Some time ago (this past summer, to be exact) I created my first instructable. A fun little trick, really, about how to quickly extract the cube roots of large integers mentally. Included in the comments section of this instructable was a very useful comment from mahi16 which suggested I incorporate Vedic Mathematics methods into my technique. Having never heard of it before, I investigated it, and found myself intrigued by the clever strategies employed in the ancient Indian system. I got a book on it this year for Christmas and am researching the different techniques and methodologies presented in it.
So, without further ado, I present to you Vedic Multiplication
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Step 1: Multiplication...The Standard Algorithm
Most of us remember the standard algorithm for multiplication (well, those of us not being brought up under the hilarious joke that is modern math education in some schools, anyway). To demonstrate it, consider the multiplication problem 52x45. This is done in the following steps (which are demonstrated in photos as well):
1) 5 times 2 is 10 (put down a 0, carry the 1)
2) 5*5 is 25, add the carry (put down a 6, carry a 2)
3) On a second line, add a 0 to the 1's place (4 has place value of 10)
4) 4 times 2 is 8 (put down an 8)
5) 4 times 5 is 20 (put down a 0, carry a 2)
6) Add together the 2 numbers (260 + 2080) to get the product (2340).
This method is general and efficient, to be sure. However, it is not hard to see that this method can involve considerable tedium and rigor when you start doing more involved multiplications (3 digits by 3 digits or more).
Read on to see how we can more efficiently multiply, the Vedic way!
Step 2: Enter the Sutra
The Vedic method (the general method, at least) is based on the Urdhva-Tiryagbhyam sutra. A very terse sutra, it simply translates in English to say "vertically and crosswise". The sutra is rather vague, so the technique, as well as an algebraic analysis of the technique, is presented in the following steps.
Step 3: An Algebraic Perspective
As most readers are hopefully aware, all numbers n the base 10 number system (and number systems of any other base, for that matter) consist of a number of digits. Each digit represents a multiple times a power of 10 (or whatever the number system's base is). So, for example, given a number like 52, we could rewrite it as 5*10+2.
Algebraically speaking, we can express any 3-digit number as:
ax+b (where a, and b are integers).
So, suppose we wanted to multiply 2 2-digit numbers. We can express them in polynomial form. Then, by foiling:
(ax+b)(cx+d) = acx2+(ad+bc)x+bd
Read on to see how we take the algebraic multiplication to higher numbers of digits.
Step 4: Algebraic Multiplication for Higher Numbers of Digits
So, for brief review, 2-digit by 2-digit algebraic multiplication goes as follows (x in all of the following examples is the base of the number system being used, which is usually 10):
(ax+b)(cx+d) = acx2+(ad+bc)x+bd
Expanding to 3-digit by 3-digit algebraic multiplication:
(ax2+bx+c)(dx2+ex+f) = adx4+(ae+bd)x3+(af+be+cd)x2+(bf+ec)x+cf
Now, 4-digit by 4-digit algebraic multiplication:
(ax3+bx2+cx+d)(ex3+fx2+gx+h) = aex6+(af+be)x5+(ag+bf+ce)x4+(ah+bg+cf+de)x3+(bh+cg+df)x2+(ch+dg)x+dh
Finally, 5-digit by 5-digit multiplication:
(ax4+bx3+cx2+dx+e)(fx4+gx3+hx2+ix+j) = afx8+(ag+bf)x7+(ah+bg+cf)x6+(ai+bh+cg+df)x5+(aj+bi+ch+dg+ef)x4+(bj+ci+dh+eg)x3+(cj+di+eh)x2+(dj+ei)x+ej
By looking carefully at the products of the above multiplications, a pattern becomes noticeable. The pattern looks something like this:
1) For the highest digit of the the product, simply write the product of the highest digits of the 2 numbers.
2) The process for the middle digits is a little tricky to explain using words. For each successive digit, you cross-multiply the highest digit of the top number by the next highest digit of the bottom number and the highest digit of the bottom number by the next highest digit of the top number and add these numbers together. You continue on for all subsequent digits by decrementing the digit on the bottom number, cross-multiplying, incrementing the bottom digit while decrementing the top digit, and summing the products. If this confuses you, fear not, it will be explained via examples on the following pages.
