Topics Covered in Chapter 4

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
• Ones complement
• Twos 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
• Ones complement
• Twos complement

12
Ones 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
Ones 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
Ones 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
Ones 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
Twos Complement Representation

• Positive numbers
• The representation of a positive number is the
  same as it is for ones complement.
• The most significant bit is set to zero.
• The remaining bits determine the size of the
  number.
• Example
• 0001 0101two 21ten

17
Twos Complement Representation

• Negative numbers
• The representation of a negative number is formed
  by taking the ones, 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
Twos 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
Twos Complement Representation

• The twos complement representation of 127 is
• 0111 1111two
• The twos 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
Twos Complement Representation

• The twos complement representation of ?128 is
• 1000 0000two
• The twos 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
Twos 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 doesnt 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 (ones complement)
• ?8 (twos complement)

23
Problems

• Find the decimal equivalent of each of the
  following binary numbers
• 1000 0110two (unsigned representation)
• 1111 1000two (ones complement representation)
• 0010 1101two (sign and magnitude representation)
• 1110 1001two (twos 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 twos 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.
• Lets look at the two examples on page 217 of
  your text.

28
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
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
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
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