CSE 111

Binary Arithmetic

Thanks to Dr. Schindler

Binary Addition

Addition Table

Using the above table, proceed as with base ten.

0

1

0

1

1

0

+ 0 1

Carry1

Binary Addition

ExampleConsider 1410 + 910 using binary addition

Sum = 101112 = 2310

1 1 1 0+ 1 0 0 1

1 0 1 1 1

1

Subtraction with 2’s Complement

Procedure (M-N)Express minuend, M, and subtrahend, N, with same

number of integer bitsAdd 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.

Subtraction with 2’s Complement

Example #1Consider 8110 - 4510 using 2's complement and 8 bits

10100012 - 001011012

2's complement of 4510 = 110100112

001010001 + 11010011 = 100100100End carry occursDiscard end carryDifference = 001001002 = 3610

Subtraction with 2’s Complement

Example #2Consider 5310 - 6010 using 2's complement and 8 bits

001101012 - 001111002 2's complement of 6010 = 110001002

00110101 + 11000100 = 11111001No end carryResult (111110012) is in 2's complement notation

Difference = 111110012 = -710

Overflow

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

Overflow

Detection of overflow for addition of signed numbersCarries into & out of MSB (sign bit) differTwo positive numbers added & negative result is

obtainedTwo negative numbers added & a positive result is

obtained

Note that overflow cannot occur if two numbers of differing signs are added

Number Ranges

Unsigned Numbers0 2n-1Example

8 Bit0255

16 Bit0 65,535

32 Bit04,294,967,295

64 Bit0 18,446,744,073,709,599,999

Number Ranges

Signed Numbers in Two’s Complement Notation-2n-1 2n-1-1Example

8 Bit-128127

16 Bit32,76832,767

32 Bit-2,147,483,6482,147,483,647

64 Bit-9,223,372,036,854,780,000 9,223,372,036,854,779,999

Overcoming Overflow

If overflow occurs, how can we overcome it?Additional bits are required

Overcoming Overflow

The ProblemAdding 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 number00100110

Extend it to 16-bits0000000000100110

Example - SignedRepresent the signed number 38 as an 8-bit number

11011010

Extend it to 16-bits and put 0’s in the new bit positions 0000000011011010The problem is that this is not -38, it is +218

To preserve the sign and magnitude, the sign must be replicated in the new positions1111111111011010

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