Tag Archives: zero

The Importance of ‘Nothing’

I’m a programmer. When I ask people their impression of what I do, the usual response is a long string of ones and zeros, said in a robot voice. Before I first started my Computer Science degree, I probably would have said the same thing. After my first semester, I would scoff at such binary answers, and feel powerful knowing I know how to write code. Halfway through my degree, I discovered that when you get down to brass tacks, zeros and ones are really all that comes down to. Finally, here at the end of my degree, I’m really happy that I don’t have to work in raw ones and zeros.

And it has always tickled my fancy that there is no Roman numeral representation for the number zero. I usually just pull this out for fun trivia, but after discovering in class that the Egyptians and Babylonians also struggled with the concept, I thought it might warrant a little extra research.

In this day and age, with our modern schooling, it seems as if zero is trivial. It literally means nothing, after all. It might have a few cool properties. For example, zero added to any number will result in the number as one example…but you can get the same behavior by just multiplying by one! For a computer scientist, zero is a boolean value. Zeroes also have a very friendly feel to them. If you see a lot of zeros at the end of a number, you know that number is a nice round one. And we like round numbers.

But being able to use zero is HUGE! Without it, we would either have an ill-defined positional notation for our numbers, or have to resort to an additive system like Roman numerals.  The lovely round number of 100,000 so cleanly represented here (with a little help from a comma) would require 100 M’s in a row using present day Roman Numerals. Even ancient cultures that used a positional notation would just use contextual clues to figure out if 216 meant 2016 or 2160 or what have you. Babylonians started to help with this problem by making two tiny stylus tick marks. So now, 2106 became 21”6. Interestingly enough, there was never any tick marks at the end of numerals, only in the middle. This leads scholars to believe that these tick marks were not an idea of zero; simply punctuation, much like our helpful little comma from before.

Zero is special in that it has two roles. It can be used for positional notation as we have just seen, but that was just as easily solved with punctation. Zero is also, of course, a number in and of itself, which brings on a whole barrel of troubles. Historically, numbers were thought of much more concretely. People used them to solve ‘real’ problems rather than abstract ones. It is a pretty far jump from for a farmer to go from five horses he owns, to five “things” in existence, to an abstract idea of ‘five’. If the farmer is solving the problem of how many more horses he needs, it is going to be “zero more horses.”

For this reason, perhaps it was lucky for earlier civilizations to miss out on zero. Working with zero can get you into a lot of trouble. There are cases of some of the brightest mathematicians of their time struggling with the concept of zero. And Indian mathematician has this to say about division:

“A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number 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.” – Brahmagupta

You can tell he is reaching when he suggests a number divided by zero is N/0.

The first known zero, found in Cambodia. Image: Amir Aczel.

What’s even more mysterious is how there isn’t some clearly defined point in history where zeros are firmly established. There are some hints and teases in the nautical readings of Greeks and odd punctuation marks in Egypt, but nothing concrete. The earliest known writing of zero is famously from a stone tablet found deep in Cambodia, where it has the date of 605 in sanskrit, with a small dot to denote the zero between the six and five.

A clean rendering of the oldest known numeral using zero. Image: Pakse, via Wikimedia Commons.

It seems odd that such a powerful and tricky number wouldn’t have a more auspicious start. Instead, somewhere, someone in India put a dot on a tablet…and the world was changed forever.

I just hope something like this doesn’t happen:

Zach Wiener, SMBC-Comics 08/29/2012.

-Fin-

Sources:

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html

http://www.sciencealert.com/search-for-the-world-s-first-zero-leads-to-the-home-of-angkor-wat/

http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM

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Getting Something from Nothing

This last week we read about the creation of zero and how it wasn’t really part of a numerical system until later in civilization. It seems weird that such an essential part of the number system is left out. The reading by the Smithsonian Magazine intrigued me enough to dig into for myself because of the heavy importance writer Amir Aczel puts on it. He talks at the end of his article about if the zero was created or discovered. For the most part it makes sense that zero wasn’t necessary because the old practical reason for a numerical system was to count things and most of my readings supported that. They did not necessarily need a symbol to represent nothing. Their notation served its purpose and was all that was needed back then. The acknowledgement of zero as not only a numerical value, but as an idea has brought mathematics to what it is today, and without math we would never be anywhere close to what we are today.

caption Babylonian numerical system circa 3100 BC. Image: Josell7, via Wikimedia Commons.

