Urdhva-tiryagbhyam happens to be a very useful sutra; it translates roughly to “Vertically and Crosswise.” This sutra happens to be so useful, that it actually has a whole book dedicated to it (click here for preview). Vertically and Crosswise is the general multiplication formula, and also sees its use in division (among many other topics). In this post I’ll be working from left to right, however you can choose to work right to left if you wish (just a matter of personal preference).
Multiplication of two 2-Digit Numbers
For our example, we’ll multiply 13 x 12.
1) The first step should be to write out the two numbers in the following way (or visualize it in your head):
1 3
1 2
———
2) Next multiply the left-hand column (1 x 1), hence “vertically.” Write down the number “1.”
1 3
1 2
———
1
3) Now multiply diagonally and add (hence “crosswise”). In this case you will multiply the ’1′ from 13 with the ’2′ from 12…(1 x 2 = 2). Then you will multiply the ’1′ in the 12, with the ’3′ in the 13…(1 x 3 = 3). Add these two numbers together (2 + 3 = 5), and write it down.
1 3
1 2
———
1 5
4) Following that, you multiply the right-hand column vertically. 3 x 2 = 6, write this down.
1 3
1 2
———
1 5 6
Answer: 13 x 12 = 156
Example #2: 33 x 21
Following the steps above…
2) 3 x 2 = 6
3) (3 x 1) + (2 x 3) = 9
4) 3 x 1 = 3
3 3
2 1
———
6 9 3
Answer: 33 x 21 = 693
What about when one of the digits ends up having a result with more than one digit?
For example 27 x 21, when you find the middle digit you end up with (2 x 7) + (2 x 1) = 16…but the spot can only hold one digit. So what do you do? Carry it over to the column to the left, other than that the process is exactly the same.
Example #3: 27 x 21
1)
2 7
2 1
———
2) 2 x 2 = 4
2 7
2 1
———
4
3) This part is a little different, you still do the crosswise method though. (2 x 1) + (2 x 7) = 16. Since this number contains two digits, but the spot only allows one, we write down the rightmost digit. In this case, we would write down ’6′, as the middle digit, and carry the left digit (’1′ in this case) to the left. The first digit (in our answer) is now 4 + 1 = 5.
2 7
2 1
———
5 6
4) 7 x 1 = 7
2 7
2 1
———
5 6 7
And to prove that this works all the time, here is the algebraic proof.
Algebraic Proof: Multiplication of two 2-digit numbers
The first Number will be represented as (ax + b), where x = 10, and ‘a’ & ‘b’ are some number between 1 and 9.
The second Number will be represented as (cx + d), where x = 10, and ‘c’ & ‘d’ are some number between 1 and 9.
(ax + b) x (cx + d) = (ac)*(x^2) + (ad)*x + (bc)*x + bd
(ax + b) x (cx + d) = (ac)*(x^2) + (ad + bc)*x + bd
There you go, that’s the whole proof! For those of you who need to visualize it…
a b
c d
———
ac | ad + bc | bd
Practice Problems:
1. 16 x 11
2. 26 x 12
3. 31 x 17
4. 45 x 45
Problem 4 could also be done by a method I showed you previously. Click Here
In Part II, I will be discussing multiplication of two three-digit numbers. Once you understand how to multiply two three-digit numbers, you can extrapolate it out to any number of digits.
Vertically and Crosswise, Part 2
WOW, this is REALLY, REALLY interesting. Is this the way everyone is taught to multiply in India? It is COMPLETELY different from how we are taught (and are teaching) in the West. We start multiplying from the LEFT side, and work right. I had NO IDEA you could start at the right side.
Maybe using entirely different methods of calculating is what gives Indians the “edge” in world mathematics!
Eileen
Dedicated Elementary Teacher Overseas (in the Middle East)
elementaryteacher.wordpress.com
Eileen,
To my knowledge it’s not part of the “standard” curriculum in India. However, students trying to get into competitive engineering schools in India tend to seek lessons in Vedic Mathematics to give them an edge on entrance exams.
“We start multiplying from the LEFT side, and work right. I had NO IDEA you could start at the right side.”
