Number Systems for class 11 computer science

Number system class 11

1. Number Systems

2. Common Number Systems
System Base Symbols
Decimal 10 0, 1, … 9
Binary 2 0, 1
Octal 8 0, 1, … 7
Hexa-
decimal
16 0, 1, … 9,
A, B, … F

3. Quantities/Counting (1 of 3)
DecimalBinaryOctal
Hexa-
decimal
0 00 0
1 11 1
2 102 2
3 113 3
4 1004 4
5 1015 5
6 1106 6
7 1117 7
p. 33

4. Quantities/Counting (2 of 3)
DecimalBinaryOctal
Hexa-
decimal
8 100010 8
9 100111 9
10 101012 A
11 101113 B
12 110014 C
13 110115 D
14 111016 E
15 111117 F

5. Quantities/Counting (3 of 3)
DecimalBinaryOctal
Hexa-
decimal
16 1000020 10
17 1000121 11
18 1001022 12
19 1001123 13
20 1010024 14
21 1010125 15
22 1011026 16
23 1011127 17 Etc.

6. Conversion Among Bases
•The possibilities:
Hexadecimal
Decimal Octal
Binary

7. Quick Example
25
10= 11001
2= 31
8= 19
16
Base

8. Decimal to Decimal (just for fun)
Hexadecimal
Decimal Octal
Binary
Next slide…

9. 125
10=>5 x 10
0
= 5
2 x 10
1
= 20
1 x 10
2
= 100
125
Base
Weight

10. Binary to Decimal
Hexadecimal
Decimal Octal
Binary

11. Binary to Decimal
•Technique
–Multiply each bit by 2
n
, where nis the “weight”
of the bit
–The weight is the position of the bit, starting
from 0 on the right
–Add the results

12. Example
101011
2=> 1 x 2
0
= 1
1 x 2
1
= 2
0 x 2
2
= 0
1 x 2
3
= 8
0 x 2
4
= 0
1 x 2
5
= 32
43
10
Bit “0”

13. Octal to Decimal
Hexadecimal
Decimal Octal
Binary

14. Octal to Decimal
•Technique
–Multiply each bit by 8
n
, where nis the “weight”
of the bit
–The weight is the position of the bit, starting
from 0 on the right
–Add the results

15. Example
724
8=> 4 x 8
0
= 4
2 x 8
1
= 16
7 x 8
2
= 448
468
10

16. Hexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary

17. Hexadecimal to Decimal
•Technique
–Multiply each bit by 16
n
, where nis the
“weight” of the bit
–The weight is the position of the bit, starting
from 0 on the right
–Add the results

18. Example
ABC
16=>C x 16
0
= 12 x 1 = 12
B x 16
1
= 11 x 16 = 176
A x 16
2
= 10 x 256 = 2560
2748
10

19. Decimal to Binary
Hexadecimal
Decimal Octal
Binary

20. Decimal to Binary
•Technique
–Divide by two, keep track of the remainder
–First remainder is bit 0 (LSB, least-significant
bit)
–Second remainder is bit 1
–Etc.

