Moonlight of Mathematics, Part 1 (1936) - Page 13

The Rule for Squaring

A square is the product of two identical numbers.

Place the square of the last digit; then multiply the last digit by the others and double it.

Place these above the respective digits; then, discarding the last digit,

shift the remaining digits and repeat the process again. ॥ 17 ॥

Alternatively: add and subtract a chosen number from the original number;

the product of these two, plus the square of that chosen number, is the square.

Or, the sum of a series of numbers starting with one and increasing by two,

or the square of the difference added to four times the product. ॥ 18 ॥

Example

O friend, tell me the squares of the numbers from one to nine,

as well as ten and seven. Also, find the square of

ten thousand added to two hundred and twenty-five, original: "sapañcavarga-dviśatī-yutasya," literally: with the square of five (25) and two hundred (200) joined [to the large number]

calculated individually and then for the total sum, if you know the method. ॥ 3 ॥

Statement: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 10225

The squares produced by these methods: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 49, 104,550,625.

104,550,625.

Thus ends the section on Squaring.

1. The sum of a number of terms in a series starting from 1 and increasing by 2 an arithmetic progression of odd numbers is equal to the square of the number of terms Padavarga: the square of the number of terms. This means: first term (Ādi) = 1; common difference (Caya) = 2; number of terms (Gaccha) = n. Using the rule "the sum of a series is the number of terms divided by two multiplied by..." etc., the sum becomes equal to the square of the number of terms. This is easily proven. For example: 1 + 3 + 5 + 7 + ... + n. The sum = n/2 [ 2(1) + (n - 1)2 ] = n/2 [ 2 + 2n - 2 ] = n/2 × 2n = n².

2. To find the square of a number, split it into two unequal parts. The square of the original number is equal to the square of the difference between those two parts added to four times their product In modern algebra: (a + b)² = (a - b)² + 4ab.