11th Lesson -Nikhilam sutra-corollaries

      

11th Lesson of  mathematics tricks and fun

 

 

 

Hello friends ,today we are going to study Nikhilam Sutra in multiplication. I am Dr Prema b sejekan.

 Nikhilam Sutra says Nikhilam Navatascharamam Dasatah.Means All from Nine and Last from Ten. Nikhilam sutra has upasutras corollaries and sub-corollaries.All of them and their application  in multiplication will be discussed here.

 Here  applications of Nikhilam Sutra in general is shown.Let us take the first among the 7 examples. In this, the first question is  9 times 9.working base is 10 . So complement of 9 from 10 is 1.   1×1 is 1 and  9-1 is 8. The answer is 81.2nd example is 7× 6 , so complement of  7 from 10 is 3. Complement of 6 is 4 so we will write 7-3× by 6-4 so ,3×4 is 12 and 7-4,is 3 .Adding the carry overs answer is 42.  Rest all examples are with different bases ie  100,1000 base  done in the similar way.

Anurupyena Sutra means proportionately.we can see some ratiowise relation and so, we  should do the calculation and adjust the  product  proportionately. We can understand this using an example. We have to find the square of 512  Here  512 is away from 1000  but  nearer  to 500 which is 1000÷2. So keeping this in mind we will do the multiplication,  taking 500 as working base we have ,  12 excess   so  we have to take square of 12 which is 144.Then we can add  12 to 512 we get 524 now we should divide this with 2 so we get 262 and the answer is 262144. 

These are 2 more examples of Anurupyena sutra.first one 235×246 Here the digits are near 250 which is 1000/4 ,235-015×  248-002 so the answer we get 233030.we should divide 233/4 the answer is 58 and ¼ ie 1000/2=250  . This 250, added to 30 gives us 280.the 38×13 answer is 58280.

In the second example we have 100 as working base.so 100×4 is 400.and the 438+38×388-12 so we get  +38×-12= -456 and 438-12 gives 426 This 42600×4 gives 170400   we   can subtract -456 from ,170400.The answer is 169944.

We can use the anurupya sutra  in Cubing. Here we have  3 examples  cube of 24. In this  tthere are 2 digits we can take 2 as " a" and ,4 as "b "  we canmultiply using the formula a3   3 a2b  3 ab2 and b3

 .cube of 2 is 8 , then 3a2b is 48, then 3ab2   is 96 and cube of ,4. Is 64.final answer is13824 same way , other examples can be done.

Yavadhunam sutra is another corollary of Nikhilam sutra.this is yavadhunam tava dhunokritya varge cha yojayet whatever the extent of its deficiency ,lessen it still further and do the squaring of the deficiency. If it is excess , add once more the excess and square of the excess.

12×12= base is 10. 12+2  and 2×2 answer is 144    same way square of 97 so complement of 97 is 03. So  again subtract 03 from 9 and square of 03 is 09 answer is 9409   the third example is ,having 3 digits    square of 989. So here 011 is less. Subtract once more ,011 from 989 and find square of 011 The final answer is 978121.

The 4th example is cube of ,104 here 4 is excess we multiply this with 3 we get 12,  so  we  put this as left  part of the answer.    then   this 12×by  previous 4 is 48 this the second part and cube of 4 is ,64 which is third part of the answer.thus the final answer is ,1124864

This Ekadhikena Purvena is the third corollary of Nikhilam sutra  which  means one more than the previous one.   45×45    here unit's place 5s and tens place same digit .  In  this  we  can add 1 to one digit in tens place and multiply.   so 5×5 is 25 and 5×4 is 20 The final answer is 2025.

 The same sutra is useful in finding recurring decimals by multiplications. Here 1/19 so one more than  one is 2 like this we continue   muliplying  previous digit with 2 to get digit and when we get 2 digits as answer , the keep tens place digit  as  carry over and unit place digit is used for multiplication .  To  the product ,the ten'splace digit is added .   Thus.  the  answer goes 1 2 4 8 16 13 7 14 9 18 17 15 ,11….

Antyayor Dasake pi means not  only for 5, the adding of can  be done ,but in all cases where the unit place digits added together gives 10.  This example will  make it clear.77×73 we increade one 7 to 8 so the answer is 8×7 and 7×3  is 5621. same way the 3  digit question.   Ten's and unit's place digits are taken together to multiply  91× 09  =  0819.  4×3 is 12.  so. The answer is  120819.

Ekanyunena purvena means  one less than the previous one.This  is Converse of ekadhikena Purvena,  this is useful in astronomy  while multiplying with a sequence of 9s.   7×9 one less than 7 is 6 and deficiency of ,   6-9 is 3. Thus the answer is 63.

234×999  reduce  one from 234 we get  233 and 999-233   gives  233766.

3rd example 78×9999 here ,  2 digits in multiplicand side and 4 digits on multiplier side.   So extra 9s  will  be written in the middle and worked as earlier and answer is  779923

When the multiplicand digits are more han 99s.    Here we multiply as shown 112×99 since 99 has two digits we will put line beffore 12 so  that two gps 1 and 12.  then we put 2  below 12 gp and 12 in the 3rd gp.   Then subtract .  12 we have to subtract from 100 and 2 from 12.  The answer is 11088

In the coming class we will study cubing of numbers.thank you.