Square Roots

In this section we are going to deal with a simple, straightforward, elegant single line Vedic method of finding the square root of any number.

In Vedic Mathematics, finding the Square Root, or Vargamoola, of a number involves using a simple, straightforward procedure.

The fundamental rules governing the extraction of the square root of a number are:

  1. First of all, the number is arranged in "two-digit" groups going from right to left. If there is a single digit remaining on the left hand side it is just placed in a group of its own.

  2. The number of digits in the square root will be the same as the number of digit-groups in the number itself. Remember, any single digit remaining on left will be in its own group. So, here are some examples: 25 will count as one group; 225 as two groups; 1089 as two groups; 10609 as three groups and so on.

  3. So, if the square root contains n digits, the square must have either 2n or 2n - 1 digits.

  4. Conversely, if the given number has n digits, the square must contain n/2 or (n+1)/2 digits.

  5. In the case of pure decimals, the number of digits in the square is always double that in the square root.

  6. The squares of the first nine natural numbers are: 1, 4, 9, 16, 25, 36, 49, 64 and 81. The implication of this is that:

    • An exact square cannot end in 2, 3, 7 or 8
    • A complete square ending in 1 must have, as the end digit of its square root, either 1 or 9 (mutual complements from 10)
    • A complete square ending in 4 must have, as the end digit of its square root, either 2 or 8 (again mutual complements from 10)
    • A complete square ending in 5 or 0 must have, as the end digit of its square root, either 5 or 0 (again mutual complements from 10)
    • A complete square ending in 6 must have, as the end digit of its square root, either 4 or 6 (again mutual complements from 10)
    • A complete square ending in 9 must have, as the end digit of its square root, either 3 or 7 (again mutual complements from 10)

In short, we can observe the following:

  • if a number ends in either 0, 1, 5 or 6 the last digit of square of the number will be same as the end digit of the number. For example 112 = 121; end digit(units place) in number 11 is 1 and the end digit of the square, 121, is also 1. For example: 162 = 256; Here we see that the end digit in the number and its square is 6....and so on.
  • The squares of the complements from 10 have the same last digit. 12 and 92; 22 and 82; 32 and 72; 4 2 and 52; 02 and 102 have the same ending, namely: 1, 4, 9, 6, 5 and 0 respectively
  • The digits 2, 3, 7 and 8 never become the final digit of a perfect square!

Vedic Method of Finding Square Roots Download Classic Square Root Method

There is also a very old, non-Vedic, method of calculating square roots that we have included for the sake of comparision.

Download "Ole Skool" Square Root Method