Finding Square of Numbers – Part 3

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.

I. N. Herstein

Welcome to the blog Math1089 – Mathematics for All.

Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy schedule to check out Math1089!

Squaring a number means multiplying that number by itself. For example, 3 × 3 = 9 is the square of 3; 11 × 11 = 121 is the square of 11 etc. This method is available for finding squares in conventional mathematics. But in Vedic mathematics, there are smarter ways available to find the square of specific numbers.

Few Important Terms

Bases are the numbers starting with 1 and followed by any number of 0’s. For example, 10, 100, 1000, 10000, . . A base number must start with 1 and should be followed only by 0’s. They are the first number for those many digits. Like, 10 for two-digit numbers, 100 for three-digit numbers, and so on.

The surplus of a number is the number obtained from subtracting the nearest base from the given number. For example,

  • surplus of 13 is 13 – 10 = 3;
  • complement of 119 is 119 – 100 = 19.

In this blog post, we are going to discuss about the shortcut rule for finding the square of numbers, near the base. The trick is very much well-known in Vedic mathematics and useful for various competitive examinations.

The rule is given below:

∎ (a) Since the given number is above the base, find its surplus;

∎ (b) The answer will come in two parts – RHS and LHS, where

(i) for RHS, write the square of the surplus (make sure that the number of digits on RHS is equal to the number of 0‘s in the base. If it is less, put zeros to the left; but if it is more, consider only that many digits [from the right only] same as the number of zeros in the base and if anything remains in the right side, take that as a carry. This carry should be added to the LHS number);

(ii) for LHS, add the surplus from its base to the given number.

The final answer looks like (LHS)(RHS) or (number + surplus)(square of the surplus).

Example 1. Find the square of 104.

Solution. Base close to 104 is 100.Therefore, surplus from the base is 104 – 100 = 4.

(i) RHS = square of the surplus = 42 = 16;

(ii) LHS = add the surplus to the number

= 104 + 4 = 108.

The final answer is (LHS)(RHS) = 10816.

Example 2. Find the square of 101.

Solution. Base close to 101 is 100.Therefore, surplus from the base is 101 – 100 = 1.

(i) RHS = square of the surplus = 12 = 1 = 01 (since, base 100 has two 0′s, so to make two digits on RHS, we write it as 01);

(ii) LHS = add the surplus to the number

= 101 + 1 = 102.

The final answer is (LHS)(RHS) = 10201.

Example 3. Find the square of 1008.

Solution. Base close to 101 is 100.Therefore, surplus from the base is 1008 – 1000 = 8.

(i) RHS = square of the surplus = 82 = 64 = 064 (since, base 1000 has three 0′s, so to make two digits on RHS, we write it as 064);

(ii) LHS = add the surplus to the number

= 1008 + 8 = 1016.

The final answer is (LHS)(RHS) = 1016064.

Example 4. Find the square of 111.

Solution. Base close to 111 is 100.Therefore, surplus from the base is 111 – 100 = 11.

(i) RHS = square of the surplus = 112 = 121 = 21 (since base 100 has two 0′s, so we take 21 here and consider the remaining digit 1 as carry);

(ii) LHS = add the surplus to the number

= 111 + 11 = 122.

Since there is a carry (equal to 1), add 122 and 1 to get the final value of LHS = 122 + 1 = 123.

The final answer is (LHS)(RHS) = 12321.

Check Your Understanding

Find the square of the following numbers:

  • 1. 108
  • 2. 106
  • 3. 1005
  • 4. 1015

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call Math1089 – Mathematics for All!“.

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