Shesanyankena Charamena: The remainders by the last digit

Shesanyankena Charamena is the twelfth sutra of Vedic mathematics. Its corollary is Vilokanam. It shares some similarities with the first sutra, Ekadhikina Purvena.

This sutra can be used to express a fraction as a decimal, to all it's decimal places.

Example: Express 1/7 as a decimal

Part One:
Add a zero to the numerator: 10
If this number is less than the denominator, add another zero. If it is greater than the denomonator, proceed to the next step.
Divide this number by the denomonator and pay attention to the remainder: 10 / 7 = 1 with a remainder of 3
Add a zero to this remainder and divide it by the denomonator: 30 / 7 = 4 with a remainder of 2
Add a zero to this remainder and divide it by the denomonator: 20 / 7 = 2 with a remainder of 6
Add a zero to this remainder and divide it by the denomonator: 60 / 7 = 8 with a remainder of 4
Add a zero to this remainder and divide it by the denomonator: 40 / 7 = 5 with a remainder of 5
Add a zero to this remainder and divide it by the denomonator: 50 / 7 = 7 with a remainder of 1
At this point, notice that this remainder (1) is the same number as the numerator. The answer is going to repeat from here on out, so we'll stop.
These remainders, written in order, will be used for step two: 3, 2, 6, 4, 5, 1


Part Two:
Multiply the denomonator by the first number in the sequence above: 7 x 3 = 21
Write down the leftmost digit of this product (1) and drop all the other digits. Unlike Ekadhikina Purvena, this sutra does not require any numbers be carried.
Multiply the denomonator by the second number in the sequence above: 7 x 2 = 14
Write down the leftmost digit of this product (4) and drop all the other digits. Multiply the denomonator by the third number in the sequence above: 7 x 6 = 42
Write down the leftmost digit of this product (2) and drop all the other digits. Multiply the denomonator by the fourth number in the sequence above: 7 x 4 = 28
Write down the leftmost digit of this product (8) and drop all the other digits. Multiply the denomonator by the fifth number in the sequence above: 7 x 5 = 35
Write down the leftmost digit of this product (5) and drop all the other digits. Multiply the denomonator by the sixth and last number in the sequence above: 7 x 1 = 7
Write down this product (7)
Write down these products in sequence: 1 4 2 8 5 7

Thus, 1/7 = 0.142857142857repeating


To learn more about this sutra, check out the resources 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
http://hinduism.about.com/library/weekly/aa062901a.htm
http://www.vedicmaths.org/
http://www.hinduism.co.za/vedic.htm
Mathemagics by Arthur Benjamin and Michael B. Shermer
http://en.wikipedia.org/wiki/Vedic_math
http://www.tifr.res.in/~vahia/dani-vmsm.pdf
http://www.sacw.net/DC/CommunalismCollection/ArticlesArchive/NoVedic.html


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