Algebra of Bhaskara II

But if one asks if that sum is in the west or east: by changing the sign of the 7-krośa debt (westward) to a positive, the sum becomes 15 + 7 = 22, representing the distance between the first and second city.

Similarly, if 15 in the eastern part is to be subtracted, then by making 15 negative, the debt of 7 + 15 = 22 krośas represents the second city located to the west of the first city. The same applies elsewhere.

If two cities are located in the same direction from Varanasi, the distance is simply their difference.

Another proof provided by modern writers is recorded here. For example, if 3 coins are to be subtracted from Devadatta's 7 coins: the 7 coins are (4 + 3). Subtracting 3 leaves 4. Similarly, if a debt of 3 coins (3̇) is to be subtracted from a debt of 7 coins (7̇): 7 coins as a debt plus the 3 coins results in a remainder of 4 coins. The result is achieved by reversing the sign of the subtrahend. Thus, if a debt of 3 coins is to be subtracted from 7 coins: by adding the 3-coin debt to the 7 positive coins, the result is 10. The sign of the subtrahend (3) was reversed. Similarly, if 3 coins are to be subtracted from a debt of 7 coins, by treating the 7 as a debt (7̇) and adding 3, the difference is 10. The sign of the subtrahend 3 was reversed to 3̇. Adding this to the 7̇ results in 10̇.

Vimalā (Commentary): A positive quantity being subtracted becomes negative, and a negative quantity being subtracted becomes positive.

Modern Proof:

Imagine a person from a merchant family who has 4 coins in profit. He wishes to subtract 2 coins, meaning he wants to give them to someone else. Thus, only 2 coins remain in his possession.
Similarly, if he had a debt of 4 coins, and after some time he gained 2 coins, and someone told him, "Pay off your debt with this," and he decided not to take any more, then in that calculation, he currently has a debt of 2 coins.
If a person had a debt of 4 coins, and he borrowed another 2 coins and spent them, he would have a debt of 6 coins.
If he had 4 coins in profit, and someone gave him a debt of 2 coins and said, "I will not accept your payment," and he kept it, then he would have 6 coins in profit. Thus, everything is consistent.