The year is 628 CE. The city is Srimala, the crown jewel of the Gurjara Empire. To the merchant, it is a city of sandalwood and silk, for pilgrims, it was a thriving hub of worship, for philosophers and astronomers, it was a centre for learning. The city crawled with administrators, courtiers, scribes, soldiers, craftsmen, merchants and priests. Everyone knew somebody who walked the corridors of power in the political centre of a vast, independent kingdom, and it was impossible to throw a stone without hitting a mathematician of some kind.
At the age of thirty years, Brahmagupta was one such mathematician. His mind operated at a different frequency, it was as if he could see the underlying order in the chaos of nature. As the director of the local observatory, his life was governed by a simple, innocuous staff called the sanku. This was a vertical rod, and Brahmagupta could decipher the geometry of the heavens by carefully monitoring the shadow it cast. As far as the Gurjara Empire was concerned, Brahmagupta was the lord of time itself, and all his power and influence was derived from a shadow cast by a stake across a dusty courtyard.
The air in the observatory was thick with the scent of old ink and palm leaves, as well as the ionized smell of a receding monsoon. Brahmagupta sat cross-legged, his fingers stained with chalk. Before him lay the Corrected Doctrine of Brahma, or the Brahmasphutasiddhanta. The work was primarily concerned with reconciling predicted planetary parameters, motion models, and eclipse computations with observations, with some discrepancies and shortcomings in the conventional understanding. It is both a revisionist treatise and a foundational text at the same time. To allow for more accurate computations, Brahmagupta suggested something revolutionary.
Brahmagupta had a fierce, competitive streak that made him provocative. He had publicly dismantled the errors of older mathematicians, mocking those who could not accurately predict the motions of planets. He challenged the notions of his predecessors, including those of Aryabhata and Varāhamihira. Most of his ideas were misguided, and history would prove them wrong, but Brahmagupta stuck by them. He maintained that the Earth was fixed, as its rotation would cause objects to rise or fall. He believed that the planets were moved by winds or divine forces. He advocated for tracking time from sunrise, instead of midnight. He also believed that it was possible to divide by zero.
Throughout the day, Brahmagupta diligently tracked the shadow of the sanku. As the Sun dipped below the horizon, Brahmagupta's true work began. He used a copper vessel filled with water, with a precise hole in the bottom, to measure the flow of time, known as a jalghati. His daily life was a rigorous choreography of data collection, and his biggest fears was of a loss of data. If the calculations of say a lunar eclipse was off by even a nimesha, or the blink of an eye, he knew that there was something wrong in the entire celestial model. He saw the universe as a grand machine, but there was a hole in it, that he would fill with nothing.
One evening, struggling to calculate the longitudinal drift of Mars, Brahmagupta closed his eyes, and drifted off in a vision. The motions of the planets in the sky, the regular rhythm of the day-night-cycle, the vast collection of stars, all became an interconnected web of ratios. The numbers and coordinates were scattered, chaotic, without a centre. A new entity arose, a circle, perfect and empty, yet infinitely heavy with potential. It was like the Sun of knowledge, illuminating the darkness of ignorance, and it shone brighter than all the stars in the sky. This was the purest form of truth and logic, the light of mathematics. In his fever-dream, this Sun anchored the universe, provided a place for infinity to start, the Zero Point.
Brahmagupta realised that mathematics was the fundamental language of reality. If you have five cows and take away five, you are left with a value, not a gap. If you are five coins in debt, and gain five coins, you reach a point of equilibrium. This state, Brahmagupta decided, needed a name, a set of laws, a seat at the table of mathematics, a point allocated on the continuum of numbers. Brahmagupta began to chant. His mathematical treatise had no equations, only Sutras. In the ancient tradition, science was oral, it had to be portable. Metred, Sanskrit poetry was easier to memorise than equations. As his chants transformed to song, the rhythm helped Brahmagupta lock the logic into memory... the memory of a civilisation.
A debt minus zero is a debt.
A fortune minus zero is a fortune.
The product of zero and a negative or positive is zero.
Brahmagupta knew the concept would provoke traditionalists who feared the very idea of non-existence. He relished the thought. He had discovered an overlooked number. As the real dawn broke over Srimala, Brahmagupta recorded the final verses of the eighteenth chapter of his treatise. His predecessors had used dots to mark empty places in columns. For example, 105 would be marked as 1•5, where there was no number in the 10s column. Brahmagupta recognised the logical friction of an incomplete system. Treating nothing as a gap was a scientific loose end.
He would never know that his Sun of Knowledge would travel the Silk Road to Baghdad, where Al-Khwarizmi would use it to birth Algebra. He would never have seen Fibonacci carrying his Hindoo Numerals to a skeptical Europe, or Leibniz and Newton using his zero to invent Calculus, that requires limits that approach zero. He certainly would never have imagined a future where his invention would be essential in the binary code of a quantum computer, or to accurately describe the singularity at the hearts of black holes.
"A mathematician among other mathematicians," he had once written with a smirk, "is like a lion among deer."
