Vedic Sutra #3:     Vertically and Crosswise

We want to multiply two numbers.
Example: 8*6 = ?

  1. First we subtract each digit from 10:
    10 - 8 = 2 and 10 - 6= 4
  2. We arrange the numbers (mentally!) like so:
    82
    64
  3. Subtract Crosswise getting 8 - 4 = 4 (or, equivalently, 6 - 2 = 4)
  4. That 4 is the first digit in the answer.
  5. Multiplying the two Vertical digits 2 * 4 gives 8, the last digit in the answer.
  6. Answer is 48
Why?
We let the two numbers be A = 10 - x and B = 10 - y
(where x and y are the deviations from the number 10).

Then AB = (10 - x)(10 - y) = 100 - 10(x+y) + x y = 10 (10 - x - y) + x y = 10 (A - y) + x y
so A - y is the first digit and x y the second.

It's the "10" out front that moves A - y into the first digit position.

We can also multiply numbers by considering their deviation from, say, 100:
We let the two numbers be A = 100 - x and B = 100 - y

Then AB = (100 - x)(100 - y)
    = 100(100) - 100(x+y) + x y = 100 (100 - x - y) + x y = 100 (A - y) + x y
so A - y gives the first two digits and x y the last.

It's the "100" out front that moves A - y into the first-two-digits position.


Example: 97*83 = ?
  1. First we subtract each digit from 100:
    100 - 97 = 3 and 100 - 83= 17
  2. We arrange the numbers (mentally!) like so:
    973
    8317
  3. Subtract Crosswise getting 97 - 17 = 80 (or, equivalently, 83 - 3 = 80)
  4. That 80 gives the first-two-digits in the answer.
  5. Multiplying the two Vertical digits 3 * 17 gives 51, the last-two-digits in the answer.
  6. Answer is 8051