Subtraction in those early days

What do you do when someone asks you to subtract 723 from 1081?? Since 3>1, for the unit's digit subtraction, you take (1+10-3)=8, and turn the ten's digit from 8 to 7(unless u know some even easier technique of subtraction!). This two operation are known as Difference and Borrow. But at the ancient times, Aryan Pundits had no such concepts of difference and borrow, yet they used to subtract with acute perfection. Well, as I told earlier, all their computations were based on the two earlier definitions, and subtraction was not an exception either!

Let us recall the Sutra I stated a little ago, that is:

"निखिलं नवतश्चरमम दशतः"

Which means, "All from nine and last from ten". So, we are going to use our pet funda of complements.
I am presenting the process point wise,

1) We take our previous example. (1081-723). In this case, we refer the digits of 1081 to be Upper digits, and the digits of 723 to Lower digits, just for easier representation.

2) In each column of subtraction(by column I refer to unit's digit,ten's digit etc.), we take the difference of upper digit and lower digit, i.e. (upper digit ~ lower digit).

3) If in some column, upper digit > lower digit, then the answer becomes (upper digit - lower digit - 1). This extra 1 is subtracted since we are coming out of complements.

4) If in some column, upper digit < lower digit, the answer becomes the complement of (lower digit - upper digit).


Let's now take our example. I am referring unit's digit as column 1, ten's digit as column 2,...etc. So, in column 1, upper digit < lower digit, which means we are to take complement. Answer for column 1 is [10-(3-1)]=8.

For the next column, upper digit > lower digit, so we come out of complements, and the answer is (8-2-1)=5.

For 3rd column, upper digit < lower digit, so we need complement. Thus, the answer is [10-(7-0)]=3.

Finally for the last column, upper digit > lower digit, so the answer becomes, (1-0-1)=0.

So, our final answer is, 0358, i.e. 358. Clearly (358+723)=1081. So, our concept of complement proved to be useful once again.

Vedic Mathematicians intended to solve a lot of problems based on this complement concepts...one cannot imagine how powerful this technique can actually be for fast calculations!! But you will start believing in my words as soon as I illustrate the multiplication techniques used by them...