Data Representation - Part I

Representing Numbers Choosing an appropriate representation is

a critical decision a computer designer has to make

The chosen representation must allow for efficient execution of primitive operations

For general-purpose computers, the representation must allow efficient algorithms for (1) addition of two integers (2) determination of additive inverse

With a sequence of N bits, there are 2N

unique representations Each memory cell can hold N bits The size of the memory cell determines

the number of unique values that can be represented

The cost of performing operations also increases as the size of the memory cell increases

It is reasonable to select a memory cell size such that numbers that are frequently used are represented

Binary Representation

The binary, weighted positional notation is the natural way to represent non-negative numbers

SAL and MAL number the bits from right to left, i.e., beginning with 0 as the least significant digit

Little-Endian vs. Big-Endian

Numbering the bits from right to left, beginning with zero is called Little Endian byte order. Intel 80x86 and DECstation 3100 use the Little Endian byte ordering.

Numbering the bits from left to right, beginning with zero is called Big Endian byte order. SunSparc and Macintosh use the Big-Endian byte ordering.

Representation of Integers

Unsigned Integer Representation Sign Magnitude Complement Representation Biased Representation Sign Extension

Unsigned Integer Representation

The representation of a non-negative integer using binary, weighted positional notation is called unsigned integer representation

Given n bits, it is possible to represent the range of values from 0 to 2n - 1

For example an 8-bit representation would allow representations that range 0 to 255

Sign Magnitude

An extra bit in the most significant position is designated as the sign bit which is to the left of an unsigned integer. The unsigned integer is the magnitude.

A 0 in the sign bit means positive, and a 1 means negative

xxx xxxxx

Given an n+1-bit sign magnitude number the range of values that it can represent is

-(2n-1) to +(2n-1) Sign magnitude representation associates

a sign bit with a magnitude that represents zero, thus it has two distinct representation of zero:

00000000 and 10000000

sign bit magnitude

Complement Representation

For positive integers, the representation is the same as for sign magnitude

For negative numbers, a large bias is added to all negative numbers, creating positive numbers in unsigned representation

The bias is chosen so that any negative number representable appears as if it were larger than the largest positive number representable

One’s Complement

For positive numbers, the representation is the same as for unsigned integers where the most significant bit is always zero

The additive inverse of a one’s complement representation is found by inverting each bit.

Inverting each bit is also called taking the one’s complement

Example 9.1

0000 0011 (3)

1111 1100 (-3)

1110 1000 (-23)

0001 0111 (23)

0000 0000 (0)

1111 1111 (0)

Note: There are two representations of zero

Two’s complement

The additive inverse of a two’s complement integer can be obtained by adding 1 to its one’s complement

The two’s complement representation for a negative number is the additive inverse of its positive representation

An advantage of two’s complement is that there is only one representation for zero

Example 9.2

010001 (17) 1101000 (-24)

101110 0010111

1 1

------ -------

101111 (-17) 0011000 (24)

take the 1’s complement

In two’s complement, one more negative value than positive value is represented - the most negative number has no additive inverse within a fixed precision.

For example, 1000000 has no additive inverse for 8-bit precision. Taking the two’s complement will yield 1000000 which seems its own additive inverse. This is incorrect and is an example of an overflow.

Note that computing the additive inverse is a mathematical operation. Taking the complement is an operation on the representation.

Biased Representation

If the unsigned representation includes integers from 0 to M, then subtracting approximately M/2 from the unsigned interpretation would shift the range from

-(M/2) to +(M/2) If a sequence has a value N when

interpreted as an unsigned integer, it has a value N-bias interpreted as a biased number

Usually the bias is either 2n or 2n-1 for an (n+1)bit representation

Example 9.3

Assume a 3-bit representation. A possible bias is 2n-1, which is 4. The following is a 3-bit representation with a bias of 4.

bit pattern integer represented

(in decimal)

000 -4

001 -3

010 -2

011 -1

100 0

101 1

110 2

111 3

Example 9.4

Given 0000 0110, what is it’s value in a biased-127 representation. Assume an 8-bit representation.

The value of the unsigned integer:

0000 0110 = 610

Its value in biased-127 is:

6 - 127 = -121

Sign Extension

For integer representations, the sizes are commonly 8, 16, 32 and 64.

It is occasionally necessary to convert an integer representation from one size to another, e.g., from 8 bits to 32 bits.

The point is to maintain the same value while changing the size of the representation

Sign Extension - Unsigned

Place the original integer into the least significant portion and stuff the remaining positions with 0’s.

xxxxxxxx

00000000xxxxxxxx

8 bits

16 bits

Sign Extension - Signed

The sign bit of the smaller representation is placed into the sign bit of the larger representation

The magnitude is put into the least significant portion and all remaining positions are stuffed with 0’s.

sxxxxxxx

s00000000xxxxxxx

Sign Extension - complement

For positive number, a 0 is used to stuff the remaining positions.

For negative number, a 1 is used to stuff the remaining positions.

0xxxxxxx

000000000xxxxxxx

1xxxxxxx

111111111xxxxxxx

The original number isplaced into the least significant portion

The original number isplaced into the least significant portion