A few days ago, while returning from a friend's house, standing half asleep in the bus, a thought came to me, “Why is the multi digit multiplication method that I use like so?” (I know, there exists many methods for multiplication, I am talking about the most commonly used one).

I started to analyse the steps – what am I actually doing with these multiplications, shifting and addition?. As I figured it out I realised that neither anyone told me nor I thought about 'WHY' - Everyone was interested in 'HOW'. Then, I wanted to know whether it was my mistake not to figure it out for this long. So, I asked some of my friends and none of them knew what they were doing. So, I decided to share it.

Let us analyse the steps by multiplying two, 3 digit numbers, say 322 and 321

Step 1: multiply 322 by 1 and write down the product

Step 2 : multiply 322 by 2 and write down the product by left shifting one place.

Step 3 : multiply 322 by 3 and write down the product by left shifting two places.

Step 4 : add them together to get the final product.


In the first step 322 is multiplied by 1. In second step 322 is multiplied by 2 and left shifted once. Actually in the second step we are multiplying by 20 and since 0 has no value during addition it is ignored and replaced with a left shift. Similarly in the third step 322 is multiplied by 300. In short, it is same as doing 
(322 x 1) + (322 x 20) + (322 x 300)
= 322 (1 + 20 + 300) .


Have you realised it before? Or is it now you are thinking about it? Please share your answers through comments.

If you haven't noticed it before, this is what we do while multiplying a multi digit number by a single digit number.