This document provides an overview of digital electronics and number systems. It discusses common number systems like decimal, binary, octal and hexadecimal. It then covers techniques for converting between these different number systems, including dividing or multiplying by the base to get place values. Binary operations like addition and multiplication are also explained. Fractions in different number systems are described at the end.

1. DIGITAL ELECTRONICS

2. What is Electronics?
Electronic Devices?
Digital Electronics?
Signal?

7. Chapter 1:
Number System
1

8. INDEX
1. Common Number Systems
2. Quantities/Counting (1 of 2)
3. Conversion Among Bases
4. Decimal to Decimal (just for fun)
5. Binary to Decimal
6. Octal to Decimal
7. Hexadecimal to Decimal
8. Decimal to Binary
9. Octal to Binary
10. Hexadecimal to Binary
11. Decimal to Octal
12. Decimal to Octal
13. Decimal to Hexadecimal
14. Binary Addition (1 of 2)
15. Multiplication (1 of 3)
16. Fractions
2

9. Common Number Systems
9
System
radix
/
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
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No

10. Quantities/Counting (1 of 2)
10
Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7

11. Quantities/Counting (2 of 2)
11
Decimal Binary Octal
Hexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

12. Conversion Among Bases
12
• The possibilities:
Hexadecimal
Decimal Octal
Binary

13. Quick Example
13
2510 = 110012 = 318 = 1916
Base

14. Decimal to Decimal (just for fun)
14
Hexadecimal
Decimal Octal
Binary

15. 12510 => 5 x 100= 5
2 x 101= 20
1 x 102= 100
125
Base
Weight
15

16. Binary to Decimal
16
Hexadecimal
Decimal Octal
Binary

17. Binary to Decimal
17
• 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

18. Example
18
1010112 => 1 x
1 x
0 x
1 x
0 x
1 x
20
21
22
23
24
25
= 1
= 2
= 0
= 8
= 0
= 32
4310
Bit

19. BINARY TO DECIMAL
CONVERSION
Convert Binary Number 110011 to
a Decimal Number:
32 + 16 + 0 + 0 + 2 + 1 = 51
1 1 0 0 1 1
Decimal
Binary

20. TEST
Convert the following binary
numbers into decimal numbers:
Binary 1001 = 9
Binary 1111 =
Binary 0010 =
15
2

21. Octal to Decimal
21
Hexadecimal
Decimal Octal
Binary

22. Octal to Decimal
22
• 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

23. Example
23
7248 => 4 x 80
2 x 81
7 x 82
= 4
= 16
= 448
46810

24. Hexadecimal to Decimal
24
Hexadecimal
Decimal Octal
Binary

25. Hexadecimal to Decimal
25
• 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

26. Example
26
ABC16 => = 12 x
= 11 x
1 = 12
16 = 176
C x 160
B x 161
A x 162 = 10 x 256 = 2560
274810
A=10,b=11,C=12,D=13,E=14,F=15

27. Decimal to Binary
27
Hexadecimal
Decimal Octal
Binary

28. Decimal to Binary
28
• Technique
• Divide by two, keep track of the remainder
• First remainder is bit 0 (LSB, least-significant bit)
• Second remainder is bit 1
• Etc.

29. Example
29
12510 = ?2
2
2
2
2
2
2
2
125
62 1
31 0
15 1
7 1
3 1
1 1
0 1
12510 = 11111012

30. DECIMAL TO BINARY
CONVERSION
Divide by 2 Process
Decimal # 13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
1 1
0
1

31. TEST
Convert the following decimal
numbers into binary:
Decimal 11 =
Decimal 4 =
Decimal 17 =
1011
0100
10001

32. Octal to Binary
32
Hexadecimal
Decimal Octal
Binary

33. Octal to Binary
33
• Technique
• Convert each octal digit to a 3-bit equivalent binary representation

34. Example
34
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012

35. Hexadecimal to Binary
35
Hexadecimal
Decimal Octal
Binary

36. Hexadecimal to Binary
36
• Technique
• Convert each hexadecimal digit to a 4-bit equivalent binary
representation

37. Example
37
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112

38. Decimal to Octal
38
Hexadecimal
Decimal Octal
Binary

39. Decimal to Octal
39
• Technique
• Divide by 8
• Keep track of the remainder

40. Example
40
123410 = ?8
8 1234
8 154 2
8 19 2
8 2 3
0 2
123410 = 23228

41. Decimal to Hexadecimal
41
Hexadecimal
Decimal Octal
Binary

42. Decimal to Hexadecimal
42
• Technique
• Divide by 16
• Keep track of the remainder

43. Example
43
123410 = ?16
123410 = 4D216
16 1234
16 77 2
16 4 13 = D
0 4

44. Binary to Octal
44
Hexadecimal
Decimal Octal
Binary

45. Binary to Octal
45
• Technique
• Group bits in threes, starting on right
• Convert to octal digits

46. Example
46
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278

47. Binary to Hexadecimal
47
Hexadecimal
Decimal Octal
Binary

48. Binary to Hexadecimal
48
• Technique
• Group bits in fours, starting on right
• Convert to hexadecimal digits

49. Example
49
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16

50. Octal to Hexadecimal
50
Hexadecimal
Decimal Octal
Binary

51. Octal to Hexadecimal
51
• Technique
• Use binary as an intermediary

52. Example
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E41
36

53. Hexadecimal to Octal
53
Hexadecimal
Decimal Octal
Binary

54. Hexadecimal to Octal
54
• Technique
• Use binary as an intermediary

55. Example
55
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148

56. Binary Addition (1 of 2)
56
• Two 1-bit values
A B A+ B
0 0 0
0 1 1
1 0 1
1 1 10
“two”

57. Binary Addition (2 of 2)
57
• Two n-bit values
• Add individual bits
• Propagate carries
• E.g.,
101110
21
+ 11001 + 25
46
1
10101
1

58. Multiplication (1 of 3)
58
• Decimal (just for fun)
35
x 105
175
000
35
3675

59. Multiplication (2 of 3)
59
• Binary, two 1-bit values
A B A B
0 0 0
0 1 0
1 0 0
1 1 1

60. Multiplication (3 of 3)
• Binary, two n-bit values
• As with decimal values
• E.g.,
1110
x 1011
1110
1110
0000
1110
10011010 51

61. Fractions
61
• Decimal to decimal (just for fun)
3.14 => 4 x 10-2 = 0.04
1 x 10-1 = 0.1
3 x 100 = 3
3.14

62. Fractions
62
• Binary to decimal
10.1011 => 1 x 2-4 = 0.0625
1 x 2-3
0 x 2-2
1 x 2-1
0 x 20
1 x 21
= 0.125
= 0.0
= 0.5
= 0.0
= 2.0
2.6875

63. Fractions
63
• 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...

64. Thank you
64