Urdhva-Tiryagbyham:

*Vertically and crosswise*
Urdhva-Tiryagbyham is the third

sutra of

Vedic mathematics. Its corollary is

Adyamadyenantyamantyena.

What follows is a

*brief* and

*incomplete* summary of the math shortcuts this sutra contains:

**Multiplying two numbers that are both two digits long**
Example: multiply 12 by 34

The answer will consist of three digits: xyz

To find x, multiply the first digit in each number: 1 x 3 = 3. So the answer is three-hundred and something. This works great if you just need a quick estimation, like figuring out if you can buy a dozen items that cost 34 cents each for less than $5.

To find y, cross multiply. Meaning you multiply the first digit of the first number by the second digit of the second number (1 x 4), and add this

product (1 x 4 =

**4****6**). 4 + 6 = 10. This means that y = 0 and you have to add the 1 to x.

To find z, multiply the second digit of each number. 2 x 4 =

**8**
Thus, 12 x 34 = 408

**Bonus shortcut: Multiplying any two-digit number by 11 in an instant.**
This sutra can be used to quickly multiply, "vertically and crosswise", any two digit number by 11. Here are two examples:

36 x 11. Place a space between the two digits of the number being multiplied by eleven:

**3_6**. Now add the two digits: 3 + 6 = 9. Place the 9 in the space and you have the answer. 36 x 11 =

**396**
69 x 11. Again, place the space: 6_9. Now add the two digits: 6 + 9 = 15. The number in the center space will be 5, and you must add the 1 to the preceeding digit. Thus, 69 x 11 =

**759**
**Multiplying two numbers that are both three digits long**
This is worked in the opposite order as the two-digit method above, and may take longer than multiplying by hand.

Example: 123 x 456

Multiply the last digit of both numbers: 3 x 6 = 18. The last digit of the answer will be

**8**. We will carry the 1 in the next step.

Multiply crosswise the last two digits of both numbers and add the 1 carried from the previous step: (6 x 2) + (3 x 5) + 1. So 12 + 15 + 1 = 28. The second to last digit will be

**8** and we will carry the 2.

Multiply the middle digit in both numbers: 2 x 5 = 10. Multiply cross wise the first and last digits: (1 x 6 = 6) and (3 x 4 = 12). Add these three products: 10 + 6 + 12 = 28. Add the 2 carried from the previous step. This yields 30. The third to the last digit will be

**0** and we will carry the 3.

Multiply crosswise the first two digits of both numbers: (1 x 5) + (2 x 4) = 13. Carry the 3: 13 + 3 = 16. The fourth to the last digit will be

**6** and we will carry the 1.

*Last step:* Multiply the first digit of both numbers: 1 x 4 = 4. Add the carried 1, making the first digit in the answer

**5**
Thus, 123 x 456 =

**56088**
Urdhva-Tiryagbyham in its

converse can be used to simplify division problems involving

algebraic functions. If you want to learn this method, 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

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