Sutra - All from 9 and the last from 10 (Explained)

This is an explanation to one of the 16 sutras of Vedic Mathematics. This sutra is popularly known as "All from 9 and the last from 10 ". This sutra (Original name- Nikhilam Navatashcaramam Dashatah) is helpful in making subtraction easy.


NOTE: This sutra is useful in cases of subtraction where a number is subtracted from a number being 10ⁿ   (example- 10, 100, 1000 and so on..)

RULE: All the digits of the number should be subtracted from 9 and the digit in the last place should be subtracted from 10. (Easy, isn't it?)

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Easy Multiplication of numbers around 100

Multiplication of two numbers around 100 is even easier than you could ever imagine.

Suppose you want to multiply any two numbers, lets take 95 and 97 for example.

Step 1: We'll find out the difference of the two numbers (95 & 97) from 100.

              95 is 5 less than 100 [100-95=5],

               97 is 3 less than 100 [100-97=3].

Step 2: Subtract the total difference from 100. Here the total difference will be 8 [Since 5 + 3 = 8]

               So after subtracting , we get 92 [ 100 - 8 = 92 ]. This becomes first part of our answer.

Step 3: We multiply the two difference amount i.e. 5 & 3 to get 15 as the second part of our answer.

So our final answer becomes 9215. Easy isn't it ?

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Mental Calculation of Subsequent number 's Square

It is practically easy to calculate the square of the subsequent number if you know the square of the number.
For example, if you know the square of 10, You can figure out the square of 11 mentally and instantaneously.
Here is how you can do it.
Continuing the above example, Square of 10 is 100, So, square of 11 will be 121.
How ???

First, we multiply 10 with 2 to get 20 and add 1 to it to get 21. [We used: 2A + 1 ( Where A is 10 here)]
Next, we add 21 to 100 (Because 100 is the square of 10)

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Easy Multiplication of a number with 5

It may be difficult for you to multiply a number with 5 , specialy larger numbers like "34638445" , "43574683", or even more larger than this. But after you learn this technique, you can even multiply larger numbers than the above mentioned numbers.
Here is how you should go:
Suppose, You want to multiply 264 with 5, then..

write or imagine mentally:

 [ Note: We have replaced the value of "5" with "10/2" ]

=>2640 / 2= 1320 [ You can mentally divide any number with 2, just move from lefthandside first number to righthandside last number, in this example, we should take "2" as the first number , as "2640" is what we have to divide with 2, so start dividing the numbers mentally, start putting the final answer in our mind, like in this example, start dividing "2" ,"6","4" and "0" with 2, and write or imagine the final answer as 1320. ]

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Easy Multiplication with 9 ( The Finger Rule)

This method is applicable for multiplying 9 with numbers upto 10 .
This method requires the use of your hands, this method is very easy to learn.
This method covers a very limited amount of numbers (i.e numbers ranging between 0 - 10 ) But You'll definately find it very interesting and easy..

Here is how to do it-

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