Nikhilam Navatashcaramam Dashatah: All from 9 and the last from 10

Nikhilam Navatashcaramam Dashatah is the second sutra of Vedic mathematics. Its corollary is Sisyate Sesasamjnah.

Nikhilam multiplication can save time when multiplying numbers that are both near base 10, or both near base 100, or both near base 1000, etcetera.

Nikhilam division can be used to quickly divide any number (regardless of size) by a number that is near a base 10, base 100, etc. (such as the numbers 9, 98, 996, etc.)

What follows is a brief and incomplete summary of the math shortcuts this sutra contains:

Nikhilam multiplication using base 10, base 100, base 1000, etc.

The easiest way to explain this is to show two examples:

Rounding to base-10: If you wanted to multiply 14 by 13 you would start by adding the two numbers and subtracting 10.
14 + 13 = 27
27 - 10 = 17.
Add one zero (for the one zero in base 10) to the end: 170.
Now calculate the difference between 14 and 13 and their common base of ten, which would be the number 10.
14 is 4 away from 10; 13 is 3 away from 10.
Multiply 4 and 3. This equals 12.
Add the 12 to the number we got before (170).
12 + 170 equals 182.
Thus 14 x 13 equals 182.

Rounding to base-100: If you wanted to multiply 97 by 89, for example, you would start by adding the two numbers (186) and subtracting 100.
186 - 100 = 86.
Now add the two zeroes (for the two zeroes in base 100) to the end: 8600.
Now calculate the difference between the two numbers and their common base and multiply those two numbers.
100 - 97 = 3
100 - 89 = 11
3 x 11 = 33.
97 x 89 = 8633.

Nikhilam division using base 10, base 100, base 1000, etc.

Nikhilam division seems, at first glance, too complicated to save time, but is actually surprisingly fast and simple to do. Below are examples of some (but not all) applications of Nikhilam division:

Dividing any two-digit number by 9: Example: 53 divided by 9. the first digit (5) is the quotient. Now add the first and second digit: 5 + 3 = 8. The remainder is 8. Simple as that.
If the first and second digit add up to more than nine, you subtract 9 from the remainder and add 1 to the quotient. Example: 69 divided by 9. The quotient (at first glance) appears to be 6. But 6 + 9 = 15. You subtract 9 from 15 to get the real remainder (6) and add one to the quotient. Thus, the quotient is 7 and the remainder is 6.

Dividing any three-digit number by 9: This one might require paper and pen but is also quite easy.
Example: 412 divided by 9.
The first digit of the quotient will be the same as the first digit of the dividend: 4.
Now add the first two digits of the dividend: 4 + 1 = 5
Place this number (5) after the first digit of the dividend (4): 45
The quotient is 45.
To find the remainder, add all three digits of the dividend: 4 + 1 + 2 = 7
The quotient is 45 and the remainder is 7. If the remainder is more than 9, add one to the quotient and subtract 9 from the remainder.

Dividing any four-digit number by 9: Example: 1234 divided by 9
The first digit of the quotient will be the same as the first digit of the dividend: 1
Add the first two digits (1 + 2 = 3) and place that after the first digit (1). Now we have 13.
Add the first three digits (1 + 2 + 3 = 6) and place that after the 13. Now we have 136. This is the quotient.
The remainder will be all for digits added together. 1 + 2 + 3 + 4 = 10
But wait, the remainder can't be more than 9, so subtract 9 from the remainder and add one to the quotient.
The quotient is 137 and the remainder is 1.

Nikhilam division can also be used to divide very large numbers by a number that is near a base, such as 98, 984, 897, etc, but this gets a little more complicated. To learn more applications of Nikhilam Navatashcaramam Dashatah, check out the resources:

Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm