Monthly Archives: October 2014

My infinity is bigger than your infinity

When I was a child, I purposely found something to think about to help me fall asleep. Usually I picked cartoons or super powers, but sometimes things just came into my head, like it or not. What was the worst? Thinking about heaven. At first, heaven seems all right. There is a lot to do, gold everywhere (though no purpose for it), people are nice (it’s a prerequisite), you get to see most of your family, and there is plenty to eat (though no one is ever hungry). Anyway, I start thinking about FOREVER.

pic1cropAt first, it is just a sensation; a weird sensation like tingling and falling and nothingness. It is not a sensation that I can make sense of really because forever doesn’t really make sense, at least not to a 10 year old. I try to get away from forever but forever is a huge part of the definition of heaven. Then, the opening credits of the Twilight Zone, with the music, and starry sky, usually appear. Fade to myself standing, looking at heaven, in the dressing room mirrors of infinity. You know, when dressing rooms have those three mirrors that are angled just perfectly so the images are smaller and smaller replicas of one another, on and on, into infinity. This picture, and thoughts of the foreverness of heaven, kept me up at night as a child.

I am glad to say that forever no longer keeps me up at night. While I still find no comfort in the foreverness of heaven, the lack of a middle ground between forever and my time on earth is what usually keeps me up at night now. However; I still can’t stand it when mirrors are angled that way. It creeps me out, and I can’t help but wonder if there is an end, or if I can find a flaw from one image to the next. In my opinion, we are not meant to look into infinity like that, squarely.

When beginning to pursue mathematics, I thought math might clarify, or in some way define, forever (or as adults call it, infinity). On the contrary, Math has actually made it stranger. Theories in math have shown numerous types of infinity, and infinities within infinities, and sizes of infinities, and calculations of infinity. None of this brings me any comfort, except to say that we obviously don’t have this figured out yet because that is just not possible. Infinity is infinity, and it is very large, incalculable and non-denumerable, and there is only one kind; it is called forever. Heaven can only exist in one, all-encompassing infinity.

pic2crop

When reading A History of Mathematics, I read about Zeno’s paradox. That led to an internet search, and then to Numberphile. I watched the video, accepted the idea, and left it alone. The solution seemed reasonable enough. Later in the semester, I was required to do a research project. By some unknown scheme, we picked Georg Cantor, whom I had never heard of. If you haven’t either, he is the creator of set theory but also perhaps the mathematical or scientific father of infinity. You just can’t shake things off in life. They follow you.

My research for that project led me to question the mathematical view of infinity. Let me start by saying, I know very little of Math’s view of infinity. It seems to be an infinite topic. This is where I am in my understanding – so please comment, post, reply, educate me, and critique my understanding. Calculus one is a prerequisite for the course, and being a rule follower, I have that. So, I had experience computing limits to infinity. That is relatively easy. BUT, those are just numbers. They aren’t real things. Numbers aren’t real. So, of course I could compute the infinity of something that isn’t really real. What numbers represent is real; like Zeno’s paradox. Zeno’s paradox applies numbers to something real – something actually happening in the world (theoretically). In other words, when I take the limit of a sequence that goes to infinity, it has no relation to time or space. It is just numbers. But, if I were taking the limit of Zeno’s paradox to see how far Zeno actually travels, or to find the time it takes to travel, or to see if he can ever catch the turtle, I would have to do so in relation to time and space. When I do that, the exact opposite answer occurs. Zeno will never catch the turtle. That mathematics isn’t computing real infinity or perhaps all of infinity is perhaps echoed by the Numberphile narrator when he asks, “What I want to ask a physicist is, can you divide space and time infinitely many times?” Similarly, Kelly MacCarthur wonders in the Calculus 2 video used for online math courses, “Can I take infinitely many steps?”

However, if all of space and time existed at one instant, forever, then Math has it right. It could calculate the infinite because it occurs all at once. There is no sequence, event after event – in essence, no time or space really because it is all at once, everywhere. Yes, there are scientific theories, philosophies, and religions which believe this is the case. Of course, this idea is contrary to most people’s understanding of infinity. Whenever math instructors talk about infinity, they always say, “Infinity is only a concept. It is not a number.” Yes, it is only a concept but is it also something real? If it is only a concept then why are we computing something real that is a concept? Why would we bother to compute a concept? It seems like Math is walking a funny line here.

