1. Number Systems Decimal (base 10) {0 1 2 3 4 5 6 7 8 9}
Place value gives a logarithmic representation of the number
Ex means
4 X 103 = 4000
3 X 102 = 300
7 X 101 =
8 X 100 =
The place also gives the exponent of the base

2. Example
432,600
105
104
103
100
101
102
Powers of ten:
100 = = = 10000
101 = = =

3. Binary (base 2) {0 1} Binary Decimal 1 10 2 11 3 100 4 101 5 110 6 1111 10 2 11 3
100 4
101 5
110 6
111 7
1000 8
1001 9
1010

4. Example 27 26 25 20 21 22 24 23

5. Decimal Equivalent 1101 1001 1 X 27 = 128 + 1 X 26 = 64 + 0 X 25 = 0
217
Notice how powers of two stand out:
20 = 1
21 = 10
22 = 100
23 = 1000

6. Decimal to Binary Conversion
Ex. 575
Find the largest power of two less than the number
29 = 512
Subtract that power of two from the number
575 – 512 = 63
Repeat steps 1 and 2 for the new result until you reach zero.
25 = – 32 = 31
24 = – 16 = 15
23 = – 8 = 7
22 = – 4 = 3
21 = – 2 = 1
20 = – 1 = 0
Construct the number

7. Another Example 144 Result 10010000 27 = 128 144 – 128 = 16
27 = – 128 = 16
24 = – 16 = 0
Result

8. Hexadecimal (base 16) {0 1 2 3 4 5 6 7 8 9 A B C D E F} Assignments8 1 9 2 10 A 3 11 B 4 12 C 5 13 D 6 14 E 7 15 F

9. Example
163
162
160
161
3 B 6 E
3 X 163 = 12288
11 X 162 =
6 X 161 =
14 X 160 = 

10. Hexadecimal is Convenient for Binary Conversion
1001 9 1
1010 A 10 2
1011 B 11 3
1100 C
100 4
1101 D
101 5
1110 E
110 6
1111 F
111 7
1 0000
1000 8
 Nibble

11. Binary to Hex Conversion
Group binary number by fours (nibbles)
Convert each nibble into hex equivalent D

12. Decimal to Hex Conversion
162 = – 256 = 28
161 = = 12 (Hex C)
Result C

13. Another Example with an Extension
1054
162 = 256
But we have several multiples of 256 in 1054
1054/256 = take integer part
This eliminates 4*256 = 1024
1054 – 1024 = 30
161 = – 16 = 14 (Hex E)
Result E

14. Truth Table Binary Decimal Hexadecimal 0000 0001 0010 0011 0100 0101
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

15. Truth Table Binary Decimal Hexadecimal 0000 0001 1 0010 2 0011 3 0100
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 10 A
1011 11 B
1100 12 C
1101 13 D
1110 14 E
1111 15 F

16. Sexagesimal (Base 60)

17. Practice Convert 212 decimal to binary 212 – 27 = 84 84 – 26 = 20
20 – 24 = 4
4 – 22 = 0
Result:

18. More Practice Convert 1101 0010 binary to hex 0010 = 2 1101 = 13 = D
Result D2

19. Notation Some books use a subscript to denote the base.
Ex: 1210 = 12 decimal
1216 = 12 hex = 18 decimal

20. Logic Gates

21. Transistors as Switches
VBB voltage controls whether the transistor conducts in a common base configuration.
Logic circuits can be built

22. Boolean Algebra

23. AND In order for current to flow, both switches must be closed
Logic notation AB = C
(Sometimes AB = C) A B C 1

24. OR Current flows if either switch is closed Logic notation A + B = C A1

25. Properties of AND and OR
Commutation
A + B = B + A
A  B = B  A
Same as
Same as

26. Commutation Circuit
A  B
B  A
B + A
A + B

27. Properties of AND and OR
Associative Property
A + (B + C) = (A + B) + C
A  (B  C) = (A  B)  C =

28. Properties of AND and OR
Distributive Property
A + B  C = (A + B)  (A + C)
A + B  C A B C Q 1

29. Distributive Property
(A + B)  (A + C) A B C Q 1

30. Notice that the carry results are the same as AND
Binary Addition A B S
C(arry) 1
Notice that the carry results are the same as AND
C = A  B

31. Inversion (NOT) A Q 1
Logic:

32. Exclusive OR (XOR)
Either A or B, but not both
This is sometimes called the inequality detector, because the result will be 0 when the inputs are the same and 1 when they are different.
The truth table is the same as for S on Binary Addition. S = A  B A B S 1

33. Getting the XOR A B S 1
Two ways of getting S = 1

34. Circuit for XOR
Accumulating our results: Binary addition is the result of XOR plus AND

35. Half Adder
Called a half adder because we haven’t allowed for any carry bit on input. In elementary addition of numbers, we always need to allow for a carry from one column to the next. 18 25
3 (plus a carry) 4

36. Half Adder

37. Full Adder
INPUTS
OUTPUTS A B
CIN
COUT S 1

38. Full Adder Circuit

39. Chaining the Full Adder
Possible to use the same scheme for subtraction by noting that
A – B = A + (-B)

40. Binary Counting
Use 1 for ON
Use 0 for OFF =
So our example has = = 43

41. Counting in Binary 1 11
1011 21
10101 2 10 12
1100 22
10110 3 13
1101 23
10111 4
100 14
1110 24
11000 5
101 15
1111 25
11001 6
110 16
10000 26
11010 7
111 17
10001 27
11011 8
1000 18
10010 28
11100 9
1001 19
10011 29
11101
1010 20
10100 30
11110

42. NAND (NOT AND) A B Q 1

43. NOR (NOT OR) A B Q 1

44. DeMorgan’s Theorem
A NAND gate is equivalent to an inversion followed by an OR
A NOR gate is equivalent to an inversion followed by and AND

45. DeMorgan Truth Table A B 1
NAND
NOR

46. Exclusive NOR A B Q 1
Equality Detector

47. Summary for all 2-input gates
Inputs
Output of each gate
 A 
 B 
AND
NAND
 OR 
NOR
XOR
XNOR 1

48. Logic Gates and Symbols
NAND

49. More Gates and Symbols OR
NOR
NOT

50. And More
XOR
NXOR

51. Multi-input Gates

52. Three input OR

53. Logic Gate ICs

54. Example 7400

55. More ICs

56. And More