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