Number Systems Decimal (base 10) {0 1 2 3 4 5 6 7 8 9} o Place value gives a logarithmic representation of the number o Ex. 4378 means 4 X 103 = 4000 3 X 102 = 300 7 X 101 = 70 8 X 100 = 8 o The place also gives the exponent of the base Example 432,600 4 3 2 6 0 0

105 100 104 101 103 102 Powers of ten: 100 = 1 102 = 100 104 = 10000 101 = 10 103 = 1000 105 = 100000

Binary (base 2) Binary Decimal 0 0 1 1 10 2 11 3 100 4

101 5 110 6 111 7 1000 8 1001 9 1010 10

{0 1} Example 1 1 0 1 1 0 0 1 27 20 26 21 25 22 24 23 Decimal Equivalent 1101 1001 1 X 27 = 128

+ 1 X 26 = 64 + 0 X 25 = 0 + 1 X 24 = 16 + 1 X 23 = 8 + 0 X 22 = 0 + 0 X 21 = 0 + 1 X 20 = 1 217 Notice how powers of two stand out: 20 = 1 21 = 10 22 = 100 23 = 1000 Decimal to Binary Conversion Ex. 575 o Find the largest power of two less than the number o 29 = 512 o Subtract that power of two from the number

o 575 512 = 63 o Repeat steps 1 and 2 for the new result until you reach zero. o o o o o o 25 = 32 24 = 16 23 = 8 22 = 4 21 = 2 20 = 1 63 32 = 31 31 16 = 15 15 8 = 7 74=3 32=1

11=0 o Construct the number o 1000111111 Another Example 144 o 27 = 128 o 24 = 16 Result 144 128 = 16 16 16 = 0 10010000 Hexadecimal (base 16) {0 1 2 3 4 5 6 7 8 9 A B C D E F} Assignments Dec Hex Dec Hex

0 0 8 8 1 1 9 9 2 2 10 A

3 3 11 B 4 4 12 C 5 5 13 D

6 6 14 E 7 7 15 F Example 3B6E 163 160

162 161 3 X 163 = 12288 11 X 162 = 2816 6 X 16 = 96 14 X 160 = 14 1 15214 Hexadecimal is Convenient for Binary Conversion Binary Hex Binary

Hex 0 0 1001 9 1 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 10 1000 8 Nibble

Binary to Hex Conversion Group binary number by fours (nibbles) o 1101 1001 0110 Convert each nibble into hex equivalent o 1101 1001 0110 D 9 6 Decimal to Hex Conversion Ex. 284 o 162 = 256 o 161 = 16 o Result 1 1 C 284 256 = 28 28 - 16 = 12 (Hex C) Another Example with an Extension 1054 o 162 = 256

But we have several multiples of 256 in 1054 o 1054/256 = 4.12 take integer part o This eliminates 4*256 = 1024 1054 1024 = 30 o 161 = 16 o Result 4 1 E 30 16 = 14 (Hex E) Truth Table Binary 0000 0001 0010 0011 0100 0101 0110 0111

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

1001 1010 1011 1100 1101 1110 1111 Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Sexagesimal (Base 60) Practice

Convert 212 decimal to binary o 212 27 = 84 o 84 26 = 20 o 20 24 = 4 o 4 22 = 0 o Result: 1101 0100 More Practice Convert 1101 0010 binary to hex o 0010 = 2 o 1101 = 13 = D o Result D2 Notation Some books use a subscript to denote the base. o Ex: 1210 = 12 decimal o 1216 = 12 hex = 18 decimal Logic Gates Transistors as Switches VBB voltage controls whether the transistor conducts in a common base configuration.

Logic circuits can be built Boolean Algebra AND In order for current to flow, both switches must be closed Logic notation AB = C (Sometimes AB = C) A B C 0 0 0 0

1 0 1 0 0 1 1 1 OR Current flows if either switch is closed Logic notation A + B = C A B

C 0 0 0 0 1 1 1 0 1 1 1

1 Properties of AND and OR Commutation oA + B = B + A oA B = B A Same as Same as Commutation Circuit AB BA A+B B+A Properties of AND and OR Associative Property A + (B + C) = (A + B) + C

= A (B C) = (A B) C Properties of AND and OR Distributive Property A + B C = (A + B) (A + C) A+BC A B C Q 0 0 0 0

0 0 1 0 0 1 0 0 0 1 1 1

1 0 0 1 1 0 1 1 1 1 0 1

1 1 1 1 Distributive Property (A + B) (A + C) A B C Q 0 0 0

0 0 0 1 0 0 1 0 0 0 1 1

1 1 0 0 1 1 0 1 1 1 1 0

1 1 1 1 1 Binary Addition A B S C(arry) 0 0

0 0 1 0 1 0 0 1 1 0 1 1

0 1 Notice that the carry results are the same as AND C=AB Inversion (NOT) Logic: Q A A Q 0 1 1 0

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 0 1 0 1 B 0 0 1 1

S 0 1 1 0 Getting the XOR Two ways of getting S = 1 A B or A B A B S 0 0 0 1

0 1 0 1 1 1 1 0 Circuit for XOR A B A B A B Accumulating our results: Binary addition is the result of XOR plus AND Half Adder

Called a half adder because we havent 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 Half Adder Full Adder INPUTS OUTPUTS A B CIN COUT

S 0 0 0 0 0 0 0 1 0 1 0

1 0 0 1 0 1 1 1 0 1 0 0

0 1 1 0 1 1 0 1 1 0 1 0

1 1 1 1 1 Full Adder Circuit Chaining the Full Adder Possible to use the same scheme for subtraction by noting that A B = A + (-B) Binary Counting Use 1 for ON Use 0 for OFF =

00101011 So our example has 25 + 23 + 21 + 20 = 32 + 8 + 2 + 1 = 43 Counting in Binary 1 1 11 1011 21 10101 2 10 12 1100 22 10110 3 11 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 10 1010 20 10100 30 11110 NAND (NOT AND) Q A B A B Q 0

0 1 0 1 1 1 0 1 1 1 0 NOR (NOT OR)

Q A B A 0 0 1 1 B 0 1 0 1 Q 1 0 0 0 DeMorgans 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 DeMorgan Truth Table NAND NOR Exclusive NOR Q A B Equality Detector A B Q 0 0

1 0 1 0 1 0 0 1 1 1 Summary Summary for all 2-input gates Inputs A

B Output of each gate AND NAND OR NOR XOR XNOR 0 0 0 1

0 1 0 1 0 1 0 1 1 0 1 0

1 0 0 1 1 0 1 0 1 1 1 0

1 0 0 1 Logic Gates and Symbols AND NAND More Gates and Symbols OR NOR NOT And More

XOR NXOR Multi-input Gates Three input OR Logic Gate ICs Example 7400 More ICs And More