NUMBER REPRESENTATION CHAPTER 3 – part 3. ONE’S COMPLEMENT REPRESENTATION CHAPTER 3 – part 3

NUMBER REPRESENTATIONCHAPTER 3 – part 3

ONE’S COMPLEMENT REPRESENTATION

CHAPTER 3 – part 3

Integer Representation

Integer Representat

ion

Unsigned Signed

Sign and Magnitude

One’s Complemen

t

Two’s Complemen

t

One’s complement integers

To represent

+ positive number same way as UNSIGNED Integer

- negative number complement the

positiveRANGE : -(2N-1 -1)…. +(2N-1 -1)


Table 3.5 Range of One’s complement integers

#of Bits ---------

81632

Range----------------------------------------------------

----127 -0 -32767 -0-2,147,483,647 -0

+0 +127

+ 0 +32767

+ 0 +2,147,483,647


There are two 0s in One’s complement

representation: positive and negative.

In an 8-bit allocation:+0 00000000

-0 10000000


In one’’s complement representation ,

the leftmost bit defines the sign of the number .

If it is 0, the number is positive.If it is 1, the number is negative


Storing One’s complement integer process:

1. The integer is changed to binary, (the sign is ignored).

2. 0s are added to the left of the number to make a total of N

3. bits4. If the sign is positive, no more action is

needed. If the sign is negative, every bit is complemented

Store +7 in an 8-bit memory location using one’s complement representation.

The integer is changed to binary (111). Add 5 0s to make a total of N (8) bits,

00000111 . The sign is positive so no more action is

needed

Example 4

Solution

Store –258 in a 16-bit memory location using one’s complement representation

First change the number to binary 100000010.

Add 7 0s to make a total of N (16) bits, 0000000100000010.

The sign is Negative so each bit is complemented

0000000100000010 1111111011111101

Example 5

Solution


Table 3.6 Example of storing one’s complement integers in two computers

Decimal

------------+7−7

+124−124

+24,760−24,760

8-bit allocation

------------00000111111110000111110010000011overflowoverflow

16-bit allocation

------------------------------000000000000011111111111111110000000000001111100111111111000001101100000101110001001111101000111


Interpretation1. If the left most bit is 0 (positive number)

A. Change the entire number from binary to decimal

B. Put a positive sign(+) in front of the number

2. If the left most bit is 1 (negative number)

A. Complement the entire number (change all 0’s to 1’s and

vice versa)

B. Change the entire number from binary to decimal

C. Put a negative sign(-) in front of the number

Interpret 11110110 to decimal if the number was stored as a one’s complement integer.

the leftmost bit is 1, so the number is negative

Complement the number 11110110 00001001

the complement in decimal is 9

So the answer is -9

Example 6

Solution

The one’s complement representation is not

used now by computers because:

Operations: such as subtraction and addition

is not straightforward for this representation.

Uncomfortable in programming: because

there are two 0s in this representation

One’s complement Applications

However..The advantage of this representation is:1. It’s the foundation of the next

representation(Two’s complement)2. It has properties that make it

interesting for data communication applications such as error detection and correction

One’s complement Applications

TWO’S COMPLEMENT REPRESENTATION

Integer Representation

Integer Representat

ion

Unsigned Signed

Sign and Magnitude

One’s Complemen

t

Two’s Complemen

t

Two’s complement integers

It’s…The most commonThe most importantThe most widely used representation today

RANGE : -(2N-1)…. +(2N-1 -1)


Table 3.7 Range of Two’s complement integers

#of Bits ---------

81632

Range----------------------------------------------------

----128 -0 -32768 -0-2,147,483,648 -0

+0 +127

+ 0 +32767

+ 0 +2,147,483,647


There is Only one 0 in Two’s complement

representation

In an 8-bit allocation:0 00000000


In two’s complement representation ,

the leftmost bit defines the sign of the number .

If it is 0, the number is positive.If it is 1, the number is negative


Storing One’s complement integer process:

1. The number is changed to binary, (the sign is ignored).

2. 0s are added to the left of the number to make a total of N bits

3. If the sign is positive, no more action is needed. If the sign is negative, leave all the rightmost 0’s and the first 1 unchanged. Then complement the rest of the bits.

Store +7 in an 8-bit memory location using two’s complement representation.

The integer is changed to binary (111). Add 5 0s to make a total of N (8) bits,

00000111 . The sign is positive so no more action is

needed

Example 4

Solution

Store –40in a 16-bit memory location using two’s complement representation

First change the number to binary 101000. Add 10 0s to make a total of N (16) bits,

0000000000101000. The sign is Negative so leave the rightmost

0’s up to the first 1

0000000000101000

1111111111011000

Example 5

Solution


Table 3.8 Example of storing two’s complement integers in two computers

Decimal

------------+7−7

+124−124

+24,760−24,760

8-bit allocation

------------00000111111110010111110010000100overflowoverflow

16-bit allocation

------------------------------000000000000011111111111111110010000000001111100111111111000010001100000101110001001111101001000


Interpretation1. If the left most bit is 0 (positive number)

A. Change the entire number from binary to decimal

B. Put a positive sign(+) in front of the number

2. If the left most bit is 1 (negative number)

A. leave the rightmost 0’s up to the the first 1 unchanged.

Complement the rest bits.

B. Change the whole number from binary to decimal

C. Put a negative sign(-) in front of the number

Interpret 11110110 to decimal if the number was stored as a two’s complement integer.

the leftmost bit is 1, so the number is negativeLeave 10 at the right alone and Complement the

rest11110110 00001010the complement in decimal is 10

So the answer is -10

Example 6

Solution

The two’s complement representation is

the standard representation used for

storing integers by computers today

because it makes the operations simple.

Two’s complement Applications

SUMMARY OF INTEGER REPRESENTATION•Unsigned•Sign & magnitude•One’s complement•Two’s complement

Two’s compleme

nt

One’s compleme

nt

Sign & magnitude Unsigned Contents

of Memory

+0 +0 +0 0 0000

+1 +1 +1 1 0001

+2 +2 +2 2 0010

+3 +3 +3 3 0011

+4 +4 +4 4 0100

+5 +5 +5 5 0101

+6 +6 +6 6 0110

+7 +7 +7 7 0111

-8 -7 -0 8 1000

-7 -6 -1 9 1001

-6 -5 -2 10 1010

-5 -4 -3 11 1011

-4 -3 -4 12 1100

-3 -2 -5 13 1101

-2 -1 -6 14 1110

-1 -0 -7 15 1111

EXCESS SYSTEM

Excess System

It's another representation that allows you to store both �positive and negative numbers in a computer

� the magic number a positive number

used in the conversion process

� = 2N-1 -1 OR 2N-1

where N is the bit allocation�The bit allocation :is the number of bits used to represent an integer

Example

If N= 8 bits .Calculate the magic number.

Answer:

=2N-1 -1 OR 2N-1

=28-1 -1 OR 28-1

=127 OR 128 EXCESS_127 OR EXCESS_128

Excess System RepresentationTo represent a number in Excess, use the

following procedure: Add the magic number to the integer� Change the result to binary and add 0s �

so that there is a total of N bits

Excess System RepresentationExample:Represent –25 in Excess_127 using an 8-

bit allocation.

Answer:127 + (-25) = 102102 1100110add 0’s to the left to make it 8 bit01100110

Excess System InterpretationTo interpret a number in Excess, use the

following procedure: Change the number to decimal � Subtract the magic number from the

integer

Excess System InterpretationExample:Interpret 11111110 if the

representation is Excess_127.

Answer:11111110 254

254 – 127 = 127

FLOATING POINT REPRESENTATION

Floating point representation

Floating point number = a number containing an integer & a fraction

Example: 14.234Integ

erFractio

n

Convert from Floating point number to binary

1. Convert the integer part to binary2. Convert the fraction to binary�3. Put a decimal point between the two �

parts


ExampleTransform the fraction 0.875 to binary

Answer:Write the fraction at the left corner. Multiply the number continuously by 2 and extract the integer part as the binary digit. Stop when the number is 0.0.fraction

binary

0.875 1.750 1.50 1.0 0.0

0 . 1 1 1


ExampleTransform the fraction 0.4 to a binary of 6 bits.

Answer:Write the fraction at the left cornet. Multiply the number continuously by 2 and extract the

integer part as the binary digit. You can never get the exact binary representation. Stop when you have 6 bits

fraction

binary

0.4 0.8 1.6 1.2 0.4 0.8 1.6

0 . 0 1 1 0 0 1

NORMALIZATION

Normalization

A fraction is normalized so that operations are

simpler

Normalization:the moving of the decimal point so that

there is only one 1 to the left of the decimal point.

Normalization

Normaliazed Move Original Number

+ 26 x 1.01000111001 6

+ 1010001.11001

-2 2 x 1.110011 2

- 111.000011

+2-6 x 1.11001 6 + 0.00000111001

-2-3 x 1.110011 3 - 0.001110011

Normalization

After the number is normalized we store 3 pieces of information about it:

SIGN EXPONENT MANTISSA

Example:

+ 26 x 1.0001110101

SIGN EXPONENT

MANTISSA

IEEE standard for floating point representation

Single-Precesion RepresentationThe procedure of storing a normalized

floating point number using single precession format is as follow:

1. Store the sign as 0 (positive) or 1 (negative)

2. Store the exponent (power of 2) as Excess_127

3. Store the mantissa as unsigned integer

EXAMPLE

Show the representation of the normalized number + 26 x 1.01000111001

ANSWER:

The sign is positive. The Excess_127 representation of the exponent is 133.You add extra 0s on the right to make it 23 bits. The

number in memory is stored as:

0 10000101 01000111001000000000000

SIGN EXPONENT

MANTISSA

Example of floating point representation

Mantissa Exponent

Sign

Number

11000011000000000000000

10000001 1 -2 2 x 1.11000011

11001000000000000000000

01111001 0 +2-6 x 1.11001

11001100000000000000000

01111100 1 -2-3 x 1.110011

Floating point Interpretation for single precision

the following procedure interprets a 32-bit floating-point number stored in memory

1. Use the leftmost bit as the sign2. Change the next 8 bit to decimal and subtract

127 from it. (this is the exponent)3. Add 1 and a decimal point to the next 23 bits

(ignore any extra 0’s in the right)4. Move the decimal point to the correct position

using the value of the exponent 5. Change the whole part to decimal6. Change the fraction part to decimal7. Combine the whole and the fraction parts

Example

Interpret the following 32-bit floating-point number

1 01111100 11001100000000000000000

ANSWER:

The sign is negative. The exponent is –3 (124 –127). The number after normalization is

-2-3 x 1.110011

Documents

NUMBER REPRESENTATION CHAPTER 3 – part 3. ONE’S COMPLEMENT REPRESENTATION CHAPTER 3 – part 3