimal
Hexadecimal to Decimal
• Technique
– Multiply each bit by 16n, where n is the
“weight” of the bit
– The weight is the posi...
Example

ABC16 =>

C x 160 = 12 x
1 =
12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810
Decimal to Binary
Decimal

Octal

Binary

Hexadecimal
Decimal to Binary
• There are two methods that can be used to
convert decimal numbers to binary:
– Repeated subtraction me...
The Repeated Subtraction method
– Step 1:
• Starting with the 1s place, write down all of
the binary place values in order...
The Repeated Subtraction method
– Step 2:
• Mark out the largest place value (it just tells
us how many place values we ne...
The Repeated Subtraction method
• – Step 3:
• Subtract the largest place value from the
decimal number. Place a “1” under ...
The Repeated Subtraction method
– Step 4:
• For the rest of the place values, try to
subtract each one from the previous r...
The Repeated Subtraction method
• – Step 5:
• Repeat Step 4 until all of the place values
have been processed.
• The resul...
The Repeated Subtraction method
Repeated division method
12510 = ?2

2 125
2 62
2 31
2 15
7
2
3
2
1
2
0

1
0
1
1
1
1
1

12510 = 11111012
Octal to Binary
Decimal

Octal

Binary

Hexadecimal
Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent
binary representation
Example
7058 = ?2

7

0

5

111 000 101

7058 = 1110001012
Hexadecimal to Binary
Decimal

Octal

Binary

Hexadecimal
Hexadecimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit
equivalent binary representation
Example
10AF16 = ?2

1

0

A

F

0001 0000 1010 1111

10AF16 = 00010000101011112
Decimal to Octal
Decimal

Octal

Binary

Hexadecimal
Decimal to Octal
• Technique
– Divide by 8
– Keep track of the remainder
Example
123410 = ?8

8
8
8
8

1234
154
19
2
0

2
2
3
2

123410 = 23228
Decimal to Hexadecimal
Decimal

Octal

Binary

Hexadecimal
Decimal to Hexadecimal
• Technique
– Divide by 16
– Keep track of the remainder
Example
123410 = ?16

16
16
16

1234
77
4
0

2
13 = D
4

123410 = 4D216
Binary to Octal
Decimal

Octal

Binary

Hexadecimal
Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits
Example
10110101112 = ?8

1 011 010 111

1

3

2

7

10110101112 = 13278
Binary to Hexadecimal
Decimal

Octal

Binary

Hexadecimal
Binary to Hexadecimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits
Example
10101110112 = ?16

10 1011 1011

2

B

B

10101110112 = 2BB16
Octal to Hexadecimal
Decimal

Octal

Binary

Hexadecimal
Octal to Hexadecimal
• Technique
– Use binary as an intermediary
Example
10768 = ?16
1

0

7

6

001

000

111

110

2

3

E

10768 = 23E16
Hexadecimal to Octal
Decimal

Octal

Binary

Hexadecimal
Hexadecimal to Octal
• Technique
– Use binary as an intermediary
Example
1F0C16 = ?8
1

F

0

C

0001

1111

0000

1100

1

7

4

1

4

1F0C16 = 174148
Exercise – Convert ...
Decimal
33

Binary

Octal

Hexadecimal

1110101

703
1AF
Don’t use a calculator!

