# MSC #2 – All from 9 and the Last from 10.

The second Vedic Math sutra or word formula is “All from 9 and the Last from 10” or Nikhilam Navatashcaraman Dasatah”. It is used for rapidly determining the “ten’s complement” of a number or how far it is from the next higher power of ten. It is also the title of the 2^{nd} chapter of our book “30 Master Strategies in Computing”

Using this Sutra, we can immediately say that the ten’s complement of 72 is 28 because 7 from 9 is 2 and 2 from 10 is 8.

We can also quickly announce the answer to 1,000 – 574 as “four hundred” (by mentally subtracting 5 from 9) twenty (7 from 9) six (4 from 10).

Like the First Sutra, “By One More Than the One Before”, it has a wide range of applications: subtraction, Nikhilam or base Multiplication, base division and even in Algebraic operations.

Our example for today is one of its application in subtraction. I first discussed this technique, which I called “subtraction by steps” in my book “25 Math Short Cuts”. It is 1 of 3 ways to avoid borrowing.

Using the traditional right to left method taught in schools, the subtraction 12, 124 – 8798 would require 4 regroupings or borrowings, which makes it very difficult to compute mentally.

However, between the minuend and the subtrahend is a very convenient number to use as a base, 10,000. Just by looking at the number 12, 124, we know that it is 2,124 more than 10,000. Now with Nikhilam, we can readily obtain how far 8,798 is from 10,000.

Hence we can mentally compute the difference and announce the answer as:

- 8 from 9 is 1 and 1 added to 2 is “three thousand”,
- 7 from 9 is 2 and 2 added to 1 is “three hundred”,
- 9 from 9 is 0 and 0 added to 2 is “twenty”,
- 8 from 10 is 2 and 2 added to 4 is “six”.

“All from 9 and the Last from 10” will be discussed by IAVM trustee Gowri Ramachandran in the opening session of Inspirational Maths from India (IMI 4) webinars of Dec 4, 2021.

Nikhilam or base mulitiplication will be explained by Ms. Adora Leonano, coach of Jeremiah Emmanuel Lubiano of Calamba Elementary School, First Place winner in the Primary Category of the 1^{st} International Vedic Math Olympiad in the afternoon session of IMI 4 on Dec 4, 2021.