Speed Math via Vedic Mathematics

Vedic mathematics is based on sixteen sutras which serve as somewhat cryptic instructions for dealing with different mathematical problems. Below is a list of the sutras, translated from Sanskrit into English. They were presented by a Hindu scholar and mathematician, Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century.

1. By one more than the previous one
2. All from 9 and the last from 10
3. Vertically and crosswise (multiplications)
4. Transpose and apply
5. Transpose and adjust (the coefficient)
6. If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero
7. By the Para-party rule 
8. If one is in ratio, the other one is zero.
9. By addition and by subtraction.
10. By the completion or non-completion (of the square, the cube, the fourth power, etc.)
11. Differential calculus
12. By the deficiency
13. Specific and general
14. The remaining by the last digit
15. The ultimate (binomial) and twice the penultimate (binomial) (equals zero)
16. Only the last terms
17. By one less than the one before
18. The product of the sum
19. All the multipliers

The first one is basically the multiplication algorithm by 11 discovered independently by Trachtenberg.

Let us look at the second one, which is used quite a bit in Vedic Mathematics: All from nine and the last from ten.

When
subtracting from a large power of ten with many columns of zeros, it is
not necessary to write the notation for borrowing from the column on
the left. One can instead subtract the last (rightmost) digit from 10
and each other digit from 9. For example:

This
the method is also used when finding the deficit from the next larger power
of ten when setting up a multiplication problem using the
cross-subtraction method.

The third one is Vertically and crosswise (multiplications). One use for this is for multiplying numbers close to 100.

Suppose
you want to multiply 88 by 98. Both 88 and 98 are close to 100. Note
that 88 is 12 below 100 and 98 is 2 below 100. This can be pictured as
follows:

We
subtract crosswise to get the first two digits of the answer. It
doesn’t matter if we do 88-2=86 or 98-12=86, both give the same number.
To get the last two digits we multiply vertically: 12 x 2=24.
Therefore, the answer is 8624.

The same strategy works for
multiplying two numbers above 100. For example, 107 times 111. Quickly
we add the surplus from 107 (which is 7) to 111 to get 118, the first 3
digits of the answer. To get the last two digits, we multiply the
surplus of 107 from 100 by the surplus of 111 from 100: namely, 7 x
11=77. Thus, the answer is 11,877.

Vedic Mathematics is all about using different formulas in a variety of ways.
In the above rule we are using:

The above notation is short for:

and is often used since it’s easier to see what the number actually is.

The
above generalizes for numbers close to a base of 1000. Note that the
second sutra becomes quite useful for when you are computing the
deficit from the base.