Quality Education | Gayathri Rajinikanth | Vedic Math | Episode - 4 @SakshiTV

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[00:00:00]Welcome to fifth class on Vic mathematics. Today we will learn one of the most powerful sutras in Vic mathematics for multiplication. The urvatriak sutra also known as vertical crosswise. Unlike the niklam sutra which depends on the base calculations and its opposutra anurupena urvatriakam is a universal or method or a general method that can be applied to all types of multiplication. This technique not only helps us multiply numbers of same lengths but also different lengths easily such as a three-digit number by a two-digit number or a four-digit number by a two-digit number and many more combinations.

[00:00:38]It is a fast, systematic and versatile method that makes multiplication simple and more efficient. Let's now understand the multiplication of two-digit numbers AB * CD using the dot diagram method.

[00:01:05]in this diagram. Each dot represents a digit in the numbers being multiplied.

[00:01:10]The upper two dots represent the digits A and B of the first number while the lower two dots represents a digit C and D of the second number. Using this arrangement we apply urvak method step by step through vertical and crosswise multiplication. Step one we will be we will be performing vertical multiplication of the digits in the ones column or the units columns. So it's going to be B * D which is equal to BD.

[00:01:51]The second step we're going to perform cross multiplication and the products of these cross multiplication we're going to add them.

[00:02:06]So, it's going to be a d a * d plus b * c, which is equal to a d + bc.

[00:02:18]And the third and the final step, we're going to perform the vertical multiplication again.

[00:02:30]So, we going to multiply a * c, which is equal to a c.

[00:02:36]So now we have step one, step two and step three. The order is very important.

[00:02:42]We need to make sure the step three, step two and step one are placed in the below order. So step three will be the hundreds place, step two, and step one will be the units of the on's place. So it's going to be a c a c a a a d plus bd and then bd.

[00:03:11]The order is the order of these results is very very important to get a correct answer. Okay. Let's see an example using this uh this uh technique. Let's multiply 37 * 54.

[00:03:27]Applying the uratra method. Step one, we are going to multiply the on's place digit. 3 7 5 4.

[00:03:39]So vertical multiplication of the on's place digit. So 7 * 4 is equal to 28.

[00:03:47]Step two, cross multiplication.

[00:03:51]of the on's place digit and the 10's place digit after which we're going to add the products. So we're going to multiply 3 * 4 + 5 * 7 12 + 35 = 47.

[00:04:10]Step three, vertical multiplication of the 10's place values. So 3 * 5 = 15.

[00:04:23]As I mentioned earlier, the order is very important. So we're going to start with the third place the step three. So step three is 15. Step two 47 followed by step one. Again, as I mentioned earlier, each place value cannot hold more than a single digit. The extra set digit needs to be added or carry forward to the next level. So we're going to apply the balancing technique to solve this. So we drop 8 which is going to be the part of the final answer. We're going to add 2 to 7. 2 + 7 gives us 9 and 4 + 5 is again a 9 and 1 we are going to put it as is. So the product of 37 * 54 is 1 1998.

[00:05:05]Let's see another example. Let's multiply um let's say 76 * 93.

[00:05:11]So applying the udatra method step one is the vertical multiplication of the on's place digit.

[00:05:25]So 6 * 3 = 18.

[00:05:30]Step two cross multiplication.

[00:05:39]So 7 * 3 + 9 * 6 21 + 54 = 75. And the final step would be the vertical multiplication of the 10's place digits. So 7 6 9 and 3.

[00:05:58]So 7 * 9 equal to 63.

[00:06:02]So now we are going to put place these numbers in an order. So 63 is going to be in the hundred's place followed by 10's place and then the unit's place.

[00:06:12]Again we have two digits in one place value. The extra digit needs to be added or carry forward to the next level. So dropping eight to be in in adding 1 to 5 gives us 6 and 7 + 3 gives us a 10. So 0 and carry of 1. 1 + 6 is 7.

[00:06:31]768 is the answer for this problem.

[00:06:35]Let's see one more example. Let's multiply 92 * 18.

[00:06:39]Applying udatra sutra. Step one results in a product.

[00:06:46]The vertical multiplication of 2 and 8 gives us a 16.

[00:06:52]Then step two yields us 92 18. So 9 * 8 and 1 * 2. 9 * 8 is 72 + 1 * 2 gives us a 2. So 72 + 2 is equal to 74. And finally step three vertical multiplication of the 10's place digits.

[00:07:17]So 9 * 1 is equal to 9. Now arranging the numbers in the order which is very very important to get the correct answer. So 9 followed by 74 followed by 16. Now apply the balancing technique since we have more than a single digit in one place value. So drop six and 1 to 4 which gives us a five 7 to 9 gives us a 16. So the answer to this problem is 16 56.

[00:07:44]If you have observed the above examples we have just worked out. You will notice an important advantage of this method.

[00:07:52]At no point did we use any base nor did we check whether the numbers are close to a particular base or a working base as this was the one of the most important requirement for niclam sutra or anurupa sutra. Unlike those techniques, this approach works purely through systematic digit wise multiplication and addition. It is a general method that applies to any pair of numbers making it simple, flexible and universally usable without any base consideration. Similarly, we can extend this method to the multiplication of two three-digit numbers A B C XYZ using the dot diagram approach.

[00:08:34]So, A B C X Y Z dot A B C X Y and Z. In this representation, each dot corresponds to a digit in the number being multiplied. The upper three dots represents the digits A, B and C of the first number while the lower three dots represents X, Y, Z of the second number.

[00:09:08]Using this arrangement we can systematically apply urutra sutra by performing vertical and crosswise multiplication across all digit position. So step one C * C = C 0.

[00:09:36]Step two, cross multiplication of the ones and 10's place. So, b * z + c * y.

[00:09:54]Bz + c y.

[00:09:59]Step three, cross multiplication of ones at 100's place. Similarly, the other 100 and on's place and the 10th's place. So, it's going to be a * z + c * x + b * y.

[00:10:26]So, it's a z + cx + b y.

[00:10:35]Step four, vertical cross multiplication of the 10th and the hundreds place and the other combination also. So it's going to be a * y + b * x a y + bx.

[00:10:56]Step five, the vertical multiplication of the 100's place digit which is a * x a x.

[00:11:10]Now arrange all these numbers in a particular order. So we start from step five followed by step four, step three, step two, and step one.

[00:11:31]So, it's going to be a y sorry, a x a y + b y bx then a z + c x + b y followed by bz + c y then c.

[00:11:58]Again we apply the balancing technique uh to get the final answer. So this is how we represent the dot diagram of a 3x3 digit multiplication. Let's see an 3x3 multiplication. So 532 * 472.

[00:12:15]So applying the udotra sutra step one will be the vertical multiplication of the on's place digit. So 5 3 2 4 7 2 * 2 gives us a 4. Step four, step two is a cross multiplication of the ones place digit with the 10's place digit and the other combination also we add the products. So 3 * 2 + 7 * 2. So 6 + 14 20.

[00:12:54]Step three 5 3 2 4 72. So the hundred's place with the on's place cross multiplication of the hundred's place again with the other combination and the 10's place with the 10's place. So 5 * 2 + 4 * 2 + 3 * 7. So 10 + 8 + 21.

[00:13:24]So 39.

[00:13:27]Step four, cross multiplication of 100's place with the 10's place.

[00:13:38]The other combination also we're going to consider. So 7 * 5 + 4 * 3. So 35 + 12 47.

[00:13:50]The last step vertical multiplication of the 100's place digit 5 * 4 = 20.

[00:14:05]Now let's arrange all the numbers in the order 20 followed by 47 followed by 39 20 and then four. So we apply the balancing technique four we drop as um it as a final answer zero two we are going to add to the 9. So 1 and 1 carry 1 + uh 3 is 4. 4 + 7 11. So carry 1 drop here. 1 + 4 uh is 5.

[00:14:38]5 + 0 is 5 again. And then 2. So the final answer for this number for this product is 25 1 04. Let's see another example.

[00:14:50]716 393.

[00:14:54]Step one, vertical multiplication of the ones place digit. So 6 * 3 = 18.

[00:15:05]Step two, cross multiplication of the on's place with the 10's place and the other combination also we're going to consider. So 1 * 3 + 9 * 6. So 3 + 54 57.

[00:15:26]Step three cross multiplication of of the hundred's place with the ones place. We're going to consider the other combination also and the 10's place with the 10's place. So 7 * 3 + 3 * 6 + 1 * 9. So 21 + 18 + 9. So it's equal to 1848.

[00:15:56]Step four, cross multiplication of the 100's place digit with the 10th's place digit. So 1 * 3 and 7 * 9.

[00:16:09]1 * 3 + 7 * 9. So 3 + 63 66.

[00:16:16]And finally step five 7 * 3 gives us 21 again now apply arrange the numbers in the order. So 21 21 followed by 66 followed by 48 57 and then 18. Apply the balancing technique.

[00:16:45]So, drop 8. Add 1 + 7. 8. 5 + 8 is 13.

[00:16:51]So, drop three. Add 1. Carry 1. 1 + 4 is 5. 5 + 6 is 11. So, carry 1, drop 1. 6 + 1 is 7. 7 + 1 is 8 8 and then 2. So, the product of 71 6 into 3 93 gives us a result of 281 388. Now, let's see another example. Let's multiply 818 * 1 192.

[00:17:21]Here I'm going to apply the udotra steps but instead of writing each and every step separately I'm going to multiply as belows I'm going to multiply the first step which is um the vertical multiplication of the ones place digit which is 8 * 8. I'm going to write it as 8 * 8 followed by a line separator.

[00:17:42]Then I'm going to multiply the 10's place with the on's place. So 1 * 2 + 9 * 8 followed by the line separator. Then I'm going to apply third step which is 8 * 2 + 1 * 8 + 1 * 9 line separator. Step four is 100's place with the 10's place. The cross multiplication of one 100's place with the 10's place. So 8 * 9 + 1 * 1 and finally the vertical multiplication of the 100's place digit which is 1 * 8.

[00:18:22]So this is 8 9 73 33 74 followed by 16. So since we have more than a single digit in one place value we're going to apply the balancing technique. So drop six, add 1 + 4, which is a 5. 7 + 3 is a 10. So drop zero, add 1. So 3 + 1 is 4. 4 + 3 is again a 7. 7 + 8 is a 15.

[00:18:57]So 818 * 192 gives us a answer of 157056.

[00:19:04]If you have seen the the all the steps like all the five steps you have reduced to a single line answer single line step single step and then we have reduced it to a we have got the answer. So this is the power of udatra sutra. Let's multiply 342 with 76.

[00:19:22]So here 342 is a three-digit number while 76 is a two-digit number. So all that you have to do is add a leading zero in front of seven which is a two-digit number to make it a look like a three-digit number. Now since both the numbers are having three digits, we can apply the udvatra sutra. So here the very first step would be multiplying the vertical multiplication of the ones place digit which is 2 * 6 followed by 4 * 6. The cross multiplication of the on's place digit with the 10th's place.

[00:19:55]Again the 10th's place digit with the on's place followed by the hundred's place with the on's place.

[00:20:03]Again 100's place with the on's place and then the 10th's place with the 10th's place. Followed by the cross multiplication of the hundred's place with the 10's place and the 10th's place with the hundred's place and finally the vertical multiplication of the hundred's place with the hundreds place. So here this is 12 24 U 24 + um 14 8 38 then this is 18 18 + um 0 18 again and um 20 uh 8 um 6 46 and then 21 0.

[00:20:51]So two carry forward 9 9 5 and then a two.

[00:21:02]So when you multiply 342 with 76 you get an answer of 25992.

[00:21:08]Let's multiply a fourdigit number.

[00:21:13]Let's say with a twodigit number.

[00:21:16]So now uh we have a four-digit with a two-digit. So add two leading zeros in front of the two-digit number so that it appears like a four-digit number. Now we'll apply the udotra sutra. So 1 * 3 which is the vertical multiplication followed by the cross multiplication 2 * 3 + 1 * 4 followed by the hundreds with the ones. Again the ones with the hundreds and the t with the t. Now here the thousands we're going to multiply with the on's place which is 4 * 3.

[00:21:55]Again the thousands with the ones place 0 + um 0 * 1 is 0. Then again the hundreds with the 10. So 3 * 4 plus hundreds with the 10 it's a zero.

[00:22:08]followed by 4 which is thousand's place into the 10's place. So 4 * 4 again a 0 * 2 which is a 0 and 0 * 3 is also a zero. So just a zero.

[00:22:24]Then um this times this 0 * 4 is a 0 and 0 * 3 is also a 0. And the vertical multiplication of the thousand's place is also 0. So we have a 0 followed by a 0. Then 16. Then 4 * 3 is 12 and 3 * 4 is also 12. So 12 + 12 is 24 followed by 3 * 3 is 9. 9 + 2 * 8 uh 2 * 4 is 8. So 17. 2 * 3 is 6 + 1 * 4 is 10. So plus and then 3. So apply the balancing technique. Drop three. Drop zero. Add 1 to 7 which gives an 8 and 1 + 4 is 5. 2 + 6 is 8 and then 1. So when you multiply 432 with 40 43 it yields a result of 185803.

[00:23:20]So the advantage of uratra it or the vertically vertical crosswise method. It is a universal method for multiplying numbers. Its primary benefit includes a massive increase in mental calculations, enhanced speed, enhanced focus and drastic reduction of errors by eliminating complex intermediate steps.

[00:23:42]With this we conclude today's session.

[00:23:44]Thank you all for joining and we'll continue in the next class. Thank you.