Math has worked something out though. I’m just not sure what it is. Math is summing an infinite process (as if infinity happened to end). Obviously, Math’s understanding of infinity has proven useful in mathematical calculations and many practical applications. To paraphrase others before Cantor, “It works. So, no need to define it. It works.” So, Math has worked something out about infinity but what has Math worked out, and is it really infinity?

pic3cropMathematicians always like to joke about engineers rounding numbers to 3 or 4 places because it doesn’t really matter to engineering after that, but is mathematics rounding off infinity or at least only capturing some aspect of infinity? After all, how can there be different types of infinity? My preferred illustration for the existence of multiple infinities is from Galileo. Galileo used a thoughtful but intuitive approach to understand infinity. He drew a circle. Then, he drew an infinite number of rays from the center of the circle. These rays filled up the space inside the circle. But then, he drew a larger circle around the smaller one and extended those rays to the larger circle. Though he drew as many rays as possible (an infinite number perhaps), the infinite number of rays did not fill up the larger circle; there were spaces between the rays. This led him to believe that first infinity was not large enough for the second circle; not even close. He would need another size of infinity to fill up the larger circle. [BAM! PHH! Did your mind just explode?] It is important to note that intuitively, his illustration makes sense. However, with today’s current understanding of infinity and better ability to calculate infinity, we now know that the infinity in the smaller circle leaves no space between the rays when extending to a larger circle. But, I liked his intuitive approach. Though intuition seems to be severely lacking when it comes to infinity.

References:

Dangerous Knowledge: http://topdocumentaryfilms.com/dangerous-knowledge/
Georg Cantor His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben
TML: The Infinities In Between (1 of 2): http://www.youtube.com/watch?v=WihXin5Oxq8
TML: The Infinities In Between (2 of 2): http://www.youtube.com/watch?v=KhgNiqI-bt0
Infinite Series: http://stream.utah.edu/m/dp/frame.php?f=f55f900bec01a3106121
Zeno’s Paradox – Numberphile: http://www.youtube.com/watch?v=u7Z9UnWOJNY

My new bumper sticker.

pic4crop

Ada Lovelace – The Enchantress of Numbers

Women in Mathematics

Reading about Sophie Germain’s contributions to the world of mathematics ignited my curiosity about the role of other women in contributing to the advancement of math throughout history. Women have made some incredible contributions to mathematics, and I was more than pleasantly surprised (as a Computer Science major) to read the story of Ada Lovelace – the only legitimate daughter of the romantic poet Lord Byron, and the woman often called the world’s first “computer programmer” – her collaboration with British Mathematician Charles Babbage began around the middle of the 19th century.

History

Ada Lovelace was born to Lord Byron and Anne Isabelle Milbanke in 1815, though she never met her famous father – he was reportedly disappointed at not having a boy, and left both mother and child when Ada was only one month old. Lord Byron passed away when she was only eight years old (Ada herself was buried next to her father after she succumbed to uterine  cancer at only 37). Her mother saw to it that Ada did not take seek her father’s “artful” pursuits, and made sure that the girl was tutored in mathematics, science and music. While not unusual for someone of her family’s elite noble class, it was unusual for a woman to study math and science in that time. These intellectual pursuits led Ada to communicate with the circle of “gentlemanly scientists” of that era (though the term scientist was not actually coined until 1836, the description applies), and it was in this circle that she met her lifelong friend, British Mathematician Charles Babbage, the inventor of the Difference Engine (the machine itself never made it past the prototype stage). They wrote extensive correspondences back and forth on mathematics and logic, among other topics. Babbage was impressed by Lovelace’s intelligence and analytical skill, referring to her as the Enchantress of Numbers.

