Bhaskara II wasn’t just a guy who liked math. He was the leading mathematician of the 12th century. Born in 1114 in Biddur, India, he died around 1185, probably in Ujjain. His work changed how we count. He wrote the first work to use the decimal number system fully and systematically.
Before him, people used symbols that didn’t flow like our digits do today. Bhaskara II standardized this. He wasn’t working in isolation either. He was the lineal successor of Brahmagupta, a famous mathematician from the 6th and 7th centuries. Bhaskara II even took over Brahmagupta’s job as head of the astronomical observatory at Ujjain. This city was the leading mathematical center of ancient India. We call him Bhaskara II to distinguish him from the original astronomer of the same name.
Solving the Hard Parts of Algebra
His main mathematical works, Lilavati and Bijaganita, are written in verse. This was standard for Indian mathematical classics at the time. But the content inside was revolutionary. He didn’t just list rules. He filled in the gaps left by Brahmagupta.
One huge gap was the Pell equation. This is a type of indeterminate quadratic equation. It looks like this: $x^2 = 1 + py^2$. Brahmagupta hadn’t found a general solution. Bhaskara II did.
Bhaskara II provided the first general solution to the Pell equation, a problem that stumped European mathematicians for centuries.
He gave specific answers too. Take $x^2 = 1 + 61y^2$. The solution is massive. $x = 1,766,319,049$ and $y = 226,153,980$. That’s a big number. Five hundred years later, in 1657, French mathematician Pierre de Fermat used this exact problem as a challenge to his friend Frenicle de Bessy. Bhaskara II had already solved it.
He also anticipated modern algebra conventions. He used letters to represent unknown quantities. This is exactly what we do now. He solved indeterminate equations of first and second degrees. He reduced quadratic equations to a single type. He solved them.
Understanding Zero and Infinity
Bhaskara II played with ideas about zero that were dangerous to contemplate. He was the first to gain some understanding of the meaning of division by zero. He stated that $3/0$ is an infinite quantity.
It wasn’t perfect. He got it wrong later on. He also stated that $a/0 \times 0 = a$. That’s incorrect. But the attempt itself was groundbreaking. He was probing the limits of logic.
He also standardized signs. Minus by minus makes plus. Minus by plus makes minus. Before him, these rules weren’t always clear in texts. He made them consistent.
Approximating Pi with Polygons
How did he find the value of pi? He investigated regular polygons. He went up to those having 384 sides. By doing this, he obtained a good approximate value of $\pi = 3.141666$.
That’s accurate to three decimal places. In the 12th century, that was sharp. He didn’t just guess. He calculated.
Astronomy and the Water Clock Legend
His astronomical works, like Siddhanta Shiromani and Karana Kutuhala, cover planetary positions, conjunctions, eclipses, cosmography, and geography. He also wrote about the mathematical techniques and equipment used in these studies. He was a noted astrologer too.
There’s a legend about Lilavati. A 16th-century Persian translation first recorded it. The story says he named his first work after his daughter to console her. She was unhappy, maybe about marriage. He tried to determine the best time for her wedding using a water clock.
The device was a cup with a small hole in the bottom. It floated in a larger vessel. The cup would sink at the beginning of the correct hour. Lilavati looked into the clock. A pearl fell off her clothing. It plugged the hole. The cup never sank. She missed her chance for marriage and happiness.
Is it true? Who knows. Some problems in Lilavati are addressed to women. They use feminine vocatives like “dear one” or “beautiful one.” That suggests he kept her in mind, even if the legend is just a story.
Bhaskara II left behind a legacy that feels surprisingly modern. We use his decimal system every day. We solve equations he defined. We look at stars using methods he refined. The connection is real. It’s not just history.