Skip answer

Answ...
Exercise – Convert …
Answer

Hexadecimal

Decimal
33

Binary
100001

Octal
41

117

1110101

165

75

451

111000011

703
...
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Number systems

  1. 1. 1. Number Systems
  2. 2. Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexadecimal 16 0, 1, … 9, A, B, … F No No
  3. 3. Quantities/Counting (1 of 3) Decimal Binary HexaOctal decimal 0 1 2 0 1 10 0 1 2 0 1 2 3 4 5 6 7 11 100 101 110 111 3 4 5 6 7 3 4 5 6 7
  4. 4. Quantities/Counting (2 of 3) Decimal Binary HexaOctal decimal 8 9 10 1000 1001 1010 10 11 12 8 9 A 11 12 13 14 15 1011 1100 1101 1110 1111 13 14 15 16 17 B C D E F
  5. 5. Quantities/Counting (3 of 3) Decimal Binary HexaOctal decimal 16 17 18 10000 10001 10010 20 21 22 10 11 12 19 20 21 22 23 10011 10100 10101 10110 10111 23 24 25 26 27 13 14 15 16 17 Etc.
  6. 6. Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal pp. 40-46
  7. 7. Quick Example 2510 = 110012 = 318 = 1916 Base
  8. 8. Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide…
  9. 9. Weight 12510 => 5 x 100 2 x 101 1 x 102 Base = 5 = 20 = 100 125
  10. 10. Binary to Decimal Decimal Octal Binary Hexadecimal
  11. 11. Binary to Decimal • Technique – Multiply each bit by 2n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
  12. 12. Example Bit “0” 1010112 => 1 1 0 1 0 1 x x x x x x 20 21 22 23 24 25 = = = = = = 1 2 0 8 0 32 4310
  13. 13. Octal to Decimal Decimal Octal Binary Hexadecimal
  14. 14. Octal to Decimal • Technique – Multiply each bit by 8n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
  15. 15. Example 7248 => 4 x 80 = 2 x 81 = 7 x 82 = 4 16 448 46810
  16. 16. Hexadecimal to Decimal Decimal Octal Binary Hexadecimal
  17. 17. Hexadecimal to Decimal • Technique – Multiply each bit by 16n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
  18. 18. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
  19. 19. Decimal to Binary Decimal Octal Binary Hexadecimal
  20. 20. Decimal to Binary • There are two methods that can be used to convert decimal numbers to binary: – Repeated subtraction method – Repeated division method • Both methods produce the same result and you should use whichever one you are most comfortable with.
  21. 21. The Repeated Subtraction method – Step 1: • Starting with the 1s place, write down all of the binary place values in order until you get to the first binary place value that is GREATER THAN the decimal number you are trying to convert. 1024 512 256 128 64 32 16 8 4 2 1
  22. 22. The Repeated Subtraction method – Step 2: • Mark out the largest place value (it just tells us how many place values we need). 853 1024 512 256 128 64 32 16 8 4 2 1
  23. 23. The Repeated Subtraction method • – Step 3: • Subtract the largest place value from the decimal number. Place a “1” under that place value. 853 - 512 = 341 512 256 128 64 32 16 8 4 2 1 1
  24. 24. The Repeated Subtraction method – Step 4: • For the rest of the place values, try to subtract each one from the previous result. – If you can, place a “1” under that place value. – If you can’ t, place a “0” under that place value.
  25. 25. The Repeated Subtraction method • – Step 5: • Repeat Step 4 until all of the place values have been processed. • The resulting set of 1s and 0s is the binary equivalent of the decimal number you started with.
  26. 26. The Repeated Subtraction method
  27. 27. Repeated division method 12510 = ?2 2 125 2 62 2 31 2 15 7 2 3 2 1 2 0 1 0 1 1 1 1 1 12510 = 11111012
  28. 28. Octal to Binary Decimal Octal Binary Hexadecimal
  29. 29. Octal to Binary • Technique – Convert each octal digit to a 3-bit equivalent binary representation
  30. 30. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
  31. 31. Hexadecimal to Binary Decimal Octal Binary Hexadecimal
  32. 32. Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation
  33. 33. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
  34. 34. Decimal to Octal Decimal Octal Binary Hexadecimal
  35. 35. Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder
  36. 36. Example 123410 = ?8 8 8 8 8 1234 154 19 2 0 2 2 3 2 123410 = 23228
  37. 37. Decimal to Hexadecimal Decimal Octal Binary Hexadecimal
  38. 38. Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder
  39. 39. Example 123410 = ?16 16 16 16 1234 77 4 0 2 13 = D 4 123410 = 4D216
  40. 40. Binary to Octal Decimal Octal Binary Hexadecimal
  41. 41. Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits
  42. 42. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
  43. 43. Binary to Hexadecimal Decimal Octal Binary Hexadecimal
  44. 44. Binary to Hexadecimal • Technique – Group bits in fours, starting on right – Convert to hexadecimal digits
  45. 45. Example 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
  46. 46. Octal to Hexadecimal Decimal Octal Binary Hexadecimal
  47. 47. Octal to Hexadecimal • Technique – Use binary as an intermediary
  48. 48. Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
  49. 49. Hexadecimal to Octal Decimal Octal Binary Hexadecimal
  50. 50. Hexadecimal to Octal • Technique – Use binary as an intermediary
  51. 51. Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
  52. 52. Exercise – Convert ... Decimal 33 Binary Octal Hexadecimal 1110101 703 1AF Don’t use a calculator! Skip answer Answer
  53. 53. Exercise – Convert … Answer Hexadecimal Decimal 33 Binary 100001 Octal 41 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF 21
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