Chapter 3 Data Representation part2

Dr. Bernard Chen Ph.D.University of Central Arkansas

Spring 2010

Outline

Subtraction of Unsigned Numbers using r’s complement

How To Represent Signed Numbers Floating-Point Representation

Subtraction of Unsigned Numbers using r’s complement

(1) if M N, ignore the carry without taking complement of sum.

(2) if M < N, take the r’s complement of sum and place negative sign in front of sum. The answer is negative.

Example 1 (Decimal unsigned numbers),  perform the subtraction 72532 - 13250 = 59282.  M > N : “Case 1” “Do not take complement of sumand discard carry”The 10’s complement of 13250 is 86750. Therefore:

M = 7253210’s complement of N =+86750

Sum= 159282Discard end carry 105= - 100000Answer = 59282 no complement

Example 2;Now consider an example with M <N. The subtraction 13250 - 72532 produces negative 59282. Using the procedure with complements, we have 

M = 1325010’s complement of N = +27468

Sum = 40718

Take 10’s complement of Sum = 100000

-40718

The number is : 59282

Place negative sign in front of the number: -59282

Subtract by Summation

Subtraction with complement is done with binary numbers in a similar way.

Using two binary numbers X=1010100 and Y=1000011

We perform X-Y and Y-X

X-Y

X= 1010100 2’s com. of Y= 0111101 Sum= 10010001 Answer= 0010001

Y-X

Y= 1000011 2’s com. of X= 0101100 Sum= 1101111

There’s no end carry: answer is negative --- 0010001 (2’s complement of 1101111)

Outline

Subtraction of Unsigned Numbers using r’s complement

How To Represent Signed Numbers Floating-Point Representation

How To Represent Signed Numbers Plus and minus signs used for decimal

numbers: 25 (or +25), -16, etc.

For computers, it is desirable to represent everything as bits..

Three types of signed binary number representations:

1. signed magnitude, 2. 1’s complement, and 3. 2’s complement

1. signed magnitude• In each case: left-most bit

indicates sign: positive (0) or negative (1).

Consider 1. signed magnitude:

000011002 = 1210

Sign bit Magnitude

100011002 = -1210

Sign bit Magnitude

2. One’s Complement Representation

The one’s complement of a binary number involves inverting all bits.

• To find negative of 1’s complement To find negative of 1’s complement number take the 1’s complement of number take the 1’s complement of whole number including the sign bit.whole number including the sign bit.

000011002 = 1210

Sign bit Magnitude

111100112 = -1210

Sign bit 1’complement

3. Two’s Complement Representation• The two’s complement of a binary

number involves inverting all bits and adding 1.

To find the negative of a signed number take the 2’s the 2’s complement of the positive number including the sign bit.

000011002 = 1210

Sign bit Magnitude

111101002 = -1210

Sign bit 2’s complement

The rule for addition is add the two numbers, including their sign bits, and discard any carry out of the sign (leftmost) bit position. Numerical examples for addition are shown below.Example:

+ 6 00000110 - 6 11111010+13 00001101 +13 00001101+19 00010011 +7 00000111

+6 00000110 -6 11111010-13 11110011 -13 11110011-7 11111001 -19 11101101

In each of the four cases, the operation performed is always addition, including the sign bits.Only one rule for addition, no separate treatment of subtraction. Negative numbers are always represented in 2’s complement.

Sign addition in 2’s complement

Arithmetic Subtraction

A subtraction operation can be changed to an addition operation if the sign of the subtrahend is changed.

(±A) - (+B) = (±A) + (-B) (±A) - (-B) = (±A) + (+B)

Arithmetic Subtraction Consider the subtraction of (-6) - (-13) =

+7. In binary with eight bits this is written as 11111010 - 11110011. The subtraction is changed to addition by taking the 2’s complement of the subtrahend (-13) to give (+13). In binary this is 11111010 + 00001101 = 100000111.

Removing the end carry, we obtain the correct answer 00000111 (+ 7).

Overflow

Overflow example:

+70 0 1000110 -70 1 0111010 +80 0 1010000 -80 1 0110000 = +150 1 0010110 =-150 0 1101010

Overflow

• An overflow cannot occur after an addition if one number is positive and the other is negative, since adding a positive number to a negative number produces a result that is smaller than the larger of the two original numbers.

An overflow may occur if the two numbers added are both either positive or negative.

Outline

Subtraction of Unsigned Numbers using r’s complement

How To Represent Signed Numbers Floating-Point Representation

Floating-Point Representation

+ 6132.789 is represented in floating-point with a fraction and an exponent as follows:

Fraction Exponent +0.6132789 +04

Scientific notation : + 0.6132789 10+4

Floating-Point Representation 32-bit floating point format. Leftmost bit = sign bit (0 positive or 1

negative). Exponent in the next 8 bits. Use a biased

representation. Final portion of word (23 bits in this example)

is the significand (sometimes called mantissa).

Example Convert the following number;37.75 into floating point

format to fit in 32 bit register. Convert the number from decimal into binary

100101.11 Normalize all digits including the fraction to

determine the exponent. 1.0010111 x 25

0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0

signsign EXPEXP Significant

Outline

Subtraction of Unsigned Numbers using r’s complement

How To Represent Signed Numbers Floating-Point Representation