Finding Square of Numbers – Part 2

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.

Robert A. HEINLEIN

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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 complement of a number is the number obtained from subtracting the number to its nearest base. For example,

  • complement of 43 is 100 – 43 = 57;
  • complement of 729 is 1000 – 729 = 271.

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 below the base, find its deficiency (or complement);

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

(i) for RHS, write the square of the deficiency (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, subtract its deficiency from the number.

The final answer looks like (LHS)(RHS) or (number – deficiency)(square of the deficiency).

Example 1. Find the square of 96.

Solution. Base close to 96 is 100.Therefore, deficiency from the base is 100 – 96 = 4.

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

(ii) LHS = subtract the deficiency from the number

= 96 – 4 = 92.

The final answer is (LHS)(RHS) = 9216.

Example 2. Find the square of 97.

Solution. Base close to 97 is 100.Therefore, deficiency from the base is 100 – 97 = 3.

(i) RHS = square of the deficiency = 32 = 9 = 09 (since, base 100 has two 0′s, so to make two digits on RHS, we write it as 09);

(ii) LHS = subtract the deficiency from the number

= 97 – 3 = 94.

The final answer is (LHS)(RHS) = 9409.

Example 3. Find the square of 992.

Solution. Base close to 992 is 1000.Therefore, deficiency from the base is 1000 – 992 = 8.

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

(ii) LHS = subtract the deficiency from the number

= 992 – 8 = 984.

The final answer is (LHS)(RHS) = 984064.

Example 4. Find the square of 89.

Solution. Base close to 89 is 100.Therefore, deficiency from the base is 100 – 89 = 11.

(i) RHS = square of the deficiency = 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 = subtract the deficiency from the number

= 89 – 11 = 78.

Since there is a carry (equal to 1), add 78 and 1 to get the final value of LHS = 78 + 1 = 79.

The final answer is (LHS)(RHS) = 7921.

Check Your Understanding

Find the square of the following numbers:

  • 1. 92
  • 2. 94
  • 3. 995
  • 4. 9985

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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