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CSE 111

CSE 111. Binary Arithmetic Thanks to Dr. Schindler. Binary Addition. Addition Table Using the above table, proceed as with base ten. Binary Addition. Example Consider 14 10 + 9 10 using binary addition Sum = 10111 2 = 23 10. Subtraction with 2’s Complement. Procedure (M-N)

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CSE 111

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  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 1410 + 910 using binary addition • Sum = 101112 = 2310

  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 8110 - 4510 using 2's complement and 8 bits • 10100012 - 001011012 • 2's complement of 4510 = 110100112 • 001010001 + 11010011 = 100100100 • End carry occurs • Discard end carry • Difference = 001001002 = 3610

  6. Subtraction with 2’s Complement • Example #2 • Consider 5310 - 6010 using 2's complement and 8 bits • 001101012 - 001111002 • 2's complement of 6010 = 110001002 • 00110101 + 11000100 = 11111001 • No end carry • Result (111110012) is in 2's complement notation • Difference = 111110012 = -710

  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  2n-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 • -2n-1 2n-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, 11th 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

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