Basic arithmetic operations to be studied in COA

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