Multiplication

Multiplication in decimal requires the memorization of a rather big table. Binary multiplication, on the other hand, only has four rules:

• $0_2 \times 0_2 = 0_2$
• $0_2 \times 1_2 = 0_2$
• $1_2 \times 0_2 = 0_2$
• $1_2 \times 1_2 = 1_2$

Now that we have discussed single digit binary multiplication, let talk about multiple digit binary multiplication. Before we discuss binary multiplication, let's talk about decimal multiplication, first.

Let us consider the following:

$$215 \times 71 = 200 \times 71 + 10 \times 71 + 5 \times 71$$ (8.16)

$$= 2 \times 71 \times 100 + 1 \times 71 \times 10 + 5 \times 71 \times 1$$ (8.17)

$$= 2 \times 71 \times 10^2 + 1 \times 71 \times 10^1 + 5 \times 71 \times 10^0$$ (8.18)

Of course, there is nothing really surprising in the previous derivation. It is, however, important to understand it anyway. Next, we present an example in binary multiplication.

$$1011_2 \times 0101_2 = 1000_2 \times 0101_2 + 000_2 \times 0101_2 + 10_2 \times 0101_2 + 1_2 \times 0101_2$$ (8.19)

$$= 0101000_2 + 0 + 01010_2 + 0101_2$$ (8.20)

The reason why $1000_2 \times 0101_2 = 0101000_2$ is exactly why $100 \times 13 = 1300$ in base 10.

When written in vertical form, the same multiplication can be done as follows:

num1						1	0	1	1
num2	$\times$					0	1	0	1
leftmost				1	0	1	0	0	0
second rightmost	+					1	0	1	0
partial sum				1	1	0	0	1	0
rightmost	+						1	0	1
product				1	1	0	1	1	1