Yaavadunam

Yaavadunam: Whatever the extent of its deficiency

Yaavadunam is the tenth sutra of Vedic mathematics. Its corollary is Samuccayagunitah. It shares similarities with the sutra Ekadhikina Purvena and with the upsutras Yavadunam Tavadunikritya Varga Yojayet and Anurupyena.

The sutra is used to cube numbers:

Cubing numbers that are near a power of ten Such as 13, 104, 1012, etc.
Example: 12³
Step One: Subtract the nearest power of ten from the number: 12 - 10 = 2
Step Two: Double this number and add the number being cubed: (2 x 2) + 12 = 16
Step Three: Subtract from this number the power of ten from step one: 16 - 10 = 6
Step Four: Multiply this number by the answer in step one: 2 x 6 = 12
Step Five: Cube the answer in step one: 2³ = 8
Step Six: Since we're cubing a 2 digit number, add two zeros to the answer in step two: 1600
Step Seven: Add 1 zero to the answer in step four: 120
Step Eight: Add the answers in steps seven and eight to the answer in step five: 1600 + 120 + 8 = 1728
Thus, 12³ = 1728

Cubing numbers that are near a multiple of a power of ten Such as 42, 507, 2008, etc.
Example: 48³
Step One: Keep in mind that the main base is 10 and our working base will be 50
Step Two: Divide the main base by the working base: 10 / 50 = 1/5
Step Three: Subtract the working base from the number being squared: 48 - 50 = -2
Step Four: Double this number and add the number being squared: (-2 x 2) + 48 = 44
Step Five: Divide this number by the square of the ratio found in step two: 44 / (1/5)² = 44 / (1/25) = 44 x 25 = 1100
Step Six: Subtract the working base from the answer in step four: 44 - 50 = -6
Step Seven: Multiply the answer in step three by the answer in step six: -2 x -6 = 12
Step Eight: Divide this number by the ratio in step two: 12 / (1/5) = 12 x 5 = 60
Step Nine: Cube the answer found in step three: -2³ = -8
Step Ten: Since we're cubing a 2 digit number, add two zeros to the answer in step five: 110000
Step Eleven: Add 1 zero to the answer in step eight: 600
Step Twelve: Add the answers in steps ten and eleven to the answer in step nine: 110000 + 600 + -8 = 110592
Thus, 48³ = 110592

To see more examples of Yaavadunum, 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