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Arithmetic -- addition. Binary similar to decimal arithmetic. No carries. Carries. 1+1 is 2 (or 10 2 ), which results in a carry. Arithmetic -- subtraction. No borrows. Borrows. 0 - 1 results in a borrow Borrow makes it (10) 2 = (2) 10. Arithmetic -- multiplication.

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Arithmetic -- addition


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    1. Arithmetic -- addition • Binary similar to decimal arithmetic No carries Carries 1+1 is 2 (or 102), which results in a carry

    2. Arithmetic -- subtraction No borrows Borrows 0 - 1 results in a borrow Borrow makes it (10)2 =(2)10

    3. Arithmetic -- multiplication Successive additions of multiplicand or zero, multiplied by 2 (102). Note that multiplication by 102 just shifts bits left.

    4. Complements • Simply Subtraction (Subtraction by addition) • R’s Complement • In Binary 2’s complement • In Decimal 10’s complement • (R-1) Complement • In Binary 1’s complement • In Decimal 9’s Complement

    5. R, complement/2,s/10,s complement For a given positive no N in base r with integer part of n digits. The r,s complement of N is defined as rn-N for N not =0 and 0 forN=0 Example 10,s Complement of(52520) is 105-52520=47480 (0.3267)=(1-0.3267=0.6733 No integer so 100=1 2,s Complement of(101100)2= 26-(101100)2=010100 0.01101 = (1-0.0110)=0.1010

    6. (r-1),s, complement/1,s/9,s complement For a given positive no N in base r with integer part of n digits and a fraction part of m digits. The( r-1),s complement of N is defined as (rn – rm- N) Example 9,s Complement of(52520) is( 105-100- 52520)=47479 (0.3267)=(1- 10-4- 0.3267=0.6732 No integer so 100=1 1,s Complement of(101100)2= (26- 1)-(101100)2=010011 0.0110 = (1-2-4)- (0.0110)=( 0.1111-0.0110) = 0.1001

    7. DUALITY METHOD • 1,S COMPLEMENT OF (10111) • REPLACE 1 BY 0 & O BY 1 • (10111)= (01000) • 2,S COMPLEMENT • REPLACE BY 1,S COMPLEMENT +1 • (10111)=01000+1=01001

    8. Subtraction with r-Complement • M-N • Add the minuend, M to r’s complement of Subtrahend, N • M+ (rn -N)= M-N+ rn • If M GTE N then sum will produce end carry . Ignore it • If M LT N (No Carry) then take r’s complement of answer (Negative)

    9. Subtraction with r’s Complement • Using 10’s complement subtract 72532-03250 • Using 10’s complement subtract 03250 -72532 • Using 2’s complement subtract 1010100 -1000011 • Using 2’s complement subtract 1000011- 1010100

    10. Subtraction with r-1 Complement • Similar to r’s complement • But since r-1 complement is 1 less than r complement, Carry is added back to get the result • If no carry, result is negative1’s complement to get the answer • 1010100-1000011 • 1000000-1010100

    11. Signed Binary Numbers • Need notation for negative values • Everything must be represented by binary digits • Signed magnitude convention • Left most bit can be used • 0  Positive • 1  Negative • 01001 is +9 and 11001 is -9 (Not 25. Convention known in advance) • Signed Complement (Store negative as comps) • Signed 1’s complement (8 bits)11110110 • Signed 2’s complement (8 bits)11110111 • Signed Magnitude (8 bits) 10001001

    12. BCD • Binary Coded Decimal • Decimal digits stored in binary • Four bits/digit (Use 10 instead of 16) • Like hex, except stops at 9 • Example 931 is coded as 1001 0011 0001 • People understand decimal system better • Written differently but decimal value is same • Decimal 15 in BCD 0001 0101 in Binary it is 1111 Since most computers store data in eight-bit , bytes • Ignore 4 extra bits • one can store two digits per byte, called "packed" BCD

    13. BCD Addition • Since each digit is max 9 Sum will always be less than 19= 9+9+1(carry) • Two BCD digits are added as binary numbers • When binary sum is more than binary (1001)2, result is invalid (unlike Hex last 6 were ignored) • Addition of 6=(0110)2 make a correct BCD and produces a carry • Binary Sum carry and Decimal Carry differ by 16-10=6 • 4+5, 4+8, 8+9 • 184+576

