Negative Numbers: Rna and Dhana (Debt and Wealth)

How Indian mathematicians embraced what others feared

Learn how Brahmagupta and Bhaskara II developed rules for negative numbers using the intuitive concepts of debt (rna) and wealth (dhana).

The Wealth and the Debt at Bhillamala

In the year 628 CE, in the walled city of Bhillamala on the western edge of the Indian subcontinent, the thirty-year-old astronomer Brahmagupta sits at his desk in the observatory of the Cāpa kings and writes a sentence that will quietly govern arithmetic for the next fourteen hundred years.

The sentence is short. The sum of two debts, he writes in compressed Sanskrit verse, is a debt. The product of two debts is a wealth. He uses the words ṛṇa for debt and dhana for wealth, the same words a merchant in the bazaar outside the observatory would use for the silver in his pouch and the grain he owed after a bad harvest. The carbon ink dries on the palm leaf. Brahmagupta moves to the next verse, gives the rule for division, then the rule for zero, and continues without pause.

He does not know it yet, but he has just settled, in a single chapter, a question that will torment European mathematicians for the next thousand years.

Nine centuries later, in 1545 CE in Milan, Girolamo Cardano will publish his Ars Magna and call quantities less than nothing numeri ficti, fictitious numbers. The German mathematician Michael Stifel will call them numeri absurdi. As late as 1758, the English mathematician Francis Maseres will declare that negative quantities 'darken the very whole doctrines of the equations' and should be driven out of algebra for the sake of clear thinking. Maseres is writing in the same language as Newton, in the same century as the Industrial Revolution, more than eleven centuries after the question has already been settled in India by a single Sanskrit astronomer.

This is the story of how Brahmagupta did it, and why everyone else took so long.