21. Example
125
10= ?
2
2 125
62 12
31 02
15 12
7 12
3 12
1 12
0 1
125
10= 1111101
2

22. Octal to Binary
Hexadecimal
Decimal Octal
Binary

23. Octal to Binary
•Technique
–Convert each octal digit to a 3-bit equivalent
binary representation

24. Example
705
8= ?
2
7 0 5
111 000 101
705
8= 111000101
2

25. Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary

26. Hexadecimal to Binary
•Technique
–Convert each hexadecimal digit to a 4-bit
equivalent binary representation

27. Example
10AF
16= ?
2
1 0 A F
0001 0000 1010 1111
10AF
16= 0001000010101111
2

28. Decimal to Octal
Hexadecimal
Decimal Octal
Binary

29. Decimal to Octal
•Technique
–Divide by 8
–Keep track of the remainder

30. Example
1234
10= ?
8
8 1234
154 28
19 28
2 38
0 2
1234
10= 2322
8

31. Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary

32. Decimal to Hexadecimal
•Technique
–Divide by 16
–Keep track of the remainder

33. Example
1234
10= ?
16
1234
10= 4D2
16
16 1234
77 216
4 13 = D16
0 4

34. Binary to Octal
Hexadecimal
Decimal Octal
Binary

35. Binary to Octal
•Technique
–Group bits in threes, starting on right
–Convert to octal digits

36. Example
1011010111
2= ?
8
1 011 010 111
1 3 2 7
1011010111
2= 1327
8

37. Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary

38. Binary to Hexadecimal
•Technique
–Group bits in fours, starting on right
–Convert to hexadecimal digits

39. Example
1010111011
2= ?
16
10 1011 1011
2 B B
1010111011
2= 2BB
16

40. Octal to Hexadecimal
Hexadecimal
Decimal Octal
Binary

41. Octal to Hexadecimal
•Technique
–Use binary as an intermediary

42. Example
1076
8= ?
16
1 0 7 6
001 000 111 110
2 3 E
1076
8= 23E
16

43. Hexadecimal to Octal
Hexadecimal
Decimal Octal
Binary

44. Hexadecimal to Octal
•Technique
–Use binary as an intermediary

45. Example
1F0C
16= ?
8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C
16= 17414
8

46. Exercise –Convert ...
Don’t use a calculator!
DecimalBinary Octal
Hexa-
decimal
33
1110101
703
1AF
Skip answerAnswer

47. Exercise –Convert …
DecimalBinary Octal
Hexa-
decimal
33 100001 41 21
117 1110101 165 75
451111000011703 1C3
431110101111657 1AF
Answer

48. Common Powers (1 of 2)
•Base 10
Power Preface Symbol
10
-12
pico p
10
-9
nano n
10
-6
micro 
10
-3
milli m
10
3
kilo k
10
6
mega M
10
9
giga G
10
12
tera T
Value
.000000000001
.000000001
.000001
.001
1000
1000000
1000000000
1000000000000

49. Common Powers (2 of 2)
•Base 2
Power Preface Symbol
2
10
kilo k
2
20
mega M
2
30
Giga G
Value
1024
1048576
1073741824
•What is the value of “k”, “M”, and “G”?
•In computing, particularly w.r.t. memory,
the base-2 interpretation generally applies

50. Example
/ 2
30
=
In the lab…
1. Double click on My Computer
2. Right click on C:
3. Click on Properties

51. Exercise –Free Space
•Determine the “free space” on all drives on
a machine in the lab
Drive
Free space
Bytes GB
A:
C:
D:
E:
etc.

52. Review –multiplying powers
•For common bases, add powers
2
6
2
10
= 2
16
= 65,536
or…
2
6
2
10
= 64 2
10
= 64k
a
b
a
c
= a
b+c

53. Binary Addition (1 of 2)
•Two 1-bit values
pp. 36-38
A B A + B
0 0 0
0 1 1
1 0 1
1 1 10
“two”

54. Binary Addition (2 of 2)
•Two n-bit values
–Add individual bits
–Propagate carries
–E.g.,
10101 21
+ 11001 + 25
101110 46
11

55. Multiplication (1 of 3)
•Decimal (just for fun)
pp. 39
35
x 105
175
000
35
3675

56. Multiplication (2 of 3)
•Binary, two 1-bit values
A B A B
0 0 0
0 1 0
1 0 0
1 1 1

57. Multiplication (3 of 3)
•Binary, two n-bit values
–As with decimal values
–E.g.,
1110
x 1011
1110
1110
0000
1110
10011010

58. Fractions
•Decimal to decimal (just for fun)
pp. 46-50
3.14 => 4 x 10
-2
= 0.04
1 x 10
-1
= 0.1
3 x 10
0
= 3
3.14

59. Fractions
•Binary to decimal
10.1011 => 1 x 2
-4
= 0.0625
1 x 2
-3
= 0.125
0 x 2
-2
= 0.0
1 x 2
-1
= 0.5
0 x 2
0
= 0.0
1 x 2
1
= 2.0
2.6875

60. Fractions
•Decimal to binary
3.14579
.14579
x 2
0.29158
x 2
0.58316
x 2
1.16632
x 2
0.33264
x 2
0.66528
x 2
1.33056
etc.11.001001...

61. Exercise –Convert ...
Don’t use a calculator!
Decimal Binary Octal
Hexa-
decimal
29.8
101.1101
3.07
C.82
Skip answerAnswer

62. Exercise –Convert …
Decimal Binary Octal
Hexa-
decimal
29.8 11101.110011…35.63…1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.50781251100.1000001014.404 C.82
Answer

63. Thank you