3) For the lowest digit of the product, multiply the lowest digits of the numbers being multiplied.
4) Throughout this process, only one digit can be kept per place, thus if your sum is higher than 9, you will have to write the ones digit only and implement a carry to the next higher place in the number (or carry to several higher places if your sum exceeds 99).
Coming up, examples!
Step 5: Vedic Multiplication Examples: 2-Digit by 2-Digit
In this step (and the following steps) we will attempt to clarify the vedic procedure through the use of several examples. Under each example, there will be a number of steps (indicated by numbers). Each step has a separate picture, so feel free to use those to follow along with what's going on.
Example 1: 42*21
1. Multiply the 2 highest digits (4 and 2), resulting in an 8.
2. For the next higher digit, cross multiply 4*1 (4) and 2*2 (4), and add together, producing the middle digit of 8.
3. For the lowest digit, multiply the 2 lowest digits (1*2) together, resulting in a 2.
4. Put all of the digits together to produce your answer (882)
A quick check on the calculator indicates our answer is indeed correct.
One thing that can and should be noted here is that the order in which you go through for the vedic process does not actually matter. So, we can similarly start with the lowest digit and work our way up to the highest digit.
Example 2: 67*23 (Starting with the lowest digit and working our way left)
1. Lowest Digit: Multiply the 2 lowest digits together (7*3), to get 21. Only one digit can be kept per place, so write a 1 and carry a 2 over to the next position.
2. Now, move over by 1 position on top, (3 will be multiplied by 6 now) and cross multiply pairs, decrementing the position on top while incrementing the position on the bottom. That is to say, multiply 3*6 (18) and 2*7 (14) and add them together (32). Write a 2, carry a 3 to the next position.
3. At this point, we have reached the end of the number, and thus incrementing the position on top is not possible. Thus, we simply increment the position on the bottom (2 will be multiplied by 6) and no cross multiplication will occur (as incrementing the bottom digit is not possible, we have reached the end of the number). So, 2*6 will be 12, so write a 2 and carry a 1.
4. Now that the multiplication has been completed, we have 2 numbers. The one on top is all of ones digits from the multiplications, and the one on the bottom is all of the carries that occurred. So, simply add these 2 numbers (221 and 1320) together, resulting in the answer (1541).
In the following steps, we will perform examples to expand the idea to higher orders of digits.
Step 6: Vedic Multiplication Example: 3-Digit by 3-Digit
Example 3: 341*562
At this point, some of the faint of heart may be reaching for their calculators. Be strong, you can handle this using the vedic technique with relative ease (and considerably less writing) than that which you probably learned in elementary school.
1) Ones Digit: 2*1 = 2
2) Tens Digit: 2*4+6*1 = 8+6 = 14 (write a 4, carry a 1)
3) Hundreds Digit: 2*3+4*6+5*1 = 6+24+5 = 35 (write a 5, carry a 3)
Notice at this point that we have reached the end of the top number, that is to say, 2 can not be multiplied by anything else, as there are no more digits. Thus, for the thousands digit we shall multiply 6 and 3 first, and then for the ten thousands digit, 5 and 3. (read on for the steps)
4) Thousands Digit: 6*3 + 5*4 = 18+20 = 38 (write an 8, carry a 3)
5) Ten Thousands Digit: 5*3 = 15 (we've reached the end of the multiplication, so feel free to simply write 15 rather than carry a 1)
6) Add in the carries to produce the final answer: 191642 (verified via calculator)
At this point it should be noted that the 2 numbers need not have the same number of digits. For example, if a 3-digit number were to be multiplied by a 2-digit number, just follow the formula for 3-digit by 3-digit multiplication and just pad out the missing digits with zeros.
Step 7: More Examples and Practice
This step is dedicated to a few additional examples. The viewer is encouraged to work these out on his or her own and verify that they have done the technique correctly. Attached are images showing my work for each of the practice problems below.
All of the problems below have been verified by calculator. Also, the vedic method should be executable much more quickly than the standard multiplication algorithm (the author of this instructable performed all of the multiplications below in under a minute each, including the last one).
and finally, for those who are gluttons for punishment...
You may not get the technique right away, but with a little practice you too can multiply quickly, efficiently, and like a pro. Happy hunting!