CS/CoE 0447 Subtraction, Multiplication, and Division Examples

Transcription

1 CS/CoE 0447 Subtraction, Multiplication, and Division Examples 1. Perform three subtractions on 9-bit 2's complement binary numbers as follows: a) b) c) For each of these subtractions, convert the numbers into 9-bit 2's complement, and for the numbers to subtract, or the negative numbers, convert them to their negative counterpart. Add those numbers and convert the resultant binary 2's complement number into decimal form. Show all your work. a) Conversion to 9-bit 2 s Complement Notation from Decimal Notation Step 1a. Convert positive part to binary. Step 1b. Include preceding 0. Step 1c. Preserve negative sign, if present. Step 2. If negative sign is present, flip bits and add 1. Not Applicable Step 3. Extend sign bit as needed Decimal Notation Two s Complement Notation 206(decimal) (decimal) (decimal)

2 b) Conversion to 9-bit 2 s Complement Notation from Decimal Notation Step 1a. Convert positive part to binary. Step 1b. Include preceding 0. Step 1c. Preserve negative sign, if present. Step 2. If negative sign is present, flip bits and add 1. Not Applicable Step 3. Extend sign bit as needed Decimal Notation Two s Complement Notation 68(decimal) (decimal) (decimal) c) Conversion to 9-bit 2 s Complement Notation from Decimal Notation Step 1a. Convert positive part to binary. Step 1b. Include preceding 0. Step 1c. Preserve negative sign, if present. Step 2. If negative sign is present, flip bits and add Step 3. Extend sign bit as needed Decimal Notation Two s Complement Notation -51(decimal) (decimal) (decimal)

3 2. Show the steps for the multiplication of b and b (unsigned) using Hardware Design 3 ( Here b is the multiplicand and b is the multiplier. Fill up the columns: Itera Multiplicand Implementation 3 -tion Step Product6-bit) b initial values b 1: 0 -> no op b 1: 0 -> no op b 1: 1 -> product = product + multiplicand b 1: 0 -> no op b 1: 1 -> product = product + multiplicand b 1: 1 -> product = product + multiplicand b 1: 0 -> no op b 1: 1 -> product = product + multiplicand *Shift in carry out bit of Step 1a in the flowchart on page 3 (slide 42) of ( If Step 1a is bypassed, shift in 0.

4 3. Convert the following 8-bit binary numbers into Booth's encoding form: , , Original: Write Implied 0: Encode each pair: Booth s encoded: Convert the following decimal numbers into 9-bit binary numbers in Booth's encoding form: 21, -217, -45 Original (dec.): Orig.(9-bit 2 s): Write Implied 0: Encode each pair: Booth s encoded: Check by multiplying each Booth s digit by the weight of its place and then adding each product. 256 = 0)+ 128 = 0)+ 64 = 0)+ 32 = 32)+ 16 = -16)+ 8 = 8)+ 4 = -4)+ 2 = 2)+ 1 = -1)= =-256)+ 128 = 0)+ 64 = 64)+ 32 = -32)+ 16 = 0)+ 8 = 8)+ 4 = 0)+ 2 = 0)+ 1 = -1)= = 0)+ 128 = 0)+ 64 = -64)+ 32 = 32)+ 16 = -16)+ 8 = 0)+ 4 = 4)+ 2 = 0)+ 1 = -1)= -45

5 5. Show the steps for the multiplication of b and b (signed) using Booth's algorithm ( Here b (-74decimal) is the multiplicand and b (-76decimal) is the multiplier. Fill up the columns: Iteration Multiplicand b (-74decimal) Booth's Algorithm Step Product7-bit) initial values b 1: 00 -> no op b 1: 00 -> no op b b b 1: 01 -> product = product + multiplicand b 1: 11 -> no op b b 1: 01 -> product = product + multiplicand (74 dec.) (-74 dec.) (74 dec.) (-74 dec.) (74 dec.) (-74decimal) times (-76decimal) = (5624 decimal = binary)

6 6. Show the steps for the multiplication of b and b (signed) using Booth's algorithm (available here: ). Here b is the multiplicand and b is the multiplier. Fill up the columns: Iteration Multiplicand b (55dec) b b b Booth's Algorithm Step Product7-bit) initial values : 01 -> product = product + multiplicand (-55 dec.) (55 dec.) b 1: 00 -> no op b 1: 00 -> no op b (-55 dec.) b (55 dec.) 1: 01 -> product = product + multiplicand b 1: 00 -> no op b (-55 dec.) (55decimal) times 11decimal) = decimal =

7 7 In a table similar to the following one, show the steps for computing b (the dividend) divided by 0101b (the divisor, both numbers are unsigned) using non-restoring division. Nonrestoring division is described online at: An example of doing non-restoring division is also shown. Iteration Divisor (4 bits) (5 dec) Step (description) Remainder Register (8 bits) Dividend Register (8 bits) initial values shift left by 1 remainder = remainder divisor 4 5 (remainder 0) shift left; r0=1 6 remainder = remainder divisor 7 8 (remainder 0) shift left; r0=1 Done shift left half of 16-bit remainder dividend register right by 1 +(-5dec) (-5dec) (-5dec) (45 dec) / 5 = 9 (decimal) = 1001 (binary) 0