Going along periodically throughout time we began to have more systematically efficient numerical systems, including the Roman numerals and the Babylonian counting systems and eventually the Hindu Base ten system. The idea of zero was much more prominent as math theory progressed. The zero was originally just a dot to represent an empty place. It took until the ninth century when Persian Mathmatician Mohammed ibn-Musa- al-Khowarizimi(780-850) started working on equations that equaled zero. Today these equations are known as algebra.

“Russel Peters-Invention of Zero”-Youtube

An Italian mathematician named Leonardo Bonnacci(c.1170-c.1250), more commonly known as Fibonacci, refined and built on Al-Khowarzmi’s work for use in an Abacus book, which spread around European merchants and became the primitive base work for accounting. He spread it as a placeholder so we may distinguish 1 from 10 and 10 from 100. By distinguishing that it exists we can now separate the negatives from the positives, we know have a universal positional notation worldwide because of the simplicity that the base 10 system has. This affected trade and progression of civilization in surrounding areas. What it also affected was the growth of math itself.

But why is zero so substantially symbolic? As an article from Joanne Sacred Scribes from numerology-thenumbersandtheirmeaning.com states that zero symbolizes of an eternity. There is no end, nor beginning. It also has to do with the concept of Heaven, that there would be no more suffering, which gave people a tangible idea of what zero was. The symbolism of the zero also came from Rene Descartes(1596-1650), the founder of the Cartesian co-ordinate system. With (0,0) being the origin of all Cartesian planes it fits in the belief that we were created from nothing and we can base ourselves on where we were at our origin.

So now that we have the basics of graphing and algebra the zero has played its importance and is just another number right? Wrong. Perhaps the greatest progressive push the zero has helped create was the creation of physics, engineering and economics through the use of calculus. How did zero play a part in this? Most people know the laws of adding, subtracting and multiplying zero but it wasn’t until the 1600’s that mathematicians Isaac Newton and Gottfried Wilhelm Leibniz understood what it meant to divide by zero in a mathematical sense. This hook in basic arithmetic was observed through the use of infinitesimal quantities, or looking at what happens to a number, or a line as it gets infinitely small towards zero, but not zero. For a better quick history on the development of calculus the author Jason Garver of From Quarks to Quasars gives a fantastic overview.

As Tobias Danzig, a German mathematician says “In the history of culture the discovery of zero will always stand out as one of the greatest achievements of the human race”. When you look back and realize that when we got a standardized numerical system is when we started making vast headway in trade, commerce, expansion and innovation it is amazing to ponder what life would be without it.

Further reading:

-John Matson: Article talks about early uses of zero and its importance to civilization: http://www.scientificamerican.com/article/history-of-zero/

-marycneville: Article talks about creation of zero and its importance to religion: https://3010tangents.wordpress.com/2014/09/08/zero-and-infinity-from-nothing-to-everything/

-malinibindra: Slideshow talking about why zero is the most important discovery: http://www.slideshare.net/malinibindra/importance-of-zero

-JJ O’conner and E F Robertson: Article talked about the progression of zero through non western civilizations: http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html

-Joanne Sacred Scribes: Talked about the theological aspect of zero and the symbolism behind it:  http://numerology-thenumbersandtheirmeanings.blogspot.com/2011/05/number-0.html

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

http://www.britannica.com/EBchecked/topic/1555771/Tobias-Dantzig

Amir Aczel: Article read in class. Gave a broad overview of the development and implementation of zero: http://www.smithsonianmag.com/history/origin-number-zero-180953392/?no-ist

Nils-Bertil Wallin: Talks thoroughly about the importance of zero throughout time and evolving civilizations: http://yaleglobal.yale.edu/about/zero.jsp

Jason Garver: Gives overview of the creation and development of Newton’s side of Calculus: http://www.fromquarkstoquasars.com/how-and-why-did-newton-develop-such-a-complicated-math/

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

Early Chinese Mathematics

Math is something that is found all throughout history.  It was used for may different reasons, in many different cultures.  What I find interesting is how these different cultures learned some of the same ideas without even having knowledge of the others’ work. These works could be anything from counting systems to Pascal’s triangle.  It can also include how one culture passed its knowledge on to another. This makes you wonder how some ideas that were known in western civilization could also be found in Asia.  As I was looking into this I found some very interesting facts about mathematics in China. Some small examples of math found in China begin with something called oracle bone scripts: scripts carved into animal bones or turtle shells. These scripts contain some of the oldest records in China.  This, like the clay from babylonian times, had many different uses including math.  Chinese culture also had something called the six arts: Rites, Music, Archery, Charioteerring, Calligraphy, and Mathematics.  Men who excelled in these arts were known as perfect gentlemen.

