Data Representation - Part I

1. Data Representation - Part I

2. 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

3. 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

4. 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

5. 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.

6. Representation of Integers • Unsigned Integer Representation • Sign Magnitude • Complement Representation • Biased Representation • Sign Extension

7. 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

8. x xxx xxxx 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. Example 9.2 010001 (17) 1101000 (-24) 101110 0010111 1 1 ------ ------- 101111 (-17) 0011000 (24) take the 1’s complement

15. 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.

16. 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

17. 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 patterninteger represented (in decimal) 000 -4 001 -3 010 -2 011 -1 100 0 101 1 110 2 111 3

18. 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

19. 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

20. xxxxxxxx 00000000xxxxxxxx Sign Extension - Unsigned • Place the original integer into the least significant portion and stuff the remaining positions with 0’s. 8 bits 16 bits

21. sxxxxxxx s00000000xxxxxxx 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.

22. 0xxxxxxx 1xxxxxxx 000000000xxxxxxx 111111111xxxxxxx 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. The original number is placed into the least significant portion The original number is placed into the least significant portion