I had no idea either, until I learned about Vedic Math. =)
I’m still in the process of learning it…but each new thing I learn just keeps impressing me more and more!
[...] 27, 2008 by Liberius In Part 1, we discussed how to multiply two 2-digit numbers together. For Part 2, we will be discussing how [...]
Wow, thank you for your reply, Liberius! I’m going to show my family tonight!
Margot, the Marrakesh Mystic
margotmystic.wordpress.com
It is true that Vertically and crosswise is a very useful Vedic sutra! It has limitations to use for mental computing!
As regards teaching concerned computing principles, knowing virtues of zero-start 2D square matrices will greatly help!
http://www.Vedicmatrix.org is an introduction to it!
When ‘y’ and ‘x’ equal-elements (units) are numbered as 0…9 we get zero start matrix 0…9. Similar matrices 00…99, 0r 000…999 or ‘n’0s… ‘n’ 9s also have properties of matrix 0…9.
Ancient Indians had studied matrix virtues dead accurately. You will find matrix-link with each Vedic sutra, which are not explored yet!
[...] have previously posted about this topic (here and here), however I believe those posts may have been too long for most people’s liking. So [...]
‘Prime base’ numbers sorting basics (yet another vertically and crosswise)
‘Prime bases’ that relate a Vedic matrix 0…9 are shown by (x). They maintain a ordered position relation in either a vedic matrix 0…9, or 00…99, or 000…999 or n0s…n9s. In Vedic matrix 0…9 slope of each line is yx=12 <– Vedic manner just merges y and x numbers (say row and column numbers).
When we apply same logic to a Vedic matrix 00…99, we have 0102 yx number.
When we apply same logic to a Vedic matrix 000…999, we have 001002 yx number.
And so on!
Said unique relations that bind all zero start 2D square matrices have been cleverly related to ancient Indian Vedic mathematics. Nothing not-scientific in it.
A learning of 'matrix by matrix Vedic computing' greatly helps!
You may learn your lessons from a least vedic matrix 0…9 and apply it in n 0s…n 9s vedic matrix. A patient study of position relations in a zero-start 2D matrix is needed to undestand medium "Vedic matrix"(both 2D and 3D), which does no exist among known 'mathematics', but we can SEE / FEEL it in Vedic mathematics. (Sense of Vedic sutras reveals 'a zero-start matrix position relating' to it )
..0…….1….…2….…3….…4….…5…….6……7….…8….…9
—— x—– —— —— —— x—– ——x—– —— ——
0………01…………………………05………….07
—— x—– ——x—– —— —— —— x—– —— x—–
1………11………….13………………………….17………….19
—— —— —— x—– —— x—– —— —— —— x—–
2
—— x—– —— —— —— x—– —— x—– —— ——
3
—— x—– —— x—– —— —— —— x—– —— x—–
4
—— —— —— x—– —— x—– —— —— —— x—–
5
—— x—– —— —— —— x—– —— x—– —— ——
6
—— x—– —— x—– —— —— —— x—– —— x—–
7
—— —— —— x—– —— x—– —— —— —— x—–
8
—— x—– —— —— —— x—– —— x—– —— ——
9
practical application.
Is 99899 is a 'prime base' number?
There are factors 353 and 283 to 99899.
I have done it by a step of 6(+/-)1 formula and a later primality check ( Any 6 digit number divided by each prime in a number zone 4 to 999). It is CAD application.
Nearest Vedic matrix to 99899 is 000…999. Position 099 899 is to be checked for primality.
1) 99899 is 'one less than' 099 900 (row 99, col 900), which is equal to 100×999 . It does not really helps. So try step of 6(+/-)1 method (as "all from 9" did not work).
2) 99 899 – 99198 (<–this is equal to 1002*99<– a row number of 99 899) is equal to 701. As 702 is a (step of 6) 99198+702-1 is a prime base. (in this case 99899).
It is either 'a prime' or 'a prime base' that could have factor (only primes).
It means a division check is mandatory, which revealed factors 353 and 283. It is not easy to find factors like these without computer help. However logic that 'prime base' has only prime number (as factors) has helped.
Above explanation 2) is incomplete to a new person but matrix position awareness will soon clear doubts!