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How Brahmagupta formalised rules for negative numbers in 628 CE — 1000 years before Europe accepted them — and why Descartes called them "false"
What Brahmagupta did in 628 CE was revolutionary: he imagined a number less than "nothing." From dhana (assets) to rina (debt) — this was a mathematical leap that would frighten Europe for another 1000 years.
Addition: dhana + dhana = dhana. rina + rina = rina. dhana + rina = the sign of the larger magnitude.
Subtraction: rina − rina = dhana or rina (whichever is larger). dhana − rina = dhana.
Multiplication: dhana × dhana = dhana. rina × rina = dhana. dhana × rina = rina.
With zero: zero + rina = rina. zero − rina = dhana. zero × rina = zero.
Indian merchants had recorded dhana (assets) and rina (debts) in account books for millennia. Two colours of ink — black for assets, red for debts — were used. "Being in the red" remains an English idiom for being in deficit, a direct inheritance of this accounting tradition.
Brahmagupta gave this practical accounting a mathematical language. "What is my net worth if I have 5 gold but owe 8 gold?" → 5 + (−8) = −3. Simple, practical, revolutionary.