1. Chapter 4.1 Introduction

2. Topics Covered in Chapter 4 • How numbers (integers, decimal numbers) are represented in the computer • How addition, subtraction, multiplication and division are really implemented in the hardware • Logical operations

3. Chapter 4.2 Signed and Unsigned Numbers

4. Representing Unsigned Numbers of Various Sizes • A 32-bit MIPS “word” can represent 232 different numbers ranging from 0  232  1, or 0  4,294,967,295ten. • A 16-bit number can represent values ranging from 0  216  1, or 0  65,535ten. • An 8-bit number can represent values ranging from 0  28  1, or 0  255ten.

5. Representing Unsigned Numbers of Various Sizes • Examples: • 0000 0000 0000 0000 1000 0010 0011 0110two = 33,334ten • 1111 1111 1111 1111two = 65,535ten • 1001 1101two = 157ten

6. Representing Signed Numbers • We need a representation that distinguishes positive numbers from negative numbers, since computer programs use both. • Some possible representations: • Sign and magnitude • One’s complement • Two’s complement

7. Sign and Magnitude Representation • This representation is implemented by assigning a specific bit to be the “sign” bit. • By convention, a “zero” in the sign bit means that the integer represented is positive or zero; a “one” in the sign bit means that it is negative or zero.

8. Examples of Sign and Magnitude Representation • Note: assume an 8-bit number for these examples. • Unsigned number representations: • 0000 1101two = 13ten • 1000 1101two = 141ten • Signed number representations: • 0000 1101two = 13ten • 1000 1101two = 13ten Sign bit Size of number

9. Sign and Magnitude Representation • Range of 8-bit numbers that can be represented: 127  +127 Or  (27  1)  27  1 Or 1111 1111  0111 1111 Sign Sign

10. Sign and Magnitude Representation • Disadvantage of this representation: • Two distinct representations for zero (0000 0000, 1000 0000) • Advantage of this representation: • Additive inverse easily formed by inverting the “sign” bit • Result of such a representation: • Increased complexity of addition and subtraction (e.g., an extra step may be required to set the sign)

11. Complement Representation of Numbers • Complement representation is motivated by the need to minimize the complexity of addition and subtraction. • There are two types of complement representation: • One’s complement • Two’s complement

12. One’s Complement Representation • Positive numbers • The representation of a positive number is the same as it is for “sign and magnitude”. • The most significant bit is set to zero. • The remaining bits determine the size of the number. • Example: 0111 1111two = 127ten(maximum 8-bit number)

13. One’s Complement Representation • Negative numbers • The representation of a negative number is formed by “inverting” each bit of the corresponding positive number; the resulting number is the bitwise complement, or additive inverse, of the positive value. • The most significant bit is equal to one. • Example: 1111 1100two = 3ten (explanation follows ...)

14. One’s Complement Representation • Example, continued: 0000 0011two = 3ten Inverting all the bits gives us: 1111 1100two = 3ten Adding the two numbers together, we get: 0000 0011two = 3ten 1111 1100two = 3ten 1111 1111two = 0ten

15. One’s Complement Representation • The range of 8-bit numbers that can be represented is the same as it is for “sign and magnitude” representation: 127  +127 • There are still two distinct representations for zero: 0000 0000two and 1111 1111two

16. Two’s Complement Representation • Positive numbers • The representation of a positive number is the same as it is for “one’s complement”. • The most significant bit is set to zero. • The remaining bits determine the size of the number. • Example: 0001 0101two = 21ten

17. Two’s Complement Representation • Negative numbers • The representation of a negative number is formed by taking the one’s, or bitwise, complement of the corresponding positive number, then adding one. • The most significant bit is equal to one. • Example: • The bitwise complement of 0001 0101two (21ten) is 1110 1010two. Adding 1 to that value gives us: 1110 1011two, the additive inverse of 0001 0101two.

18. Two’s Complement Representation • The range of 8-bit numbers that can be represented is: 128  +127 Note that there is one Or more negative value  (27)  27  1 than there are positive values. • There is now only one representation for zero, however: 0000 0000two

19. Two’s Complement Representation • The two’s complement representation of +127 is: 0111 1111two The two’s complement representation of its opposite, 127, is: 1000 0001two Adding +127 and 127 together, we get: 0111 1111two = 127ten 1000 0001two = 127ten 1 0000 0000two = 0ten Ignored (Why?)

20. Two’s Complement Representation • The two’s complement representation of 128 is: 1000 0000two The two’s complement representation of its opposite, +128, is: 0111 1111two + 1two 1000 0000two ????? • Conclusion: • The most negative 8-bit number has no additive inverse within a fixed precision. This is an example of overflow.

21. Two’s Complement Representation • The range of 32-bit numbers that can be represented is: 2,147,483,648  +2,147,483,647 Or  (231)  231  1 • Once again, there is one more negative value than there are positive values. • No, it doesn’t have an additive inverse.

22. Problems • Give the designated 8-bit binary representation of each of the following decimal numbers: • 121 (unsigned) • 15 (sign and magnitude) • 89 (one’s complement) • 8 (two’s complement)

23. Problems • Find the decimal equivalent of each of the following binary numbers: • 1000 0110two (unsigned representation) • 1111 1000two (one’s complement representation) • 0010 1101two (sign and magnitude representation) • 1110 1001two (two’s complement representation)

24. Converting an Integer Representation from One Size to Another • It is sometimes necessary to convert a binary number represented in n bits to a number represented with more than n bits.

25. Converting an Integer Representation from One Size to Another • For example, the immediate field in the I-Format instructions contains a two’s complement 16-bit number representing 32,768 (215) to 32,767 (215  1). • To add the contents of the field to a 32-bit register, the computer must convert that 16-bit number to its 32-bit equivalent.

26. Converting an Integer Representation from One Size to Another • How is this conversion implemented? • Two steps are required: • The sign bit of the smaller quantity is replicated to fill the new (unoccupied) bits of the larger quantity. • The bits making up the original (smaller) value are simply copied into the right half of the new word.

27. Converting an Integer Representation from One Size to Another • This conversion technique is commonly called sign extension. • Let’s look at the two examples on page 217 of your text.

28. 43 29 9 8 Converting an Integer Representation from One Size to Another More Examples  • Convert the value in the “immediate” field of the following instruction to its 32-bit binary representation. sw $t1, 8($sp)$t1  $9 $sp  $29

29. 43 29 9 8 Converting an Integer Representation from One Size to Another More Examples  • Conversion of 8, the value in the immediate field, to its 32-bit binary representation. 8ten  0000 0000 0000 1000two (16-bit) 8ten  0000 0000 0000 0000 0000 0000 0000 1000two (32-bit) “new” bits original bits

30. 8 29 29 4 Converting an Integer Representation from One Size to Another More Examples  • Convert the value in the “immediate” field of the following instruction to its 32-bit binary representation. addi $sp, $sp, 4$sp  $29

31. 8 29 29 4 Converting an Integer Representation from One Size to Another More Examples  • Conversion of 4, the value in the immediate field, to its 32-bit binary representation. 4ten  1111 1111 1111 1100two (16-bit) 4ten  1111 1111 1111 1111 1111 1111 1111 1100two (32-bit) “new” bits original bits