Brahmagupta's fortunes and debts

The sign rules we worked out from tokens and patterns are old, and they were first written down in India.

Just as for addition and subtraction of integers, Brahmagupta in his Brāhmasphuṭasiddhānta of 628 CE articulated explicit rules for integer MULTIPLICATION and DIVISION.

He did not use our words 'positive' and 'negative'. He used the notions of FORTUNE — dhana — for positive values, and DEBT — ṛṇa — for negative ones. The idea is easy to feel: money you have, against money you owe.

In verses 18.30 to 18.32 he wrote four statements. 'The product or quotient of two fortunes is a fortune.' That is our rule that positive times positive is positive.

'The product or quotient of two debts is a fortune.' That is negative times negative giving a positive — the rule that most people find surprising, stated plainly nearly fourteen hundred years ago.

'The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt.' Those two are our rule that unlike signs give a negative, stated both ways round — which is also a quiet statement that the order does not matter.

This represented the FIRST time that rules for multiplication and division of positive and negative numbers were articulated, and it was an important step in the development of arithmetic and algebra. The rules you proved with red and green tokens are the same rules, arrived at the same way: by thinking carefully about what the operations must mean.