    14. Binary Codes for Numbers • Binary codes for decimal digits require 4 bits per digit • Many codes use 4 bits in 10 distinct possible combinations (out of 16) • 2421 and Excess 3 are self complementing (1 and 0 9’s Comp of decimal) • Contents can be interpreted differently. • What decimal value does 1100001111001001 represent in different binary codes? Dec Binary BCD Excess-3 2421 84-2-1 Octal Hexadecimal 0 0 0000 0011 0000 0000 000 0000 1 1 0001 0100 0001 0111 001 0001 2 10 0010 0101 0010 0110 010 0010 3 11 0011 0110 0011 0101 011 0011 4 100 0100 0111 0100 0100 100 0100 5 101 0101 1000 1011 1011 101 0101 • 110 0110 1001 1100 1010 110 0110 • 111 0111 1010 1101 1001 111 0111 • 1000 1000 1011 1110 1000 - 1000 • 1001 1001 1100 1111 1111 - 1001 • 1010 - - - - - 1010 A • 1011 - - - - - 1011 B …

    15. Other Codes Exist Gray Code/Reflected Code • Only one bit changes at a time • 0000,0001,0011,0010,0110,0111,0101,0100,1100,1101,1111,1110,1010,1011,1001,1000 • Why is this useful?The no changes by one digit. It is used to represent the digital data converted from Analog data.Where as in Binary all numbers changes. • 01111000 (All Four bits need to be changed)

    16. Character Codes • ASCII • Many applications require handling of not only numbers but letters and special characters • Stands for American Standard Code for Information Interchange • 7 Bits to store 128 characters • In ASCII, every letter, number, and punctuation symbol has a corresponding number, or ASCII code • This encoding system not only lets a computer store a document as a series of numbers, but also lets it share such documents with other computers that use the ASCII system.

    17. 11000011 11001001 If ASCII with even parity then “C” and “I”. If odd Parity then error

    18. Even Parity • Sometimes high-order bit of ASCII coded to enable detection of errors • Even parity – set bit to make number of 1’s even • Examples A (01000001) with even parity is 01000001 C (01000011) with even parity is 11000011

    19. Odd Parity • Similar except make the number of 1’s odd • Examples A (01000001) with odd parity is 11000001 C (01000011) with odd parity is 01000011

    20. Error Detection • Note that parity detects only simple errors • One, three, etc. bits • More complex methods exist • Some that enable recovery of original info • Cost is more redundant bits

    21. Binary Storage and Registers • Physical existence in information storage medium for storing individual bits • A binary cell is a device that posses stable stages and is capable of storing one bit of information • A Register is a group of binary cells. • Can store any discrete quantity of information that contains n bits. • 1100001111001001 is a 16 bit register • 2n possible states to store 0 to 2n -1 number • Contents can be interpreted differently

    22. Register Transfers • Basic Operation in digital systems • When Key is pressed 8 bit alphanumeric character code in to Input Register • Contents of Input Register are transferred to eight least significant cells of a Processor Register • After every transfer input register is cleared for new keystroke • Each eight bit character transfer to the processor register is preceded by shift of previous character to next eight cells on its left • When Processor Register is full, its contents are transferred to the Memory Register

    23. Manipulation of binary variable • Adding two 10 bit binary numbers • Memory Unit • Processor Unit

    24. Binary Logic • Variables have two possible distinct values, 0 and 1 • Three Logic operations • AND “ . ” If and only if all variable are 1 • OR “ + ” If any one or more of the variable is 1 • NOT “ ’ ” Complement (Reverse) • Unlike binary arithmetic 1 +1 is not 10 but 1 • Truth Table of Logical Operations

    25. Representation of Binary variables • Different Digital Systems represent 0 and 1 differently • Logical 0 as 0 volts. Logical 1 as 4 volts • Range

    26. Logic Gates Logic Gates are electronic circuits that operate on one or more input signals to produce an output signal

    27. Gates with Multiple Inputs

    28. LOGIC CCTS CHARACTERISTICS • Fan out. It is the max no of iutputs that can be connected to the output of a gate. • Power Dissipation.It is the power dissipated in the gate(mw). • Propagation Delay.It is the transition delay time from i/p to o/p, expressed in nano second. • Noise Margin.It is the noise which does not cause change in the o/p