In China, like in India, one can find the use of a base ten numeral system.  This is quite different from the Babylonians, which makes it seem like there must have been some conduit of knowledge between India and China.  In China, around 200 BCE, they used something called “rod numerals.”  Rod numeral counting is very similar to what we use today.  This counting system consisted of digits that ranged from one to nine, as well as 9 more digits to represent the first nine multiples of 10.  The numbers one through nine were represented by rods going vertically, while the numbers of the power of 10 were horizontal.  This means that every other digit was horizontal while its neighbor was vertical.  For example 215 would be represented like this ||—|||||.  If one wanted to use a zero you would have to use an empty space.  The empty space is also something that can been seen in the Babylonian counting system.  As with the Babylonians, a symbol was eventually used for zero.  Interestingly enough, before there was a symbol for zero, counting rods included negative numbers. A number being positive or negative depended on its color: black or red.  This idea of having negative numbers didn’t come about in another culture until around 620 CE in India.  It seems quite apparent that several ideas that originated in China could possibly have been passed on to a neighboring country. 

Rod numerals. Image: Gisling, via Wikimedia Commons.

Rod numerals. Image: Gisling, via Wikimedia Commons.

The use of counting rods as a counting system brought about another very interesting mathematical concept, the idea of a decimal system.  China first used decimal fractions in the 1st century BCE.  Fractions were used like they are today, with one number on top of another.  For example, today if you used a faction for one half, it would be written like this: 1/2.  Using rod numbers you can do the same thing like this: | / ||.  Not only could this be represented as a fraction but it could also be written as a decimal.  To do this one would simply write the number out and insert a special character to show where the whole number started.  For example, if you wanted to say 3.1213, you would write it as a whole number like this: |||—||—|||.  To show where the left side of the decimal starts, you would mark it with a symbol under the number to the left of the decimal point, in this case under the first 3.  To me the use of rod numbers is so similar to how we use our numbers today that even the arithmetic that was used can be done easily by someone in our culture.  Addition is done almost the same except they would work from left to right.  Multiplication and division were used as well.   The use of base ten as well as using rod numerals made complicated equations much easier to attain, such as the use of polynomials and even Pascals triangle.

The Yanghui Triangle. Image: Public domain, via Wikimedia Commons.

The triangle known as “Pascal’s” in the west, in a Chinese manuscript from 1303 CE. Image: Public domain, via Wikimedia Commons.

Centuries before Pascal, the Chinese knew about Pascal’s triangle.  Shen Kuo, a polymathic Chinese scientist was known to have used Pascal’s triangle in the 12th century CE.  It appears that knowledge of Pascal’s triangle begins even before this. The first finding of Pascal’s triangle was in ancient India around 200 BCE.  We can see that this idea was sprouting around and found evidence in different cultures, from Persia to China and to Europe.  This again makes one wonder how this knowledge base was passed around from one culture to another.  Lacking historic details, it is hard to see if this idea of Pascal’s triangles was thought up individually or if this concept was somehow passed from one culture to another.

It seems that in all cultures there is a need for counting, which in turn brings about the need for math.  The cultural implications can mean that you are a “perfect gentlemen” by having mathematical knowledge, or it could lead a greater knowledge that can be passed on to other cultures.  In China, we see that many ideas of numbers and mathematics were thought up on their own without having other culture’s ideas intervening.  We can also see that the knowledge that was passed on was able to thrive and turn into something even more intriguing.  It is apparent that we can always learn and teach others to help our knowledge grow.

Source:

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

http://en.wikipedia.org/wiki/Decimal#History

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

http://nrich.maths.org/5961

http://en.wikipedia.org/wiki/Pascal’s_triangle

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.

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.

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