Binary Arithmetic, ASCII, & Boolean Algebra

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1 Binary Arithmetic, ASCII, & Boolean Algebra Today: First Hour: Computer Arithmetic, Representation of Symbols Representing Symbols the ASCII code Appendix A.2 & A.3 of Katz s Textbook In-class Activity # Second Hour: Boolean Algebra Section 2. of Katz s Textbook In-class Activity #2

2 Recap X Y X OR Y X Y OR Z Z = X + Y Decimal Binary Hex Octal 2

3 Binary Arithmetic You already know the rules for decimal addition and subtraction (how to handle sums, carries, differences, and borrows). Analogously, we develop the rules for binary addition and subtraction. Taken from Appendix A.3 3

4 Decimal Addition Refresher 95 9 x + 5 x x + 6 x x + x Summary = x 2 + (+) x + x Column carries

5 Binary Addition This table calculates the sum for pairs of binary numbers + = + = + = + = with a carry of of Also known as the Half Adder Table 5

6 Binary Addition with Carry This table shows all the possible sums for binary numbers with carries carry carry addend addend augend augend sum sum + + = + + = + + = + + = with a carry of of + + = + + = with a carry of of + + = with a carry of of + + = with a carry of of Also known as the Full Adder Table 6

7 Binary Addition Similar to the decimal case Example: Add 5 and 3 in in binary (carries) 2 = = 3 2 = 8 7

8 Decimal Subtraction Refresher = 9x + 5x = 9x + 5x = -x + -6x = -x + -6x borrow = -x 7x + 9x Note: borrows are shown as as explicit subtractions. 8

9 Binary Subtraction This table calculates the difference for pairs of binary numbers -- = -- = with a borrow of of -- = -- = Also known as the Half Subtractor Table 9

10 Binary Subtraction with Borrow This shows all the possibile differences for binary numbers with borrows minuend minuend subtrahend subtrahend borrow borrow difference difference = = with a borrow of of = with a borrow of of = with a borrow of of = = = = with a borrow of of Also known as the Full Subtractor Table

11 Binary Subtraction Similar to the decimal case Example: Subtract 3 from 5 in in binary 2 = 5-2 = 3 - (borrows) 2 = 2

12 Representing Symbols American Standard Code for Information Interchange 7-bits per symbol 2 7 = 28 different symbols 26 uppercase letters (A-Z) 26 lowercase letters (a-z) digits (-9) blank space (SP) 32 special-character symbols 32 non-printing control characters delete character (DEL) 2

13 3 The 7 The 7-bit ASCII Code bit ASCII Code DEL o _ O? / US SI F ~ n ^ N >. RS SO E } m ] M = - GS CR D l \ L <, FS FF C { k [ K ; + ECS VT B z j Z J : * SUB LF A y i Y I 9 ) EM HT 9 x h X H 8 ( CAN BS 8 w g W G 7 ' ETB BEL 7 v f V F 6 & SYN ACK 6 u e U E 5 % NAK ENQ 5 t d T D 4 $ DC4 EOT 4 s c S C 3 # DC3 ETX 3 r b R B 2 " DC2 STX 2 q a Q A! DC SOH p ` SP DLE NUL Leftmost 3 bits Rightmost 4 bits

14 ASCII Example Example string: "I am RPI\r\n" I SP a m SP h e r e SP R P I CR LF NUL D D A (hexadecimal notation) 4

15 Control Codes Control codes are non-printing How do you type them on a conventional keyboard? For example, how do you get the ESC control if your keyboard doesn t have such a key? The Ctrl key forces the 2 most significant bits to Hold the Control key, Ctrl, and type [ to get Ctrl-[ [ = $5B = = $B = ESC Similarly, Ctrl-c changes to (ETX) 5

16 Do Activity # Now Learn to add and subtract binary numbers Get to know the ASCII code 2 3 Leftmost 3 bits 2 NUL DLE SP SOH DC! STX DC2 " ETX DC3 # A B C 5 P Q R S 6 ` a b c 7 p q r s 4 EOT DC4 $ 4 D T d t Rightmost 4 bits A B C ENQ ACK BEL BS HT LF VT FF NAK SYN ETB CAN EM SUB ECS FS % & ' ( ) * +, : ; < E F G H I J K L U V W X Y Z [ \ e f g h i j k l u v w x y z { D CR GS - = M ] m } E F SO SI RS US. / >? N O ^ _ n o ~ DEL 6

