I’ve always wanted to travel to India, and I’m finally getting a chance to visit Chennai (along with some other places) this winter break. I’ll be teaching my company’s Chennai, India team about service oriented architecture automation – aka boring computer stuff. However, I’ve also set some time aside to go sightseeing on the company’s dime! We always seem to bring up India-birthed math topics, or mathematicians in class, so I thought it would be very fitting to blog about how India has impacted us! Make sure you get your Tetanus, Diphtheria, and Typhoid booster shots, this journey may get a little out of hand!
*Spoiler alert: You can’t contract any foreign diseases from a blog post.
When I think of India, computer software, call centers, spicy food, and the Taj Mahal come to mind. After making my way past these generalizations, I started to see how crucial this South Asian country’s mathematical contributions have been to mankind. India has been credited with giving the world many important mathematical discoveries and breakthroughs – place-value notation, zero, Verdic mathematics, and trigonometry are some of India’s more noteworthy contributions. This country has bred many game-changing mathematicians and astrologists. Over the course of my research I identified the “big three” mathematicians. The first, and arguably most important mathematician and astronomer (Ancient astronomers are similar to modern day astrologist!) in India’s history, was Aryabhata. Soon after Aryabhata, came Brahmagupta. Brahmagupta followed in Aryabhata’s footsteps and built upon some of his more groundbreaking theories. Nearly 500 years later Bhaskara II (Not to be confused with Bhaskara I.) was born. While building upon the mathematical and astronomical work of his forefathers, Bhaskara II also paved his own way to become one of the “greats”. The “big three’s” findings, laid down some of the most vital building blocks in the history of mathematics, but how has that impacted us?
We will start off on this journey with Aryabhata (sometimes referred to as Arjehir), a well-known astrologist and mathematician, born in the Indian city of Taregana sometime between 476-550 AD. He lived during a time period we now refer to as “India’s mathematical golden age” (400-600 AD), and it is of no surprise why historians recognize this time period; Aryabhata’s achievements really were golden. He is most noted for dramatically changing the course of mathematics and astronomy through many avenues, which he recorded in a variety of texts.
Over the course of many wars and centuries, only one of Arybhata’s works survived. Aryabhatiya, which was written in Sanskrit at the age of 23, recorded the majority of his breakthroughs. Oddly enough, he only referenced himself 3 times throughout his work. Within this text, Aryabhata formulated accurate theories about our solar system and planets, all without a modern-day telescope. He recognized that there were 365 days in a year. He developed simplified rules for solving quadratic equations, and birthed trigonometry. Aryabhata’s original trigonometric signs were recorded as “jya, kojya, utkrama-jya and otkram jya” or sine, cosine, versine (equivalent to 1-cos(θ) ). He worked out the value of as well as the area of a triangle. Directly from Aryabhatiya he says: “ribhujasya phalashariram samadalakoti bhujardhasamvargah”. This translates to: “for a triangle, the result of a perpendicular with the half side is the area”. Most importantly, in my opinion, he created a place value system for numbers. Although in his time, he relied on the Sanskritic tradition of using letters of the alphabet to represent numbers. Aryabhata did not explicitly use a symbol for zero however. It kind of hard to conceptualize, but none of these things had ever been done, at least to this extent, before.
Brahmagupta was born in Bhinmal, India presumably a short time after Aryabhata’s death in 598 AD. He wrote 4 books growing up, and his first widely accepted mathematical text was written in 624 when he was only 26 years old! I find it funny that most of the chapters in his texts were dedicated to disproving rival mathematicians’ theories. Brahmagupta’s most notable accomplishments were laying down the basic rules of arithmetic, specifically multiplication of positive, negative, and zero values. In chapter 7 of his book, Brahmasphutasiddhanta (Meaning – The Opening of the Universe), he outlines his groundbreaking arithmetical rules. In the context below, fortunes represent positive numbers, and debts represent negative numbers:
A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
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.
However it seems Brahmagupta made some mistakes when explaining the rules of zero division:
Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.
Since our early teens we’ve know anything divided by zero is not zero. When zero is the denominator, the fraction will always “fall over” – that’s how I learned it as a youngin! However, we still have to give Brahmagupta credit, he was so close to getting it all right.
Bhaskara II is similar to the other mathematicians we’ve discussed in this post. He was born in 1114 AD, in modern day Karnataka, India. He is known as one of the leading mathematicians of India’s 12th century. He blessed the world with many texts but Siddhanta Shiromani, and Bijaganita (translates to “Algebra”) are the ones that have shined through the centuries. These specific texts documented some of his more important discoveries. In Bijaganita, Bhaskara demonstrated a proof of the Pythagorean theorem, and introduced a cyclic chakravala method for solving indeterminate quadratic equations:
y = ax2 + bx + c
Coincidentally, William Brouncker was credited for deriving a similar method to solve these equations in 1657, however his solution is more complex. From Siddhanta Shiromani, Bhaskara gave us these trigonometric identities:
sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
sin(a – b) = sin(a) cos(b) – cos(a) sin(b)
If I had a dollar for every time I relied on these identities, or any of their variations throughout my mathematical career, I’d probably have enough money for a new laptop! Although Newton and Leibniz are credited for “inventing” calculus, Bhaskara had actually discovered differential calculus principles and some of their applications.
A World Without Aryabhata, Brahmagupta and Bhaskara II
I know this is a long shot, but let’s entertain the idea of a world without any of Aryabhata’s, Brahmagupta’s, or Bhaskara’s work. Granted, future mathematicians would have undoubtedly discovered a portion of the “big three’s” breakthroughs, at least in one way or another. While it’s pretty obvious someone else would’ve invented a number system with a placeholder, or a zero equivalent, it’s not as clear with more complex things such as trigonometry. The foundation built by the “big three” could’ve altered slightly. This alteration could’ve given us a Leaning Tower of Pisa rather than an Eiffel tower – metaphorically speaking, that is. The main point you have to realize is: without the “big three” the progression of mathematics would have been slowed in one way or another, thus effecting our world today. If the “big three” didn’t exist there’s no telling how far back it could’ve set humanity.
That being said, these mathematicians’ theories, methods, and proofs served as building blocks for other mathematicians (globally). If you want to build out a brilliant theorem or proof, you have to start with, or at least incorporate the basics, at some point. Without these basics, the world would have been set back, at least in the realm trigonometry and algebra. It’s hard to imagine using any other number system than what we use today, especially without a numerical placeholder! Young children would be less eager to learn math because writing down large numbers would be a tedious process. What would we have used in place of zero? What about math with negative numbers?
Trigonometry electrifies our lives and rings in our ears. I think it is the biggest part of Aryabhata’s work that we take for granted. Without his trigonometric discoveries we wouldn’t have useful conventional electricity. The natural flow of alternating current, or AC current, is represented by the sine function. Electrical engineers and scientists use this function to model voltage and build the electronics we use every day. Alternating current primarily comes from power outlets, but it can also be synthesized in our electronic devices. Trigonometry is also extremely relevant today in music. Sine and cosine functions are used to visualize sound waves. This is especially important in music theory and sound production. A musical note or chord can be modeled with one or many sine waves. This allows sound engineers to morph voices and instruments into perfect harmony. However, Aryabhata is to blame for all that auto-tuned, T-Pain nonsense we hear on the radio! Lastly, trigonometry has a strong presence in modern day architecture. It’s a necessity when building complex structures and designs. We’d have to say goodbye to beautiful architecture and reliable suspension bridges if it weren’t for Aryabhata.
History of Mathematics – BBC: