Chapter 1 - Abridged Cosmic Computer

"Mathematics is the Queen of the Sciences and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all circumstances the first place is her due."

- Carl F. Gauss (1777-1855)
GERMAN MATHEMATICIAN

DIGIT SUMS AND THE NINE-POINT CIRCLE

The three main branches of Mathematics are Arithmetic, Algebra and Geometry.
Arithmetic deals with number and begins with the number 1 which represents unity. Unity and wholeness are everywhere.

The system of Vedic Mathematics is based on 16 formulas (or Sutras). For example By One More Than the One Before (Ekadhikena Purvena in the Sanskrit language).
Click here to display the list of Sutras.

Starting from number 1 all whole numbers can be generated by using By One More Than the One Before: 2 is one more than 1; 3 is one more than 2 and so on. We will see many other applications of this simple formula as we go on.

1 2 3 4 5 6 7 8 9 10 . . .

Whole numbers are also called natural numbers.
The word digit means the single figure numbers- the numbers 1 to 9 and 0.
Sum means addition.

The digit sum of a number is found by adding the figures in the number.

Example 1
Find the digit sum of a 17 b 401

a For 17 we get 1 + 7 = 8. b For 401 we get 4 + 0 + 1 = 5.

The digit sum is found by adding the digits in a number, and adding again if necessary.

Example 2
Find the digit sum for 761.

For 761 we get 7 + 6 + 1 = 14. And as 14 is a 2-figure number we add the figures in 14 to get
1 + 4 = 5.

So the digit sum of 761 is 5.

This means that any natural number of any size can be reduced to a single digit: just add all the digits, and if you get a 2-figure number, add again.

The number nine is very important in the Vedic system. As we will see it has many remarkable properties which make it very interesting and very useful.

THE NINE POINT CIRCLE

The nine digits 1,2,3,4,5,6,7,8,9 used in our number system can be usefully displayed around a circle. As shown below.

This is known as the 9 point circle. Now suppose we continue numbering round the circle.

It seems reasonable to put 10 after 9, where we have already got number 1. So we can continue numbering around the circle as shown below.

Continue numbering around the circle up to 30 in the boxes provided, numbers 1 to 16 are already shown.

1. Write down (in order) all the numbers you get on the `branch` that begins with 1,
the first one has been done for you.

2. What do you notice about their digit sums?

3. Predict the next 3 numbers on this branch.

4. What are their digit sums?

5. What can you say about the digit sums of all the numbers on this branch?

6. Do the same thing for two other branches.

   Enter the first digit of the first branch you chose
   and then fill in the other numbers on that branch below


   Enter the first digit of the second branch you chose
   and then fill in the other numbers on that branch below


   What do you notice?

7. Starting at 3 what do you have to add on to get back to the 3 branch?

8. What would you have to add on to any one number to get back to the branch you started at?

Adding 9 to a number does not affect its digit sum:
so 7, 70, 79, 97, 979 all have a digit sum of 7 for example.

Example 3
Find the digit sum of 39409

We can cast out the nines and just add up the 3 and 4. So the digit sum is 7.

Or, using the longer method we add all the digits: 3+9+4+0+9 = 25 = 7 again.

Looking again at the 9-point circle, if we count backwards round the circle we see that since 0 comes before 1 it is logical to put zero at the same place as 9. In terms of digit sums 9 and 0 are equivalent and this also explains why we can cast out nines.

As digit sums, 9 and 0 are equivalent.

This casting out the nines can be used in another way:

Any group of figures in a number that add up to 9 can also be "cast out".

Example 4
Find the digit sum of a 24701 b 21635

a In 24701 we see 2 and 7 which add up to 9.
We can therefore cast them out and add up only the other figures. 4+0+1 = 5.

b In 21635 we can see that 1, 3 and 5 add up to 9 so we cast these out.
The sum of the remaining figures is 8 so this is the answer.

DIGIT SUM PROBLEMS