Puranapuranabyham:

*By the completion or non-completion*
Puranapuranabyham is the eighth

sutra of

Vedic mathematics. Its corollary is

Antyayor Dasakepi.

Puranapuranabyham is used to simplify or solve

algebra problems.

**Example:** Solve for x in the equation x

^{3} + 6x

^{2} + 11x + 6 = 0

Start by estimating simple

quadratic equations that may be similar to the left hand side of this equation. Since this equation includes a

cubed variable, incorporate cubing into the estimated quadratic equations. Look at the numbers that aren't variables in the problem: 1, 6, 11, and 6. So we know that if we cube a large number, our estimate will be way off. We also know that the cubed variable in the equation is multiplied by 1, and only 1. So we know the simple quadratic will be in the form of (x + __)

^{3}
*estimate #1: (x + 1)*^{3} We don't even have to work out the answer to know this is going to be too low. 1 cubed is still just 1.

*estimate #2: (x + 2)*^{3} This works out to x

^{3} + 6x

^{2} + 12x + 8

We'll stop there, because estimate #2 is very close to the equation we're solving.

Subtract from our estimated quadratic equation the left hand side of the problem, (x

^{3} + 6x

^{2} + 12x + 8) - (x

^{3} + 6x

^{2} + 11x + 6) = x + 2

So, add (x + 2) to both sides of the problem, which leaves us with

**x**^{3} + 6x^{2} + 12x + 8 = x + 2
We can further simplify this as

**(x + 2)**^{3} = (x + 2)

Now we have a common term,

*(x + 2),* on both sides of the equation. Set up another variable, y, to equal (x + 2)

y

^{3} = y

We can then infer that y must equal 0 or 1 or -1

If x + 2 = 0 then x = -2

If x + 2 = 1 then x = -1

If x + 2 = -1 then x = -3

Thus,

**x = -1,-2,-3**
This sutra can also be used for simplifying or solving

quadratic equations and

biquadratic equations. To learn more about Puranapuranabyham, 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