Shunyam Saamyasamuccaye

When the samuccaya is the same that samuccaya is zero

Shunyam Saamyasamuccaye is the fifth sutra of Vedic mathematics. Its corollary is Vestanam.

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

The Many Definitions of "Samuccaya":

Any Common Term: The most simple (but incomplete) definition of samaccaya is to call it any term that is common to all the parts of a problem, such as Z in the equation 1z + 3z = 5z + 6z. Z would be the samaccaya, and in using this sutra we would assume z = 0
Another example would be the equation 3(z + 1) = 5(z + 1) The common term here is (z + 1), so in using this sutra we would assume z + 1 = 0, meaning z = -1

The Product of the Independent Terms:
We'll use this equation as our example: (1 + z) (2 + z) = (3 + z) (4 + z)
Using this sutra, we would simplify the problem as 1 x 2 = 3 x 4 then as 2 = 12 (which, obviously, is not true), so we would then assume z must equal zero for this equation to make sense.

The sum of the two Denominators with the same Numerators:
Example: 1/(3z - 1) + 1/(z - 1) = 0
Because the numerator is 1 in both fractions, this definition of summaccaya tells us to combine the denominators and rewrite the equation as 4z - 2 = 0
The next step is to solve it like any algebra equation:
4z = 2
z = 1/2

"When the samuccaya is the same that samuccaya is zero" can also be applied to some other algebra problems and some problems involving cubes. To learn more about Shunyam Saamyasamuccaye, check out the resources below:

Vedic Mathematics by Sri Bharati Krisna Tirthaji
Mathemagics by Arthur Benjamin and Michael B. Shermer