17 Boolean Algebra A set of theorems for manipulating Boolean variables. They are useful because they help simplify circuits to reduce cost help debug circuits help us with reverse engineering help us with re-engineering Most of them are similar to ordinary algebra, but a few are very different. 7

18 Theorems Set # Operations with and. X + = X D. X = X 2. X + = 2D. X = Idempotent Law 3. X + X = X 3D. X X = X Involution Law 4. X = X Laws of Complementarity 5. 5D. X + X = X X = Commutative Law 6. X + Y = Y + X 6D. X Y = Y X Associative Law: 7. (X + Y) + Z = X + (Y + Z) 7D. (X Y) Z = X (Y Z) = X + Y + Z = X Y Z 8

19 Duality The The dual dual of of a Boolean equation is is derived by by replacing AND AND operations by by ORs, ORs, OR OR operations by by ANDs, ANDs, constant s s by by s, s, and and s s by by s s (literals are are left left unchanged). For For any any equation that that is is true, true, its its dual dual is is also also true! true! Example: X + = X Dual Dual equation: X = X Use Use duality duality to to derive derive laws laws D D 7D 7D from from 7!! 9

20 Theorems Set #2 Freaky!! Distributive Law 8. X (Y+ Z) = (X Y) + (X Z) 8D. X + (Y Z) = (X + Y) (X + Z) Simplification Theorems 9. 9D. X Y + X Y = X ( X + Y ) ( X + Y) = X. X + (X Y) = X D. X (X + Y) = X (X + Y ) Y = X Y (X Y ) + Y = X + Y. D. 2

21 De Morgan's Laws Theorems Set #3 X Y = (X + Y ) NOR X + Y = (X Y ) NAND In general 2. (X + Y + Z +...) = X Y Z... 2D. (X Y Z...) = X + Y + Z F(X,X 2,...,X n,,, +, ) = F(X, X 2,...,X n,,,, +) 2

22 DeMorgan's with Bubbles Two bubbles cancel each other out All NAND circuit has the same topology as as the AND-OR X + Y = (X Y ) 22

23 DeMorgan's Law Use to convert AND/OR expressions to OR/AND expressions Examples: Z = A BC + ABC + ABC + ABC Z = ( ABC) ( ABC) ( ABC) ( ABC) Z = (A + B + C) (A + B + C) (A + B + C) (A + B + C) 23

24 Theorems - Set 4 Duality 4. (X + Y + Z +...) D = X Y Z... 4D. (X Y Z...) D = X + Y + Z {F(X,X 2,...,X n,,, +, )} D = {F(X,X 2,...,X n,,,, +)} Multiplying and Factoring Theorems 6. 6D. (X + = Y) (X + X Z + Z) X Y X Y + X Z) = ( X + Z) ( X + Y) Consensus Theorem 7. 7D. XY + YZ + = XY + XZ XZ ( X + Y)( Y + Z)( X + Z) = ( X + Y)( X + Z) Watch out! Cancellation does not work in Boolean algebra! Qn: How do we know if these theorems work? 24

25 Proving Theorems Boolean Algebra: E.g., prove the theorem: X Y + X Y = X distributive law (8) complementary law (5) identity (D) X Y + X Y = X (Y + Y) X (Y + Y) = X () X () = X E.g., prove the theorem: X + X Y = X identity (D) X + X Y = X + X Y distributive law (8) identity (2) identity () X + X Y = X ( + Y) X ( + Y) = X () X () = X 25

26 Other Useful Functions There are 6 possible unique functions of 2 variables X Y F F F2 F3 F4 F5 F6 F7 F8 F9 F F F2 F3 F4 F5 X Y X Y X + Y XOR XNOR Y X NAND NOR XOR X Y = X Y + X Y X Y Z XNOR X Y = X Y + X Y X Y Z New and useful gates 26

27 Do Activity #2 Now Due: End of Class Today RETAIN THE LAST PAGE (#3)!! For Next Class: Bring Randy Katz Textbook Electronic copy on website will disappear Friday. Required Reading: Sec 2.2 & 2.3 of Katz This reading is necessary for getting points in the Studio Activity! 27

Memory is implemented as an array of electronic switches

Memory Structure Memory is implemented as an array of electronic switches Each switch can be in one of two states 0 or 1, on or off, true or false, purple or gold, sitting or standing BInary digits (bits)