Ada_Lovelace_portrait

Watercolor portrait of Ada King, Countess of Lovelace (Ada Lovelace) by Alfred Edward Chalon. Alfred Edward Chalon [Public domain], via Wikimedia Commons

Babbage and the Analytical Machine

In the 19th century, complicated mathematical computations had to be done by hand, or using the shortcut of published tables, which also had to be written by hand and were prone errors themselves. In 1834, Babbage began work on a new kind of calculating machine, the Analytical Engine – a machine that would automate calculations of addition, subtraction, multiplication and division. In 1842, Lovelace was enlisted by Babbage to translate from French the work of Italian mathematician Luigi (or Louis) Menabrae. Her work in this translation, and her copious notes written during the process, are responsible for her enduring fame.

Her contribution

Lovelace’s notes on Menabrae’s work were larger than the translation itself, and in them she describes how the Analytical Engine itself works, as well as an algorithm, or what she called a “plan”, for using the Analytical Engine to compute Bernoulli numbers. This is said to be the first algorithm specifically suited for a computational machine to carry out a series of instructions, using the “specific ability of a calculating device to make control decisions based on the data”. In her work, she also noted the grand potential of the Analytical Engine to solve problems of any complexity, and even projected that this type of machine could be used to “compose elaborate and scientific pieces of music”.

There is some controversy about the extent of Lovelace’s contribution. However, in their article published in Scientific American (issue 280, p 71-75), Eugene Eric Kim and Betty Alexander Toole noted that while Babbage wrote several small programs, none of his approached the complexity of Lovelace’s Bernoulli numbers program, and that this program was her idea; it also cannot be disputed that her vision for the potential of such machines to extend functionality beyond simple value computations was visionary, especially for the 19th century.

Conclusion

Women have contributed extensively to the advancement of mathematics and computer science (please read any biography you can get your hands on about the incomparable Admiral Grace Hopper, and look for my upcoming third blog post all about this amazing woman!), and it is important for these contributions to be recognized. This is not only to give credit where it is due, but also to give the many girls and women thinking about a future in the sciences and mathematics role models to look up to. Ada Lovelace is recognized in a number of ways: a day in early to mid October (the actual date varies) is celebrated now as Ada Lovelace day, a day to celebrate the achievements of women in science, technology, engineering and mathematics (STEM), and the computer programming language Ada was created by the Department of Defense around 1980, named in her honor.

Sources:

https://www.sdsc.edu/ScienceWomen/lovelace.html

http://womenshistory.about.com/cs/sciencemath1/a/bio_lovelace.htm

http://mathworld.wolfram.com/FermatsLastTheorem.html

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

http://findingada.com/

http://www.linuxvoice.com/history-of-computing-part-1/

http://www.conservapedia.com/Ada_Lovelace

 

Invented or Discovered?

A philosophical question about math that has been asked since the times of the ancient Greeks (and possibly even before then) is whether mathematics is discovered or is it invented by man. People seem to think it has to be one or the other, but what if it is actually both?

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Math is just a language, and like any other language that uses words to describe something (strings of symbols), math also uses symbols. Written language was developed both independently and simultaneously in ancient times. One person got an idea to use a written symbol to represent a tangible object. Sometimes multiple people got this same idea independently of each other, and other times a person would see this writing, it would spark the idea in their heads, and they would go on to develop their own written language. The same language was not developed by different people, rather each person used different symbols to represent different words. (Guns, Germs and Steel- Jared Diamond Chapter 12) The same can be said for math. Calculus was developed simultaneously, but independently by Isaac Newton and Gottfried Leibniz. Both developed different ways of doing calculus and each way gave the same results. Other times mathematicians have relied on the work of others to further their results.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

The fact that math has been developed independently and yet yielded the same results shows that math is discovered. Math is the language used to describe the natural world and as long as the world exists someone can, at any time, develop a language to describe it. It may not be the same math that we use today (the Babylonians used an arithmetic system very different from our modern one), but it would still yield the same results. Given enough time, one would think, they would eventually be able to build the same skyscrapers and the same rocket ships that we have.

On the other hand, math was invented. We invent the symbols and decide what they represent; we invent the axioms and the particular system that we use. Newton invented infinitesimals in the use of calculus while Leibniz invented his own notation for calculus. The ancient Egyptians invented a different way to calculate the area of a circle than the one we use today. (A History of Mathematics- Uta c. Merzbach and Carl B. Boyer) Math does not exist without someone to invent the symbols we use to describe it.

