1. Number Systems

2. Number Representation
Every number like ‘a’ can be represented as

3. Common Number Systems System Base Symbols Used by humans?
Used in computers?
Decimal 10
0, 1, … 9
Yes No
Binary 2
0, 1
Octal 8
0, 1, … 7
Hexa- decimal 16
0, 1, … 9,
A, B, … F

4. Quantities/Counting Decimal Binary Octal Hexa- decimal 1 2 10 3 11 41 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
Decimal
Binary
Octal
Hexa- decimal 10
1010 12 A 11
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F
10000 20
10001 21 18
10010 22 19
10011 23

5. Conversion Among Bases
The possibilities:
Decimal
Octal
Binary
Hexadecimal

6. The others to Decimal

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

8. Weight
12510 => 5 x 100 = x 101 = x 102 =
Base

9. Binary to Decimal
Decimal
Octal
Binary
Hexadecimal

10. 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

11. Example
Bit “0”
=> 1 x 20 = x 21 = x 22 = x 23 = x 24 = x 25 = 32
4310

12. Octal to Decimal
Decimal
Octal
Binary
Hexadecimal

13. 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

14. Example
7248 => 4 x 80 = x 81 = x 82 =

15. Hexadecimal to Decimal
Octal
Binary
Hexadecimal

16. 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

17. Example
ABC16 => C x 160 = 12 x 1 = B x 161 = 11 x 16 = A x 162 = 10 x 256 = 2560
274810

18. Decimal to Binary
Decimal
Octal
Binary
Hexadecimal

19. 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.

20. Example
12510 = ?2
12510 =

21. Octal to Binary
Decimal
Octal
Binary
Hexadecimal

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

23. Example
7058 = ?2
7058 =

24. Hexadecimal to Binary
Decimal
Octal
Binary
Hexadecimal

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

26. Example
10AF16 = ?2
A F
10AF16 =

27. Decimal to Octal
Decimal
Octal
Binary
Hexadecimal

28. Decimal to Octal
Technique
Divide by 8
Keep track of the remainder

29. Example
= ?8 8
19 2 8
2 3 8
0 2
= 23228

30. Decimal to Hexadecimal
Octal
Binary
Hexadecimal

31. Decimal to Hexadecimal
Technique
Divide by 16
Keep track of the remainder

32. Example
= ?16
77 2 16
= D
0 4
= 4D216

33. Binary to Octal
Decimal
Octal
Binary
Hexadecimal

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

35. Example
= ?8
= 13278

36. Binary to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

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

38. Example
= ?16
B B
= 2BB16

39. Octal to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

40. Octal to Hexadecimal
Technique
Use binary as an intermediary

41. Example
10768 = ?16 E
10768 = 23E16

42. Hexadecimal to Octal
Decimal
Octal
Binary
Hexadecimal

43. Hexadecimal to Octal
Technique
Use binary as an intermediary

44. Example
1F0C16 = ?8
1 F C
1F0C16 =

45. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 1110101
703
1AF
Don’t use a calculator!
Skip answer
Answer

46. Exercise – Convert … Decimal Binary Octal Hexa- decimal 33 100001 41
Answer
Decimal
Binary
Octal
Hexa- decimal 33
100001 41 21
117
165 75
451
703
1C3
431
657
1AF

47. Common Powers (1 of 2) Base 10 Power Preface Symbol pico p nano n
10-12
pico p
10-9
nano n
10-6
micro 
10-3
milli m
103
kilo k
106
mega M
109
giga G
1012
tera T
Value
.001
1000

48. Common Powers (2 of 2) What is the value of “k”, “M”, and “G”?
Base 2
Power
Preface
Symbol
210
kilo k
220
mega M
230
Giga G
Value
1024
What is the value of “k”, “M”, and “G”?
In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

49. Example
In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties
/ 230 =

50. Exercise – Free Space
Determine the “free space” on all drives on your personal computer!
Drive
Free space
Bytes GB A: C: D: E:
etc.

51. Binary Addition (1 of 2)
Two 1-bit values A B
A + B 1 10
“two”

52. Binary Addition (2 of 2) 10101 21 + 11001 + 25 101110 46
Two n-bit values
Add individual bits
Propagate carries
E.g., 1 1

53. Multiplication (1 of 3)
Decimal (just for fun)
35 x

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

55. Multiplication (3 of 3)
Binary, two n-bit values
As with decimal values
E.g.,
Exercise- Search about binary division? x

56. Fractions Decimal to decimal (just for fun)
3.14 => 4 x 10-2 = x 10-1 = x 100 =

57. Fractions Binary to decimal
=> 1 x 2-4 = x 2-3 = x 2-2 = x 2-1 = x 20 = x 21 =

58. Fractions Decimal to binary 3.14579 11.001001...
x x x x x x
etc.

59. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 29.8
3.07
C.82
Don’t use a calculator!
Skip answer
Answer

60. Exercise – Convert … Decimal Binary Octal Hexa- decimal 29.8
Answer
Decimal
Binary
Octal
Hexa- decimal
29.8 …
35.63…
1D.CC…
5.8125
5.64
5.D
3.07
3.1C
14.404
C.82

61. Signed Binary number Sign-magnitude 1’s Complement 2’s Complement
In all the three methods, the left digit for the positive numbers is zero and for the negative numbers is one!
....
Sign bit
n - 1

62. Sign-magnitude The left bit is the sign Positive numbers
Negative numbers
0000=+0
1000=-0
0001=+1
1001=-1
0010=+2
1010=-2
0011=+3
1011=-3
0100=+4
1100=-4
0101=+5
1101=-5
0110=+6
1110=-6
0111=+7
1111=-7

63. Sign-magnitude(cont.)
For a number with n bits:
the biggest number is
the smallest one is
We have two values for zero: +0 and -0
Adding two numbers is not easy! ??

64. 1’s Complement (r-1)’s complement of a (with n digits) is:
Positive numbers
Negative numbers
0000=+0
1111=-0
0001=+1
1110=-1
0010=+2
1101=-2
0011=+3
1100=-3
0100=+4
1011=-4
0101=+5
1010=-5
0110=+6
1001=-6
0111=+7
1000=-7

65. 1’s Complement to Decimal
= = - 4

66. 1’s Complement(cont.) We have two values for zero: +0 and -0
Summation of two numbers:
if a carrier value is generated, add it with the result
Exe: What is the biggest and smallest numbers which can be represented by this method?

67. 2’s Complement (r)’s complement of a (with n digits) is:
Positive numbers
Negative numbers
0000=0
1111=-1
0001=+1
1110=-2
0010=+2
1101=-3
0011=+3
1100=-4
0100=+4
1011=-5
0101=+5
1010=-6
0110=+6
1001=-7
0111=+7
1000=-8

68. 2’s Complement to Decimal
= – 26 = - 53

69. 2’s Complement(cont.)
We have one value for zero

70. Multiplication (Example 1)
Sign extension

71. Multiplication(Example 2)

72. Exercise- Search about:
How compute the 1’s complement multiplication/summation?
How compute 1’s and 2’s complement division?