Question

1. What are w-bit words in computer architecture ?
2. For two 7 bit words

1011001 = A 
1101011 = B , how does multiplication returns

10010100110011 ?

Isn’t there simple binary multiplication involved in these ?
Please provide an example.

Answer 1

Both adding and multiplying are done just the same as in decimal (base 10). You just need to remember this truth table:

Multiplying
-----------
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1

Adding
-----------
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (w/ carry)

First adding. To add, you add just like you would in normal arithmetic, except follow the truth table above:

  00000101 = 5
+ 00000011 = 3
--------------
  00001000 = 8

How this works is that you start from the right and work left. 1 + 1 = 0, but you carry a 1 over to the next column. So the next column is 0 + 1, which would be 1, but since you carried another 1 from the previous column, its really 1 + 1, which is 0. You carry a 1 over the next column, which is 1 + 0, but really 1 + 1 because of the carry. So 0 again and finally move the 1 to the next column, which is 0 + 0, but because of our carry, becomes 1 + 0, which is 1. So our answer is 1000, which is 8 in decimal. 5 + 3 = 8, so we know we are right.

Next, multiplying:

  00000101 = 5
x 00000011 = 3
----------
       101 = 5
+     1010 = 10
----------
      1111 = 15

How this works is you multiply the top number 00000101 by the right most digit in the second row. So 00000011 is our second row and 1 is the right most digit, so 00000101 times 1 = 101. Next you put a 0 placeholder in the right most column below it, just like in normal multiplication. Then you multiply our top original number 00000101 by the next digit going left in our original problem 00000011. Again it produce 101. Next you simply add 101 + 1010 = 1111 …That is the answer