Many people ask, and for good reason, if this question is even important, and it just may be. What if the concept of zero or negative numbers were never invented? Without these simple concepts would we still be able to build the same skyscrapers and rocket ships? It is possible that someone could have invented a concept similar to these but using different concepts? It is even possible that someone may have invented a way around them so we could avoid them altogether and this new invention could have even lead to a much simplified math system.

The Golden Ratio

Often times, students ask themselves, “How will this apply to me in the real world?”, or  “I never use math, why do I have to learn this?”. Doubt may exist about the usefulness of mathematics, but there is no evidence that supports this doubt. As a student studying math in higher education, I know that math is all around us. I know that everything relies, and consists of some form of mathematics. The most prevalent discovery of math, that is both commonly seen, and unseen, is the Golden Ratio.  The reason I say the Golden Ration is the most prevalent, is because it can be found in nature, science, architecture, and most other aspects of life.

The golden ratio is a number found by dividing a line into two parts, say A and B. The segment A would be longer than segment B, and when A is divided by B, it is equal to the whole length of the line divided by A. The golden ratio is numerically defined as the number 1.6810339887498948420…., and continues on into infinity. It is also represented by the greek letter phi. The golden ratio has been found in nature, and the physical universe. It is not known when it was first discovered by mankind, but can be seen in the works of man dating from ancient times.

The human face is an example of how the golden ratio can be found in nature. When looking at the face, the mouth and nose are positioned in between the eyes and the chin, which represents a golden section. A “perfect” face will have golden proportions between the length and width of the face, the length of the lips and the width of the nose, and the distance between pupils and the eyebrows. Even though all humans are unique, the average of these distances between features is close to phi. It has also been said, that the closer ones features adhere to phi, the more “attractive” society finds them. The reason being, as originally stated, is because the golden ratio is thought to be the most pleasant to the eye. We can even find the golden ratio in our DNA. The molecules in our DNA have been measured to be 34 angstroms in length, and 21 angstroms wide at each full cycle of its double helix spiral.

Parthenon, Athens, Greece The Parthenon without tourists! Athens, Greece (1989/379) From flickr.com, posted by GothPhil

Architecture is another place where the golden ratio is put to use. One example is the Parthenon, found in Athens, Greece. The length and height of the structure,  the spacing between the columns, and the tip of the roof are all contained in the golden ratio. Within the building many golden rectangles can be found as well. The dimensions of the Great Pyramid of Giza, in Egypt, is also based off the golden ratio. The Egyptians believed the golden ratio to be sacred, and it was an important part of their religion. For the Pyramid of Giza, the ratio of the height of the slant compared to one half the length of the base, is the golden ratio. In modern architecture, the golden ratio still applies. In Toronto, Canada stands the CN tower. The CN tower is the tallest freestanding structure in the world. The observation deck of the tower is found at 342 meters, where the total height is 553.33 meters. This ratio between these two lengths is 0.618, or phi.

Not only is the golden ratio used for architecture and design, but also for beauty. Artists for centuries have applied the golden ratio to their work to create the most beautiful pieces of art known to the world. These examples are found in pieces such as The Last Supper by Leonardo Da Vinci, the ceiling of the Sistine Chapel by Michelangelo, and Bathers by Seurat. In the painting of The Last Supper, Leonardo Da Vinci used the golden ratio determine where Christ would sit in relation to the table, and his apostles, as well as, the proportions between the walls and the windows. The golden ratio can be found on the ceiling of the Sistine Chapel where the finger of God, and the finger of Adam meet.  This point, is found by taking the ratio of the height, and width of the area that they are contained in. All of these works of art apply the golden ratio to convey the most realistic shapes and dimensions.

The next time someone says that math does not apply to them or the real world, I would simply suggest they look in the mirror, or visit the nearest museum. The golden ratio can be found in all aspects of life, and will continue to be used to improve the beauty of our surroundings. As humans, we desire organization and structure in our lives. Math provides such needs, and the golden ratio is just one method that is used universally.

