Algebra of Bhaskara II

The Rule for Addition of Positive and Negative Numbers:

Addition is the union of two of the same kind; the difference of positive and negative is the sum.

Commentary: Now, using the first half of the Upajātikā a specific poetic meter verse, the text explains the addition of positive and negative numbers. When both are negative, or both are positive, they become a combined negative or positive total. In this case, one should perform the addition of these numbers in the ordinary way—either sequentially or inversely—and that is the sum. It should be understood that for karaṇī surds/radicals, addition or subtraction must be performed according to the rule: "Having designated the larger as the sum of the surds." The same applies to many numbers. This explains the addition of like kinds. Where one number is positive and the other negative, the rule is: "The difference of positive and negative is the sum." Whatever the remainder is, that is the sum of the positive and negative numbers. One must understand that the sum is also positive or negative depending on the nature of the remainder.

Proof:

If Devadatta borrows a debt of 3 coins from Yajnadatta, and then borrows another 4 coins, a total debt of 7 coins is created. Similarly, if one earns 3 coins and then earns another 4 coins, the sum of the two is a profit of 7 coins. If one has a debt of 4 coins and earns 3 coins, the remainder is a debt of 1 coin. If one has 4 coins in profit and incurs a debt of 3 coins, there is 1 coin of profit remaining. This logic is easy to understand.

Vimalā (Commentary): Methods for Adding Algebraic Quantities:

The addition of two positive or two negative quantities should be performed according to the formula "Perform the addition sequentially or inversely," as established in the Līlāvatī a classic mathematical text. If one quantity is positive and the other negative, the remainder obtained by finding their difference is the sum. However, if a positive remainder stays, the result is positive, and if a negative remainder stays, the result is negative. In this way, one should add two, three, or four or more quantities. Like quantities (such as adding yāvat unknown x to yāvat, or kālaka unknown y to kālaka) are added together. If one wishes to add two or more karaṇī surds, it should be done according to the rule: "Having designated the larger as the sum of the surds."

Modern Proof:

If a person had a debt of 5 coins from before, and after some time borrowed another 5 coins, he now has a debt of 10 coins. Thus, it is proven that the addition of two negatives is a union.
Now, if a person had 1 coin and earned 10 coins, he has 11 coins in profit; thus, it is proven that the addition of two positives is a union.
Now, if a person had a debt of 5 coins inherited from his father, and after some time earned 10 coins and paid off the debt, he now has 5 coins remaining; thus, it is proven that the difference of positive and negative is the sum.