# Using the Ten’s Complement in Subtraction

In our 30 MATH-Inic Specials for Christmas series, which is sponsored by Palawan Pawnshop Express Pera Padala, we will publish one Vedic Math technique daily for 30 days. Each of this post, which shows a different approach to common problems, is taken from our book “30 Master Strategies of Computing”. which, we finally decided, to release in e-book format, one master strategy at a time.

An abridged e-book version of the first 10 “master strategies” will be given free to those who will purchase a set of books – 25 Math Short Cuts, Algebra Made Easy as Arithmetic and Inspirational Maths from India. See details at https://www.facebook.com/MATHInicPhils.

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, base Multiplication, base division, bar numbers and even in Algebraic operations.

Our example for today is in using the ten’s complement in the subtraction, 9234 – 7869.

We will need three regroupings – or borrowings – if we will use the conventional right to left subtraction method taught in schools. Let us consider this alternative method to avoid numerous regroupings:

- Using all from 9 and the Last from 10, we can easily determine the ten’s complement of 7,869: a) 7 from 9 is 2,

b) 8 from 9 is 1,

c) 6 from 9 is 3, and

d) 9 from 10 is 1.

2) We can then replace 7,869 by (10,000 – 2,131).

3) Since 9,234 – (10,000 – 2131) = (9,234 + 2,131) – 10,000

Now here are the steps to solve 9,234 – 7,869 mentally:

- Add 2 (the nine’s complement) of 7 to 9 to get 11 thousand
- 11 thousand minus 10 thousand is 1 thousand
- Add 1 (the nine’s complement of 8) to 2 to get 3 hundred
- Add 3 (the nine’s complement of 6) to 3 to get sixty, and
- Add 1 (the ten’s complement of 9) to 4 to get five.

More examples for this kind of subtraction can also be found in “Inspirational Maths from India – A Teacher’s Handbook”, pp. 14-15.

Watch out for the release of **Master Strategy in Computing #2 – “All from 9 and the Last from 10”** with problems and solutions from national and international Vedic Math competitions!