Ekadhikina Purvena: By one more than the previous one

Ekadhikina Purvena is the first sutra of Vedic mathematics. Its corollary is Anurupyena. The division method contained in Ekadhikina Purvena shares some similarities with the twelfth sutra, Shesanyankena Charamena.

What follows is a brief and incomplete summary of the math shortcuts this sutra contains:

Shortcut to square numbers that end in "5", such as 35, 95, 185, etc.

This is one of the easiest and widely-known shortcuts.

As an example, if you want to square the number 95 in your head you would start by subtracting five from 95, yeilding 90. You would also add five to 95, yeilding 100. You then multiply those two numbers to quickly, easily yield 9000. You now square the number you added and subtracted (five). The square of 5 is 25. You add 25 to 9000. The square of 95 is 9025.

You can apply this method to any numbers ending in 5. To square 35, for example, you can quickly multiply 3 and 4 in your head (12), add the two zeros (1200), then add 25 (1225). The square of 35 is 1225. This method is sometimes faster than pressing buttons on a calculator.

This method can be applied to other numbers that do not end in 5 using the method explained by the corollary of this sutra, Anurupyena.

Shortcut to calculate, to several decimal places, vulgar fractions that end in 1, 3, 7, or 9, such as 1/19, 1/17, 1/23, 1/69, 1/87, etc.

There are two methods in approaching this shortcut: the Division Method and the Multiplication Method. I recommend substituting a calculator for this "shortcut".

Multiplication method using 1/19 as an example
To start subtract the numerator from the denominator: 19 - 1 = 18
This means the answer will go to 18 decimal places before repeating.
The first digit of denominator (1) will be the 18th decimal place. (Ekadhikina Purvena calculates the answer backwards) Add 1 to this to get the 17th decimal: 2
This number (first digit of denominator plus 1) will be used in every step henceforth and will be called the multiplier.
So far we have the last two of the 18 decimal places of the answer: 0.xxxxxxxxxxxxxxxx21
To find the 16th decimal place, multiply 2 by itself. 2 x 2 = 4
The 16th decimal place is 4
To find the 15th decimal place, multiply 2 by 4. The 15th decimal place is 8.
To find the 14th decimal place, multiply 2 by 8. This yields 16. The 14th decimal place is 6, and the 1 will be carried to the next step.
To find the 13th decimal place, multiply 2 by 6 and add 1. This equals 13, which means the 13th decimal place is 3 and the 1 is carried.
The answer we have so far is 0.xxxxxxxxxxxx368421
12th decimal: 2 x 3 + 1 = 7
11th decimal: 2 x 7 = 4 (carry the 1)
10th decimal: 2 x 4 + 1 = 9
9th decimal: 2 x 9 = 8 (carry the 1)
8th decimal: 2 x 8 + 1 = 7 (carry the 1)
7th decimal: 2 x 7 + 1 = 5 (carry the 1)
6th decimal: 2 x 5 + 1 = 1 (carry the 1)
5th decimal: 2 x 1 + 1 = 3
4th decimal: 2 x 3 = 6
3rd decimal: 2 x 6 = 2 (carry the 1)
2nd decimal: 2 x 2 + 1 = 5
1st decimal: This has to be a zero because the numerator (1) is smaller than our multiplier (2).
Thus 1/19 = 0.052631578947368421

I will not explain how the division method works, because long division tends to take less time. If you want to learn the method anyway, check out the resources listed below:

RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm