Category Archives: Indian mathematics

The Development of Zero

How many elephants are in the same room as you right now? Most people would answer zero to that question (if you answered something else, we should be friends). The concept of zero is familiar to us. Earlier today, my two-year-old cousin told me that his baby sister is zero years old. I filed sales taxes for my business and typed up countless zeros. Today, zero is part of daily life. Even a two year old understands the concept of zero.

Zero is nothingness — a void. If you think deeper, it’s fairly amazing that we throw around such a profound term. I can see, touch and count the number of teabags left in a box, but I can’t see, touch or count the number of elephants in my bedroom. There are also zero storm troopers, zero cookies and zero dinosaurs in my bedroom. In my bedroom, there are an infinite number of zeros. Our number zero, symbolized by “0,” enables us to do calculus, and it’s even half of the reason my computer works right now. In the early days of math, zero didn’t exist — there wasn’t even a word for it, which made even simple arithmetic a bit complicated. Thankfully, ancient Babylonian, Mayan and Indian mathematicians developed the concept of zero and paved the road for truckloads of discovery and innovation.

Just like ours, the Babylonian number system (2000 BC) was positional. In our base 10 system, having a positional number system simply means you have a position for ones, tens, hundreds, etc. Babylonians used the same concept except their ones position included the numbers 1-59 instead of 1-9. Regardless of base, the problem with having no zero is the numbers ‘11’ and ‘101’ suddenly both look like ‘11’. Most people can’t read minds, so that makes understanding other people’s writings a bit difficult. The Babylonians developed a place holding symbol to solve this dilemma. For example, if we used a period as a placeholder, those numbers would look like ‘11’ and ‘1.1’. It dispersed some confusion, but the placeholder could only be used between numbers, so ‘1’ and ‘100’ both looked like ‘1’. Without a zero, modern mathematics had no chance of developing.

Mayan placeholder symbol. Image: public domain via Wikimedia Commons.

Similarly to the Babylonians, the Mayans developed a placeholder symbol that stood for zero. They developed the notion completely independently of the Babylonians — after all, they were half way around the world and didn’t have texting. Their symbol for zero supposedly looks like a shell. To me, it looks more like a spaceship, but I digress. They had the concept of a placeholder, but like the Babylonians, they didn’t use the symbol on its own. Again, its a start, but you can’t add, subtract or multiply using a placeholder.

Brahmagupta, an Indian mathematician and astronomer. Image: public domain, via Wikimedia Commons.

A 19th century image of Brahmagupta. Image: public domain via. wikimedia commons.

The hero of this story is a Hindu astronomer by the name of Brahmagupta. Around 628 AD, Brahmagupta wrote down rules for getting to zero using addition and subtraction and the results of using zero in equations. There are earlier traces of zeros in Cambodia and various parts of India, but Brahmagupta’s account is primary because it gave the rules behind using zeros. Brahmagupta called zero ‘sunya’ or ‘kha’ which mean ‘empty’ and ‘place’ respectively. His rules included things like ‘the sum of two zeros is zero’, ‘the product of a zero and any other number is zero’, and ‘zero divided by a zero is zero’. These rules were revolutionary. As simple as they seem, this one list of rules effectively changed the entire human world. You may have noticed something wrong with one of those rules — our modern mathematics don’t allow you to divide by zero. Brahmagupta’s rules about dividing by zero may have been flawed, but that just means he left something for G.W. Leibniz and Isaac Newton to work on later!

After zero became a fully formed number, it spread like wildfire. Along with spices and other tradable goods, Arabian voyagers brought zero back from India. A hundred years after Brahmagupta discovered zero, it reached Baghdad. In the 9th century, a man named Mohammed ibn-Musa al-Khwarizmi started to develop algebra by working on equations that equaled zero. He called zero ‘sifr’ which turned directly into our word ‘cipher’ and eventually developed into our word ‘zero’. Come 879 AD, people wrote zero almost exactly like we do today; the only difference between our zero and theirs was size. They used an oval that was smaller than the other numbers — it became ‘1’, ‘1o’ and ‘1oo’. Finally, when the Moors invaded Spain they brought zero to Europe, and by the mid-1900s, Al-Khowarizmi’s work reached England at last.

Zero is universal; it transcends culture, space and time. It is part of our global language and is one of the most fundamental ideas in calculus, physics, engineering, computers, and a lot of financial and economic theory. Our lives are full of zeros. Plus, after traveling around the entire world and changing the course of human history, zero inspired this brilliant little video. Enjoy!


References:

http://yaleglobal.yale.edu/about/zero.jsp

http://www.livescience.com/27853-who-invented-zero.html

http://en.wikipedia.org/wiki/0_%28number%29#Americas

http://www.mediatinker.com/blog/archives/008821.html

http://blogs.discovermagazine.com/crux/2013/05/20/how-i-rediscovered-the-oldest-zero-in-history/#.VORI8_nF98E

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Ancient India’s Mathematical Impact On The World

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, Diphtheriaand 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?

Aryabhata

An artist’s rendition of Aryabhata. Image: Public domain, via Wikimedia Commons.

Aryabhata

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.

Sanskrit writing. Image: Diggleburnz, via Flickr.

Sanskrit writing. Image: Diggleburnz, via Flickr.

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 workWithin 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

Brahmagupta, an Indian mathematician and astronomer. Image: public domain, via Wikimedia Commons.

Brahmagupta. Image: public domain, via Wikimedia Commons.

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

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.

References:

http://www.shalusharma.com/aryabhatta-the-indian-mathematician/

http://en.wikipedia.org/wiki/Aryabhata

http://www-history.mcs.st-and.ac.uk/Biographies/Brahmagupta.html

http://www.famous-mathematicians.com/brahmagupta/

http://www.clarku.edu/~djoyce/trig/apps.html

http://www.winentrance.com/general_knowledge/scientists/bhaskara-ii.html

History of Mathematics – BBC:

https://www.youtube.com/watch?v=pElvQdcaGXE

Marveling at the Indian Genius: Ramanujan

Hardy left the cab. He was visiting an ill Ramanujan at Putney. Ramanujan asked about his trip. Hardy remarked the ride was dull; even the taxi number, 1729, was dull to the number theorist. He hoped the number’s dullness wasn’t an omen predicting Ramanujan’s declining health. “No Hardy,” Ramanujan replied, “it is a very interesting number.” The integer 1729 is actually the smallest number expressible as the sum of two positive cubes in two different ways. Indeed 1729 = 13 + 123 = 93 + 103.

leaf graphic
Ramanujan. Picture courtesy of Konrad Jacobs. From Wikimedia Commons

The mathematicians Srinivasa Iyengar Ramanujan and G.H. Hardy comprise the characters of this story. Ramanujan’s casual discovery of the smallest so-called taxicab number was no fluke. Ramanujan and Hardy’s most famous result was an asymptotic formula for the number of partitions of a positive integer. A partition of a number n is a way of writing n as the sum of positive integers. Reordering terms doesn’t change a partition. Thus the partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1; hence 5 has 7 seven partitions. Counting the partitions becomes difficult as n grows. Ramanujan had made several conjectures based on numerical evidence, and Hardy credits many of the needed insights to Ramanujan. The formula is complicated and counter-intuitive. It involves values of √3, π, and e, all very strange numbers for a counting formula. Near his death, Ramanujan discovered mock theta functions, which mathematicians are still “rediscovering” today.

These are only the achievements of later Ramanujan. The early life of Ramanujan is even more surprising. He spent his childhood in the poor south Indian town, Kumbakonam. In school, he scored top marks. He probed college students for mathematical knowledge by age 11. When he was 13, he comprehended S.L. Loney’s advanced trigonometry book. He could solve cubic and quartic equations—the latter method he found himself—and finished math tests in half the allotted time. If asked, he could recite the digits of π and e to any number of digits. He borrowed G.S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics by age 16 and worked through its many theorems. By 17, he independently investigated the Bernoulli numbers, a set of numbers intimately connected to number theory.

So How did Ramanujan do it? No brilliant teacher—excluding Hardy and J.E. Littlewood when Ramanujan was already an adult—taught Ramanujan. The odds were against Ramanujan from the start. Ramanujan often used slate instead of paper because paper was expensive; he even erased slate with his elbows, since finding a rag would take too long. He lost his first college scholarship by neglecting his every subject that wasn’t math. The first two English mathematicians he sent letters to request publication did not respond; was it luck that Hardy did?

Ramanujan credits Namagiri, a family deity, for his mathematics. He claimed the goddess would write mathematics on his tongue, that dreams and visions would reveal the secrets of math to him.

Ramanujan was undoubtedly religious. Before leaving India, he respected all the holy customs of his caste: he shaved his forehead, tied his hair into a knot, wore a red U with a white slash on his forehead, and refused to eat any meat. In the Sarangapani temple, Ramanujan would work on his math in his tattered notebook.

Hardy, the ardent atheist, didn’t believe that gods communicated with Ramanujan. He thought flashes of insight were just more common in Ramanujan than in most other mathematicians. Sure, Ramanujan had his religious quirks, but they were just quirks.

Looking at Ramanujan’s work, I find it hard to deny that some god aided the man. I cannot even imagine Ramanujan’s impact had he lived longer or had better teachers.


Sources and cool stuff:

biographical stuff: http://www-history.mcs.st-andrews.ac.uk/Biographies/Ramanujan.html, Christopher Syke’s documentary Letter’s from and Indian Clerk, Robert Kanigel’s The Man Who Knew Infinity

Ramanujan and Hardy’s original paper on partitions: http://plms.oxfordjournals.org/content/s2-17/1/75.full.pdf

Numberphile on taxicab numbers: https://www.youtube.com/watch?v=LzjaDKVC4iY

Wolfram on taxicab numbers: http://mathworld.wolfram.com/TaxicabNumber.html

Wolfram on Hardy-Ramanujan number: http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

Wolfram on Bernoulli numbers: http://mathworld.wolfram.com/BernoulliNumber.html

Ramanujan’s papers and notebooks: http://www.imsc.res.in/~rao/ramanujan/contentindex.html

mock theta functions: http://www.ams.org/notices/201011/rtx101101410p.pdf

G.S. Carr’s book: http://books.google.com/books?id=JLmCAAAAIAAJ&hl=en

S.L. Loney’s book: https://archive.org/details/planetrigonomet00lonegoog

Numberphile on Ramanujan constant: https://www.youtube.com/watch?v=DRxAVA6gYMM

Exploring Limits

In calculus, a limit is defined as the value of a function as it approaches some point. Sometimes, a function has no finite limit at a point because it just keeps growing, and we say the limit is infinite. In this case, the function never reaches the limit but the value grows arbitrarily large as it gets nearer and nearer to the limit. In our reading, I have been considering limits in a different light. I have been thinking about the limits of civilizations as they progress in their development of mathematics. Some civilizations seem to reach a limit of understanding and because of cultural restraints, their limited number systems, or even because they outwardly reject an idea, they stop progressing. Fortunately, sometimes their discoveries shape and influence other cultures and, as a whole, progression continues. I would like to explore different limits in the progress and development of mathematics and consider what limits us today.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

In ancient Mesopotamia, more than 4000 years ago, the Babylonians used the base of 60 to develop a high level mathematical system. They developed positional notation and could use fractions as well as whole numbers. They developed systems to figure square roots. Clay tablets from that time show tables with logarithms, multiplication facts and reciprocal pairs. There is information about calculating compound interest and solving quadratic equations. Writings on the tablets suggest that math was a subject that was taught and studied. In many ways, they seem to have exceeded the capabilities of other civilizations that came much later in history. No one can question that their accomplishments were amazing, to say the least, and perhaps influenced other cultures. However, because most of their mathematics were only for very practical purposes like conducting business, surveying land and constructing buildings, they stopped short of exploring some of the deeper meanings of things. For example, our text points out, “In the Babylonian square-root algorithm, one finds an iterative procedure that could have put the mathematicians of the time in touch with infinite processes, but scholars of that era did not pursue the implications of such problems.” (Merzbach and Boyer, pg. 26) What might have been the implications if they had? As they approached the limit, they stopped rather than exploring the infinite possibilities. They stood on the brink of even greater discovery, but did not pursue it.

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

One of the most dramatic examples of cultural influences limiting the progress of mathematics is the example of the progression of Indian positional decimal arithmetic to Europe. Mathematicians in India had developed a number system with ten digits, including zero, and used it to develop methods of computing fractions, square roots and π. In the tenth century, Gerbert of Aurillac attempted to introduce the system to Europe. He had learned the system first hand from Arab scholars in Spain.   However, he was rejected and during this time of the Crusades in Europe, he was rumored to be sorcerer. He died after a short reign as Pope Sylvester II. “It is worth speculating how history would have been different had this remarkable scientist-Pope lived longer” (Bailey and Borwein, 6).” The Indian system was reintroduced 200 years later by Leonard of Pisa, but was rejected again and considered “diabolical”. It wasn’t until the beginning of the 1400’s that scientists began using the system. “It was not universally used in European commerce until 1800, at least 1300 years after its discovery” (Bailey and Borwein, pg. 6). While many other areas of the world were able to do complicated computations using the Indian system, Europe, because of its cultural restraints, was still laboring with Roman numerals. Imagine what the brilliant minds of the Europeans might have discovered or developed if they had the ease of the Indian number system? In this case their culture may have created a limit that kept them from infinite discoveries.

Today in our world we have amazing tools to help us progress. Not only do we have the combination of a well-developed number system, thousands of theorems and laws and the knowledge of centuries of learning, we also have technology that assists in remarkable ways. Indeed we have all the tools of the past plus the technology of our day. However, are there things yet to be discovered, or have we reached a limit? Are there obstacles in our society or ways of thinking that limit us? As recently as the early 1900, women had a difficult time pursuing their mathematical interests. Even today, women and minorities continue to be underrepresented in the math and science fields. What might have been the result if woman had been afforded the same educational opportunities as men over the years? Do we limit ourselves by the way we approach math? Are there different number systems or “languages of math”? In recent years, computer scientists have given us other “languages” for coding. Are there similar languages for math? The challenge for our day is to not be content and accept that what has been learned is all there is.

In our reading for class I have been amazed at how often a group or civilization is on the brink of great mathematical discovery, but because of varying reasons they stop short of the mark. Sometimes cultural influences limit the progress and other times it seems individuals do not look far enough to find deeper meaning or answers. It is true that hindsight may be twenty/twenty, but I can’t help wondering what future civilizations may look back on and see that we barely missed. What are we on the brink of discovering if only we would look forward and push closer and closer to the undefined limits?

Sources:

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Bailey, David H., and Johnathan M. Borwein, “The Greatest Mathematical Discovery?,” 2011.

Basic Mathematical Notations

Mathematics seems so complete and packed with already complicated abstractions that we sometimes forget the fact that the most essential parts of math, symbols and notation, also started as a blank and share the entire history with math. They did not happen to exist from the top for one to conveniently play with numbers, or technically numerical values. Early mathematicians had to come up with their own notation and symbols and often tried to standardize them as means of communication for abstract operations in the arithmetic, algebra, and geometry. As it took many years (it is an understatement) to establish Hindu-Arabic number system in 8th to 9th century, considering that the first numeral system was invented by Sumerians in 3400 BC, the historical process by which the notation has reached current fashion is an exhaustively long journey filled with creativity and stubborn struggles.

The attempts to systematize the notation came from several directions across the world, and each of small contributions facilitated the study of math, especially in arithmetic and algebra, and eventually built the foundation for advanced topics.

The movement to what we now see as “the most basic” notations involved three stages. The first was rhetorical notation, and the second syncopation-based notation. They were the earliest, and probably the easiest approach to denote arithmetic relations, operations, and values. The two were used from ancient to 15th to 16th century, basically until the first symbolic notation emerged.

The rhetorical notation was a series of drawings and descriptive words to indicate an operation, and generally used in ancient mathematics like Egypt and Mesopotamia. Also in ancient Greek mathematics, there was no major move to systematize notation, and mathematicians mostly relied on natural language. Well, except that Aristotle denoted general number quantities by capital letters, and Euclid adopted this method in geometric algebra where he specified segments by letters with respect to their lengths.

The first syncopation-based notation appears in Diophantus’ work. A letter or syllable of the words denoted the unknown and its power.

math symbols

Image: encyclopedia2.thefreedictionary.com.

Image taken from encyclopedia2.thefreedictionary.com.

However, Diophantus’ notation had limitations of lacking generality and clarity because when a problem had more than one unknown, he had to individually point out first unknown, second unknown, and so on.

India joined this hot trend of the syncopation method by adopting the abbreviations of the words as signs. Addition was represented by yu from the word yuta, subtraction by xa from xaya, mutiplication by gu from guna, division by bha from bhaga, square root by mu from mula, and equality by pha from phalah.

In order to solve the issue Diophantus had faced, India interestingly used “colorful” notation for unknowns. The second unknown appears as ca from the word calaca meaning ‘black,’ the third unknown as ni from nilaca meaning ‘blue’, the fourth unknown as yellow, the fifth as red, and so on. The notation of powers was a combination of signs of a square and a cube, which were va and gha respectively. So, for example, the fourth power was va-va.

Mathematical Notations from Florian Cajori

Image: A History of Mathematical Notations by Florian Cajori.

Despite the tediousness, the syncopation method could have been an easy way to represent the abstraction at the time and communicate within a community, but apparently it was so inseparable to culture and language that unification across the world couldn’t possibly be achieved.

It is actually surprising that full symbolism of basic arithmetic and algebra was achieved not very long ago. Almost all math was written in the rhetorical and syncopation methods before late 15th century. After experimenting with creative notations, mathematicians started to acknowledge the simplicity and efficiency of the symbols, and symbolism grew into another topic of math. They continued to experiment with various graphic marks, which they modified, sometimes changing them completely, until they found the most successful symbols to use. Apart from the continual progress in study of math, it was a slow, collaborative process rather than an individual’s eureka-y invention that everyone happily agrees to follow. Thus, regarding the fact that progress in developing notations was not in the same speed as other branches of math, it should not be a huge surprise that the symbol of equality that we use today was not used in print before 1757 (basically around the time when everybody was talking about trigonometry and differential analysis!), when the Welsh mathematician and physician Robert Recorde designed the symbol ===== to avoid writing “is equal to” over and over in his book Whetstone of Witte.

Nicole Oresme first used the plus sign + in 1360, which was an abbreviation for et, meaning ‘and’ in Latin, and in 1489, Johannes Widmann introduced the minus sign – and used both + and – in his work, Mercantile Arithmetic. Now they are known to be the most widely used arithmetic symbols. A German mathematician named Christoph Rudolff introduced the radical symbol for square root in 1525, and Albert Girard came up with the radical signs for nth root in 1629. Nicolas Chuquet introduced first potential notation of exponent in 1484, and Rene Descartes in his book The Geometry in 1637 simplified the notation of powers, unknown variables, and constants, which converges to modern fashion. Superscript letters or numbers were applied to denote the exponentiation like xy, a2, and a3. The unknown variables were denoted by the small letters, x, y, z, w from the end of the alphabet, and known constants by a, b, c, d, e from the beginning of the alphabet. The symbols bleached out the vagueness of the verbal expression and also decisively, or I would say economically, facilitated the process of formulating problems and prepping the operations for effective solution. It is truly marvelous that every part of math, even including all tiny details I took for granted and overlooked, is a written form of humanity’s most abstract ideas to explain the world a little better.

Sources:

http://www.storyofmathematics.com/16th.html

https://archive.org/details/historyofmathema031756mbp

http://math.unipa.it/~grim/mathnotation.pdf

http://en.wikipedia.org/wiki/History_of_mathematical_notation

Faith and Math: the Origins of Math

But to us there is but one God, plus or minus one. —1 Corinthians 8:6±2.

Religions. Image: Randall Munroe.

Religion and math are oft thought of as being separate and often in opposition, at least within western society. We recently learned about the connections between math and religion in India (http://www.bbc.co.uk/programmes/b03c2zvr) but did not explore where else faith has had an impact on mathematics.

Where does math come from?

The main two answers to this are as follows: humans discover math, or humans create math. In the case of the first, it is accepted that all of math exists, has existed, and will always exist, regardless of whether or not we are aware of it. Even though the ancient Greeks were unfamiliar with negative numbers, negative numbers existed, but had simply yet to be discovered. This mode of thought is described as mathematical realism, and can be defined as the belief that our mathematical theories are describing at least some part of the real world (http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf pg. 36). There are several subdivisions among this group and more detail is given to this later. The second statement, that humans create math, is characteristic of mathematical anti-realism. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true.

The Realists

The realists should probably be subdivided into two main groups: Platonists and everyone else, with the “everyone else” being a minority, so we should probably have a definition for mathematical Platonism. According to both Stanford’s and the internet’s encyclopedias of philosophy, mathematical Platonism is based on the following theses: Existence, Abstractness, and Independence. Basically, mathematical objects exist they are also abstract, and your language, thoughts, religion, or anything else doesn’t change what they are.  I should probably also mention that there are also subcategories amongst the Platonists, like traditional Platonism, full blooded Platonism, and some others, but I don’t want to get into that.  There are, however, mathematical realists who do not subscribe to Platonism. One such group is the physicalists. A strong proponent of physicalism was John Stuart Mill. The argument for this is that math is the study of ordinary physical objects and is therefore an empirical science. According to this mathematics is basically meant to discover laws that apply to all physical objects. For instance, 1+1=2 gives us the law of all physical objects that when you have 1 of the object and you and another of the object you have 2 of the object instead. This differs from Platonism in that these objects are no longer abstract, but rather describe all objects. These are not the only two categories of realists. The main problem I have with this is that it means if all objects were to vanish math would cease to be true. This is because physicalism is not based on the abstractness of mathematical objects which means that the objects themselves must exist.

The Anti-realists

Anti-realism is in general the belief that Math does not have an ontology. As with mathematical realism there are a lot of different subcategories of mathematical anti-realism. I’ve chosen to talk a bit more about conventionalism and fictionalism because they seemed interesting.

Conventionalism holds that mathematical statements are true only because of the very definitions of the statements. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true. The statement “pi is the ratio between the circumference of a circle and its diameter” is true only because we define a circle as being a shape with a radius r, a diameter 2r, and a circumference 2*pi*r, and not because the universe made it so. In this sense, the above statement makes about as much sense as “all bachelors are unmarried”; both are obviously true, however this is because of their definitions rather than being the result of some universal laws.

Fictionalism argues that statements like 1+1=2 make about as much sense as “Harry Potter’s owl was Hedwig”. Yeah it’s true, but only within its given context. It is important to note that statements such as 1+1=3 make about as much sense as “You’re NOT a wizard, Harry”, because given the context of the story, or fiction, these statements simply make no sense. There are some interesting similarities about Fictionalism and Platonism. The biggest one is that both of them take mathematical statements at face value. This is to say that both of them take 1+1=2 to mean that to add the mathematical object 1 and adding it to another mathematical object 1 will result in the mathematical object 2. The difference then is that where Platonism takes this to also mean that these abstract objects exist, Fictionalism does not accept that these objects exist. This is different from conventionalism in that conventionalism doesn’t even accept that you are referring to objects, regardless of their existence. The thing about Fictionalism is that the subject doesn’t technically actually even a little exist. By this I mean that Harry Potter doesn’t actually exist (probably) and that therefore he isn’t actually a wizard (probably) and that since he doesn’t exist he doesn’t actually own an owl named Hedwig (probably), and that by that same logic 1 doesn’t actually exist, and neither does 2, and 1+1 doesn’t equal 2 because none of them exist.

Implications of these schools of thought

Mathematical realism, in a certain sense, seeks to prove truths about the universe. This is most obvious when you consider modes of thought like physicalism, under which math would be a really general science, but even under Platonism you are seeking to find laws that govern these abstract objects you are finding. So for instance, when you have one of some object, and you add another of that object to that first object, you now have two of that object and according to mathematical realists, this is true. It is a fact. According to mathematical anti-realists, if you remove the humans, or whatever it is that is observing this addition, then there is no longer a group, one of the things, or two of the things. These concepts existed only because the humans said they existed, and when the humans stopped existing and thus stopped observing this these things lost the properties of being one, being grouped, and finally being two. The exact way in which this is argued depends on what subcategory one subscribes to. (https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA)

How this relates to faith

Regardless of whether you believe that the statement “pi is the ratio between the circumference of a circle and its diameter” is true because of universal laws or because of human created definitions, the statement is still true. The importance of this is that it means that there, at least at this point in time, is no way to verify whether the reason for math existing is tied to the very nature of the universe or whether it is simply the product of the human mind. As a result of this, the belief in either of these theories is, at least in a certain sense, a leap of faith.

