1. CSE 111 Binary Arithmetic Thanks to Dr. Schindler

2. Binary Addition Addition Table Using the above table, proceed as with base ten.

3. Binary Addition Example  Consider 14 10 + 9 10 using binary addition  Sum = 10111 2 = 23 10

4. Subtraction with 2’s Complement Procedure (M-N)  Express minuend, M, and subtrahend, N, with same number of integer bits  Add minuend, M, to 2’s complement of subtrahend, N.  If an end carry occurs, discard it.  Indicates positive result  If not, the result is a negative value represented in r’s complement notation.

5. Subtraction with 2’s Complement Example #1  Consider 81 10 - 45 10 using 2's complement and 8 bits  1010001 2 - 00101101 2  2's complement of 45 10 = 11010011 2  001010001 + 11010011 = 100100100  End carry occurs  Discard end carry  Difference = 00100100 2 = 36 10

6. Subtraction with 2’s Complement Example #2  Consider 53 10 - 60 10 using 2's complement and 8 bits  00110101 2 - 00111100 2  2's complement of 60 10 = 11000100 2  00110101 + 11000100 = 11111001  No end carry  Result (11111001 2 ) is in 2's complement notation  Difference = 11111001 2 = -7 10

7. Overflow An overflow occurs when the result of an arithmetic operation falls outside the available range that can be stored.

8. Overflow Detection of overflow for addition of signed numbers  Carries into & out of MSB (sign bit) differ  Two positive numbers added & negative result is obtained  Two negative numbers added & a positive result is obtained Note that overflow cannot occur if two numbers of differing signs are added

9. Number Ranges Unsigned Numbers  0  2 n -1  Example  8 Bit 0  255  16 Bit 0  65,535  32 Bit 0  4,294,967,295  64 Bit 0  18,446,744,073,709,599,999

10. Number Ranges Signed Numbers in Two’s Complement Notation  -2 n-1  2 n-1 -1  Example  8 Bit -128  127  16 Bit 32,768  32,767  32 Bit -2,147,483,648  2,147,483,647  64 Bit -9,223,372,036,854,780,000  9,223,372,036,854,779,999

11. Overcoming Overflow If overflow occurs, how can we overcome it?  Additional bits are required

12. Overcoming Overflow The Problem  Adding more bits is fine for an unsigned number, but how about a signed number?  Example - Unsigned  Represent the unsigned number 38 as an 8-bit number 00100110  Extend it to 16-bits 0000000000100110  Example - Signed  Represent the signed number 38 as an 8-bit number 11011010  Extend it to 16-bits and put 0’s in the new bit positions 0000000011011010 The problem is that this is not -38, it is +218  To preserve the sign and magnitude, the sign must be replicated in the new positions 1111111111011010

13. References J. Glenn Brookshear, Computer Science - An Overview, 11 th edition, Addison-Wesley as an imprint of Pearson, 2012 Daniel Hillis, The Pattern on the Stone, Basic Books (Perseus Books Group), 1998 M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals, Prentice Hall, Inc., 2000 Victor P. Nelson, H. Troy Nagle, Bill D. Carroll, and J. David Irwin, Digital Logic Circuit Analysis and Design, Prentice Hall, Inc., 1995 Donald D. Givone, Digital Principles and Design, McGraw-Hill, 2003