Algebra of Bhaskara II

Commentary: Now, using the first half of the Bhujangaprayāta a specific poetic meter verse, the text explains multiplication.

"Two positives or two negatives": when two positives or two negatives are multiplied—meaning the repetition of the multiplicand equal to the multiplier—the product becomes positive. In the product of positive and negative, it is negative.

This means that when both the multiplicand and multiplier are positive or both negative, the resulting product is positive. When one is positive and the other negative, the resulting product is negative.

Here, only the positive or negative nature of the product is established. All other methods of multiplication should be understood as stated in arithmetic.

Proof:

Multiplication is the repetition of the multiplicand equal to the multiplier. The product of two positive numbers is positive. The product of a positive and a negative is negative because the sum of repeated negative quantities is negative.

Alternatively, the sum of two positive and negative quantities (e.g., 2 and 2̇) is zero, as the sum of a positive and negative is their difference.

Even if these are both multiplied by an equal number, the sum of the resulting quantities must be zero. Here, if 2 and 2̇ are multiplied by 3 (positive): in the first case, the product of two positives is 6; in the second case, if the product of a positive and negative is not accepted as negative, then how can the sum of 6 and 6̇ be zero? Therefore, it is correct that the product of positive and negative is negative.

Similarly, if the quantities 2 and 2̇ are multiplied by a negative 3 (3̇): in the first case, the product of a positive and negative is 6̇. In the second case, if the product of two negatives is not accepted as positive, how can the sum of the multiplied 6̇ and 6̇ be zero? Therefore, it is established that the product of two negatives is positive.

This rule is also proven by the method of multiplying by a quantity augmented or diminished by an arbitrary number. For example: multiplicand 15, multiplier 8̇. Multiplier diminished by arbitrary number 3 is 11̇. Multiplicand multiplied by this: 165̇. This is considered negative due to the product of positive and negative (165̇).

Then, if we add 45 (the multiplicand multiplied by the arbitrary number 3) to it: the sum of the positive and negative is their difference, resulting in 120̇.

If one did not accept that the product of positive and negative is negative (165̇), this result would be 210, which is incorrect, as 15 times 8 is 120.

Similarly, multiplicand 15, multiplier 8̇. Here, the multiplier is augmented by 3 to get 5̇. The multiplicand multiplied by this is 75̇. Subtracting 45 (the multiplicand multiplied by the arbitrary 3) from this results in 45, and adding it to the previous 75̇ results in 120̇. This is only possible if the product of two negatives is accepted as positive. Thus, it is proven.