1. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU1 Binary additon & subtraction

2. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU2 Class 18 – Subtraction  Binary Addition and Subtraction  Subtraction circuits  Incrementer, Decrementer  Material from section 4-3 and 4-4 of text

3. Binary Subtraction  Have previously looked at the subtraction operation. A quick review.  Just like subtraction in any other base Minuend 10110 Subtrahand -10010 Difference 00100  And when a borrow is needed. Note that the borrow gives us 2 in the current bit position. . 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU3

4. And a full example  And more ripple - 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU4

5. In General  When there is no borrow into the msb position, then the subtrahend in not larger than the minuend and the result is positive and correct.  If a borrow into the msb does occur, then the subtrahend is larger than the minuend. This was seen back in lecture 2. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU5

6. Consider  Now do the operation 4 – 6  Correct difference is -2 or -0010  Different because 2 n was brought in and made the operation M-N+2 n 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU6

7. Desired  Actual desired magnitude is N-M  To get this need to do 2 n – (M-N+2)= N-M  Doing the subtraction from 2 n gives the correct result. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU7

8. Two’s compliment  But how do you represent a minus sign electronically in a computer?  How can you represent it such that arithmetic operations are manageable?  There are two types of compliments for each number base system. Have the r’s complement Have the (r-1)’s complement  For base 2 have 2’s complement and 1’s complement 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU8

9. 1’s Complement  1’s complement of N is defined as (2 n -1)-N. If n=4 have (2 n -1) being 1 0000 - 1 = 1111  So for n=4 would subtract any 4-bit binary number from 1111.  This is just inverting each bit.  Example: 1’s compliment of 1011001  is 0100110 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU9

10. 2’s complement  The 2’s complement is defined as 2 n -N  Can be done by subtraction of N from 2 n or adding 1 to the 1’s complement of a number.  For 6 = 0110 The 1’s complement is 1001 The 2’s complement is 1010 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU10

11. Operation with 2’s complement  Add 4 and -6  Will use the 2’s complement of -6 or 1010 4 0100 -6 1010 1110  And taking the 2’s complement of 1110 get 0001 + 1 = 0010 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU11

12. A 2’s complement table for 4 bits  Listing the values represented. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU12

13. A circuit that does +/-  A general adder subtractor  OP=0 for addition/ =1 for subtraction 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU13

14. Another number format  Signed magnitude – use the MSB to indicate the sign. The remaining bits indicate the magnitude. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU14

15. Overflow  When adding 2 n-bit numbers it is possilbe to get a n+1 bit result if there is a carry out.  On paper it is easy just add another bit.  In 2’s complement add a msb 0 for a positive or a msb 1 for a negative.  In a computer the number of bits that can be used is fixed. 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU15

16. Overflow indication.  In 8-bit 2’s complement notation the range that can be represented is -127 to +127.  Then the operation to add +70 to +80 is Carries 0 1 +70 0 100 0110 +80 0 101 0000 +150 1 001 0110  Also look at the addition of -70 and -80 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU16

17. The other addition  The addition of -70 and -80 Carries 1 0 -70 1 011 1010 -80 1 011 0000 -150 0 110 1010  The rule – if the carry into the msb position differs from the carry out from the msb position then an overflow has occurred.  The circuit . 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU17

18. Class 18 assignment  Covered sections 4-3 through 4-4  Problems for hand in none  Problems for practice 4-3, 4, 5, 6, 7, 8,16  Reading for next class: sections 5-1, 5-2 9/15/09 - L15 Decoders, Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU18