Question

Perform the following binary multiplication. Assume that all values are 2's complement numbers. Indicate the result and whether there is overflow or not. 1011* 1101

Answer 1

Before we are performing this multiplication we should understand the concept of binary multiplication and addition.

Binary multiplication

0 * 0 = 0

0 * 1 = 0

1 * 1 = 1

1 * 0= 0

Binary Addition

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10 (0 is written as the sum and 1 is used as the carry)

1+1+1= 11( sum is 1 and carry is 1)

1+1+1+1 = 100 ( sum is 0 and carry is 10)

Multiplication is shown below table

						1	0	1	1
		1(carry)			X	1	1	0	1
1		1	1	1	1	1	0	1	1
2		0	0	0	0	0	0	0	x
3		1	1	1	0	1	1	x	x
4		0	0	1	0	1	x	x	x
5		1(carry)	0(carry)	1(carry)	1(carry)
Result =1	0	​0	0	0	0	1	1	1	1
		x7	x6	x5	x4	x3	x2	x1	x0

For better understanding I have use a table for showing this multiplication.

Note the red marking 1 to 5, it is the result of second numbers each bit multiplied with the first number.

1)1011x1= 1011 it is clearly written as the first four LSB bits. The last four one's 1111 is used to show this number in 8 bit form and it is extended by the current MSB bit of 1011, that is 1. So it looks like 11111011

2)1011x0 =0000 the first four LSB bit shows the product it extend with current MSB value of 0000, that is 0.

3)1011x1=1011, four LSB is the product and other 1's are extended by the MSB bit.

4) There is an important thing to remember.

When we multiplied by the number by a number with its MSB value is 1, (that is it is the sign bit is 1), then we have to take the 2's compliment of that product.

Here 1011x1=1011 take its 2's compliment

0100

+1

ie, 0101 is the 2's compliment representation. Then write this number with 0 as extention there. Note the line noted with 4 in red colour.

5)After these, we have to do normal multiplication processe like addition.

Look at the addition process,

1)add the numbers in x0 position,we will get 1.

2)add numbers in x1 position, we will get a 1 again

3)add numbers in the x2 position, its again a 1

4)add numbers in x3 position, 1+1+1=11, write 1 as result and write the other 1 as carry.

5)add numbers in x4 position, 1+carry 1=10. Write 0 I'm the result and 1 as carry.

6)add numbers in x5 position, 1+1+1+ carry 1=100 write 0 in result, and then write carry 0 in x6 position and 1 in x7 postion.

7)add numbers in x6 position, then 1+1=10 write 0 in result and add 1 as carr to x7 bit.

8)add numbers in x7 position, 1+1carry+1+1 carry= 100. Write 0 in result. 10 in carry.

Note the works shown below.

loll loll 1011x1 1011 oooox We have to take 1 от 1 XX 2's compliment of olo IX X X this number So, Oloot Olol Next step is to extend the each number's into its MSB position bit. ie 10ll llol lou ooooooox 11 10 11 x x Oolorxxx

Da ② Next step is addition 100 o 110 ||900 ooooooox 111011xx roo 101XXX o 임 Idioooo!!!! Ally no So the result is = 00001111 00001111

Check all the notes that I have prepared here, if you cannot understand the concept feel free to comment. Otherwise don't forget to upvote.