My thoughts on this

My personal opinion on this leans towards mathematical realism and more specifically Platonism. I agree that mathematical objects exist, but that they do not by necessity have a real world counterpart and thus are abstract, and I believe that regardless of whether or not humans exist, the mathematical concepts we have found to be true will still be true, even if no-one is around to appreciate, understand, or use them. One big reason I have for thinking this way is because of how various isolated cultures ended up discovering the same mathematical principles. By this I mean that counting systems, simplistic though they may have been, were not a unique event to just one area, but rather a common feature. I mean the Mayans had a counting system, so did the Greeks, Egyptians, Babylonians, Indians, etc. It seems somewhat unlikely to me that all these isolated cultures would create a method for defining something that doesn’t exist.

Additional reading/sources

Idea channel’s episode titled “Is Math a Feature of the Universe or a Feature of Human Creation?”

https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA

Mark Balaguer’s “Realism and Anti-Realism in Mathematics”

http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf

Stanford’s Encyclopedia of philosophy entry on Platonism and mathematics

http://plato.stanford.edu/entries/platonism-mathematics/

The Internet’s Encyclopedia of philosophy entry on Platonism and mathematics

http://www.iep.utm.edu/mathplat/#H1

Wikipedia’s entry on philosophy of mathematics. No, this was not used as a source; it is however, useful for additional reading.

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Mayans count as well

http://maya.nmai.si.edu/maya-sun/maya-math-game

Greeks count as well

A History of Mathematics, Merzbach and Boyer, pages 52-55

For Babylonian counting see

Plimpton 322

For Indians having a number system click the bbc  thing below

For the link to the bbc story thing

http://www.bbc.co.uk/programmes/b03c2zvr

for the comic

http://xkcd.com/900/

Zero and Infinity: from Nothing to Everything

This is the Jain Temple of Gwalior. Unfortunately I was unable to attain photos of the inside. But in the temple, there are what historians believe to be the first known recorded zeros. Image: Tom Maloney, via Flickr.

This is the Jain Temple of Gwalior. Unfortunately I
was unable to attain photos of the inside. But in the temple, there
are what historians believe to be the first known recorded zeros. Image: Tom Maloney, via Flickr.

One of our recent homework assignments that I found interesting was the BBC radio excerpt called Nirvana by Numbers. This was fascinating for a few reasons. First of all, I was astounded to learn that India had contributed so much to mathematics and I had not heard about it until now. That was mind blowing. What is happening in the education world, that so few people know about their remarkable achievements? Secondly, I could really appreciate the idea of math being something spiritual. The view of math as something fluid and moving, rather than something stagnant appeals to me. People tend to have negative opinions about mathematics and it can be hard relating math to other mediums, like art, music, or religion. When in reality, math adds value to these things, and they all have mathematical elements.

If we consider India around the time of 800 CE, we begin to understand what this middle ground of math and religion really is. We also come to learn about all of the phenomenal discoveries they have made. This component of mathematics containing spirituality (and vice versa) inspired the idea of nothingness or what we would today call zero. And what the Ancient Indians would call “Sunya.” According to the BBC article, “sunya” means void. In the ancient Temple of Gwalior, historians and archeologists have found what could be the first recorded zeros. As they began to do more math, zero became an important concept at the time because it made the point that nothing actually is something, and in some cases nothing is everything. What I mean by this, is that in certain religious beliefs like Hinduism, their word for creator, “Brahma” is equivalent to zero. And as our narrator points out, this is very different from Western culture, because our creator would typically be equivalent to infinity. Another initiative they started that we continue today, is to denote zero as a circle. They did this because a circle is symbolic to the sky, a circle of the heavens. The circle is also empty in the middle which is figurative of a void. So in their eyes there was a lot of overlap in terms of their belief system and math.

For instance, one goal in life was to reach nirvana. Nirvana is the highest state a person can achieve where there is no suffering and no desire They would even go as far to say that reaching a state of nirvana is equivalent to zero. This too could have helped establish the concept of zero.  Because of nirvana, they had an idea of “no” suffering, which meant there had to be a way to describe “none.” And thus the tangible idea of zero had blossomed.

One idea that I was interested in exploring more, was the idea of Vedic Mathematics. The Vedas are ancient Hindu texts, that contain spiritual works. They possess instructions on how to do the basic operations like addition, subtraction, multiplication, and division. But not only that, they had processes in which one could determine area of a geometric shapes. Historians have even found early forms of Pythagorean’s Theorem. According to the people interviewed, as well as an expert on Vedic Math Gaurav Tekriwal, who instructs a TED-Talk, Vedic Math can be very easy. For instance in the TED-talk the general idea for multiplying two two-digit numbers is with a vertical and crosswise pattern. First we take the numbers in the one’s place and multiply them together. Then we cross multiply the one’s and ten’s places and add those products together. Lastly we multiply the ten’s place. The example he gives is 31×12, but let’s try our own. Say we have 24×20. Step one is to multiply 0x4, which is 0. This will be the one’s place of our answer. Next we take (0x2)+(2×4), this equals 8. This is the ten’s place of our answer. Finally we multiply 2×2 to get our hundreds place. This yields 480 as our answer.

There is a very special case for multiplying with the number 11. The basic idea for multiplying any number with 11 is such: we separate them, put their sum in the middle and that gives us the answer.  Let me demonstrate with 26×11.

We take 26 and separate it, so that there is a space between the two numbers. We then add 2 and 6 and put the answer to their sum (8) in the space we left when we separated them. This gives our answer to be 286. Multiplying by 11 is a special case, it is just an extension of the general idea for multiplication in the Vedic sense. It uses all of the same ideas we used in the first example. However because 11 is comprised of all one’s, we can skip the cross multiplication and go straight to the addition. So as we can see, multiplying with these rules is quite simple and fairly straightforward.

In the BBC post, there are two men who have differing opinions about Vedic math. One thinks that it makes math more fun, whereas the other thinks that the ideas and concepts of math do not get taught, just the routine does. And based off the TED Talk I watched by Tekriwal, multiplication does seem much easier, but I can see how the notions could get lost on a student.

I read another article that discusses Vedic Math in terms of the Jain religion. According to this article, the Jains had formulas for circles, like circumference and area, and in some cases could determine answers from quadratic formulas/equations. Another great contribution the Jains made was the concept of a positional number system. In other words, putting all the one’s digits in the same place, all the ten’s digits in the same place etc. They also loved large numbers and contrast  to the establishment of zero, this was the start of infinity. One such large number was 10 to the 53rd power! Wow! That’s big! This article states that the Jains had five kinds of infinity. And those were: infinity in one direction, two directions, area, everywhere, and perpetually. The article also talks about how the early Jains were developing permutations, combinations, and had early stages of Pascal’s triangle in the works. It was called the Meru Prastara.

This article unfortunately did not go into the religious aspects I was hoping it would. But nonetheless, from what I learned through the BBC clip, religion in ancient India played a key role in the root of  their mathematics. From zero to infinity, math was being incorporated into their sacred texts and their lives.

This is something we can all bring into our personal life, even if you are a nonreligious person like me. Knowing that math is beautiful, and sacred, and has an element of spirituality to it, makes me much more excited to do my math homework. It seems less dreary, less gloomy. I will start treating math more like a combination of art and science. I think this could not only benefit me, but how we teach kids math. If we start telling them it’s a creative process, maybe more students will be excited about doing tedious algebra problems.

Works

Bellos, Alex. “Nirvana by Numbers.” BBC Radio. British Broadcasting Corporation, 28 Oct. 2013. Web. 1 Sept. 2014. http://www.bbc.co.uk/programmes/b03c2zvr

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Šafránková, Jana. “Part 1, Ancient Indian Mathematics.” 15th Annual Conference of Doctoral Students, WDS’06 “Week of Doctoral Students 2006”, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, June 6, 2006 to June 9, 2006:. Praha: Matfyzpress, 2006. 7-12. Web. 4 Sept. 2014. http://www.mff.cuni.cz/veda/konference/wds/proc/pdf06/WDS06_101_m8_Sykorova.pdf

Tekriwal, Gaurav. “The Magic of Vedic Math – Gaurav Tekriwal.” TED-Ed. N.p., n.d. Web. 04 Sept. 2014. http://ed.ted.com/lessons/the-magic-of-vedic-math-gaurav-tekriwal