10th lesson-urdhwa-Tiryak-sutra

10th lesson of Mathematics Tricks and Fun

In this lesson on Urdhwa Tiryagbhyam Sutra, we will study multiplication of big numbers where digits are in random. This is also useful for squaring of such numbers . Now what is Urdhwa Tiryak Sutra?

Urdhwa Tiryak Sutra is the general formula applicable to all cases of multiplication . It is also useful in the division of large numbers by another large number . Only multiplication part will be taught in this lesson. All arithmetic numbers are algebraic expressions in X with X equals 10. The calculations can be explained with algebraic principle, which willbe done at a later stage .Urdhwa Tiryak process of vertical and crosswise multiplication can be mastered easily, after studying a few examples.

in the first example, 28 x 46 first we multiply the digits on the tens place straight. 2 x 4 is 8. we are multiplying totally four digits the answer will have three or four digits . since 8 is a single digit we will write as 08 at the edge. Next cross multiply 4×8 and 2 × 6 the answer is 440 , this we will write below 8 onwards. . Then the last digit 8 x 6 straight multiplication and the answer for this is 048. write this below 440. the final answer is 1288.

 Here comes the multiplication with 3 digits . We can start from whichever side . First vertically, second crosswise. Then we include all 3 digits in the crosswise multiplication as in the picture .Fourth, the last two digits crosswise. Fifth last 2 digits vertically straight. But in all these quick multiplications, we should note how many digits should be there in the answer. For this we should count the number of digits on the multiplicand side and multiplier side . Thus for example, 2 each-- means the answer will have three of four digits . We will start on the unit digits place . We willnot make mistakes when only three digits in the answer . If you are starting from the lateral side of the multiplicand and then we should put 0 in front of the third digit,in the place of the fourth digits . When we multiply vertically in the example 1 X 1 is 1,1 x 0 + 1 X 0 is 0, 1 × 6 + 1 × 8 + 0 is equal to 14

0×6 + 0 x 8 = 0, 6 x 8 is 48 . All the together we get the answer 11448.

Here four digits multiplication shown. First vertical two digits , then two digits crosswise , 3 digit crosswise,4 digits crosswise, then again 3 digits crosswise,then two digits crosswise, and then vertical . Thus only 7 steps. Another thing to note here is,there are totally 8 digits involved and the answer also has 8 digits. the answer is 10, 44,44, 37,69, 3 and 24 and after adding the carry overs , the final answer is 1488 3954.

 This example is of 5 digit multiplication. Totally there are 10 digits involved in this and 9 steps for the multiplication. Thus answer is 24, 37,28 ,37,92, 67,25, 47,35 .The final answer is 28026 90005.

This picture looks beautiful but complicated.Actually it is easy, when understood. To make each step clear,I have drawn like this . Six digit square here. I will start from the lateral side of the multiplicand. Thus the steps are 5 x 5 = 25, next double of 

(5 x 6 ).

3rd step. Double of (7×5)+6×6

4th step. Double of(5×8)+double of(6×7)

5th step double of (5×9)+ double of (6×8)+ 7×7

6th step. Double of (5×6)+ double of (6×9)+double of (7×8)

7th step. Double of (6×6)+double of (7×9)+8×8

8th step. Double of (7×6)+ double of(8×9)

9th step. Double of (8×6)+ 9×9

10th step. Double of (9×6)

11th step 6×6

I have arranged the digits correctly and shown. Adding the carry overs , the answer is 322505866816. When we put this to practice we will become used to mental calculation..

Next class We will  refresh Nikhilam Sutra and corollaries . Thank you.