The document discusses arithmetic unit operations including addition, subtraction, multiplication, and division of signed and unsigned numbers. It covers integer representation using sign-magnitude, one's complement, and two's complement methods. Two's complement is identified as the best approach for integer representation as it simplifies arithmetic operations and overflow handling. The key hardware components for performing addition and subtraction of signed integers are also summarized.

1. UNIT - 2
Arithmetic Unit

2. Table of Contents
 Addition and subtraction of signed numbers
 Design of fast adders
 Multiplication of positive numbers
 Signed operand multiplication,
 Fast multiplication- Bit pair recoding of Multipliers,
 Carry Save Addition of summands
 Integer division – Restoring Division and Non
Restoring Division
 Floating point numbers and operations

3. Integer Representation
• Only have 0 & 1 to represent everything
• Positive numbers stored in binary
– e.g. 41=00101001
• No minus sign
• No period
• Sign-Magnitude
• One's Complement
• Two’s compliment

4. • Sign Magnitude: One's Complement Two's
Complement
000 = +0 000 = +0 000 = +0
001 = +1 001 = +1 001 = +1
010 = +2 010 = +2 010 = +2
011 = +3 011 = +3 011 = +3
100 = -0 100 = -3 100 = -4
101 = -1 101 = -2 101 = -3
110 = -2 110 = -1 110 = -2
111 = -3 111 = -0 111 = -1
• Issues: balance, number of zeros, ease of operations
• Which one is best? Why?
Integer Representation (cont’d)

5. • Negating a two's complement number: invert all bits and
add 1
– remember: “negate” and “invert” are quite different!
• Converting n bit numbers into numbers with more than n
bits:
– MIPS 16 bit immediate gets converted to 32 bits for arithmetic
– copy the most significant bit (the sign bit) into the other bits
0010 -> 0000 0010
1010 -> 1111 1010
– "sign extension" (lbu vs. lb)
Two's Complement Operations

6. Signed and unsigned additions
• Unsigned addition
in 4-bit arithmetic
( carry) 11_
1011
+ 0011
1110
• 11 + 3 = 14
(8 + 4 + 2)
• Signed addition in
4-bit arithmetic
( carry) 11_
1011
+ 0011
1110
• -5 + 3 = -2

7. Signed and unsigned additions
• Same rules apply even though bit strings
represent different values
• Sole difference is overflow handling

8. Sign-Magnitude
• Left most bit is sign bit
• 0 means positive
• 1 means negative
• +18 = 00010010
• -18 = 10010010
• Problems
– Need to consider both sign and magnitude in
arithmetic
– Two representations of zero (+0 and -0)

9. Range of Numbers
• 8 bit 2s compliment
– +127 = 01111111 = 27 -1
– -128 = 10000000 = -27
• 16 bit 2s compliment
– +32767 = 011111111 11111111 = 215 - 1
– -32768 = 100000000 00000000 = -215

10. Conversion Between Lengths
• Positive number pack with leading zeros
• +18 = 00010010
• +18 = 00000000 00010010
• Negative numbers pack with leading ones
• -18 = 10010010
• -18 = 11111111 10010010
• i.e. pack with MSB (sign bit)

11. Addition and Subtraction
• Normal binary addition
• Monitor sign bit for overflow
• Take twos compliment of substahend and add
to minuend
– i.e. a - b = a + (-b)
• So we only need addition and complement
circuits

12. Hardware for Addition and
Subtraction