Sources:

https://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html#Great Pyramid

http://www.geom.uiuc.edu/~demo5337/s97b/art.htm

http://www.goldennumber.net/art-composition-design/

http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature

http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html

http://www.livescience.com/37704-phi-golden-ratio.html

Math Tricks and Fermat’s Little Theorem

So you think you’re a math whiz. You storm into parties armed with math’s most flamboyant tricks. You can recite the digits of π and e to 50 digits—whether in base 10 or 12. You can calculate squares with ease, since you’ve mastered the difference of squares x2 – y2 = (x + y)(x – y). In tackling 572, simply notice that 572 – 72 = (57 + 7)(57 – 7) = 64*50 = 3200. Adding 72 to both sides gives 572 = 3249.

Tricks
Image by Hashir Milhan from
Wikimedia Commons under
Creative Commons.

You can also approximate square roots using the truncated Taylor series x ≈ c + (x – c2)/(2c) where c2 is the closest perfect square less than or equal to x. So √17 ≈ 4 + (17 – 16)/(2*4) = 4.125, whereas √17 = 4.123105 . . ..

But do you know what number theory is? It’s not taught in high school, and everyone’s repertoire of math tricks needs some number theory. Mastering modular arithmetic—the first step in number theory—will make you the life of the party. Calculating 83 mod 7 just means find the remainder after dividing by 7: 83 = 11*7 + 6, so 83 ≡ 6 mod 7. But it’s actually easier since 83 = 7*12 + (-1), so 83 ≡ -1 ≡ 6 mod 7. Modular arithmetic reveals the secrets of divisibility. Everybody knows the trick to see whether 3 divides a number; you just add the digits and check if 3 divides that number. But the reasoning is obvious when you write m = 10nan + 10n-1an-1 + . . . + 10a1 + a0 where the ai are the digits of m. Each 10k has a remainder of 1 modulo 3 so man + an-1 + . . . + a1 + a0 mod 3. Using this method generates tricks for other integers.

For instance, if 13 divides m, then 13 divides a0 – 3a1 – 4a2a3 + 3a4 + 4a5 + a6 – 3a7 – 4a+ . . . and the pattern continues. This is because

10 ≡ -3 mod 13,

102 ≡ 10*10 ≡ (-3)(-3) ≡ 9 ≡ -4 mod 13,

103 ≡ 102*10 ≡ (-4)(-3) ≡ 12 ≡ -1 mod 13,

104 ≡ 103*10 ≡ (-1)*(-3) ≡ 3 mod 13,

105 ≡ 104*10 ≡ 3*(-3) ≡ -9 ≡ 4 mod 13,

and 106 ≡ 105*10 ≡ 4*(-3) ≡ -12 ≡ 1 mod 13.

From 106 and onwards the pattern repeats. In fact, calculating 10n mod k for successive n will reveal the divisibility rule for k.

Then comes Fermat’s little theorem, the key to solving seemingly impossible calculations.

Pierre de Fermat
Fermat. Image from Wikimedia
Commons. Under public domain.

The theorem states for a prime p and integer a that aa mod p. If p doesn’t divide a, then  ap -1 ≡ 1 mod p. I’ll illustrate the power of this little result in a computation. Let’s find 2371 mod 5. We’ll be using 24 ≡ 1 mod 5, which we get from Fermat’s little theorem. Now 2371 = 236823 =(24)9223, so by the theorem, 2371 = (24)9223 ≡ 19223 ≡ 1*23 ≡ 8 ≡ 3 mod 5. Exploiting Fermat’s little theorem can impress your friends, but try to avoid questions. Computing residues modulo a composite number—calculating b mod n for a composite number n—may require paper and ruin the magic.

Leonhard Euler proved a more general version of Fermat’s little theorem; it’s called the Euler-Fermat theorem. This theorem isn’t for parties; explaining it to the non-mathematically inclined will always require paper and some time. Nonetheless, it will impress at dinner if you have a napkin and pen.

Understanding this theorem requires Euler’s totient function φ(n).

Leonhard Euler
Euler. From Wikimedia
Commons. Under public domain.

The number φ(n) for some n is the number of positive integers coprime with n that are less than or equal to n. Two numbers a and b are coprime if their greatest common factor is one. Hence 14 and 3 are coprime because their biggest shared factor is 1, but 21 and 14 aren’t coprime because they have a common divisor of 7. Moreover, φ(14) = 6 because 14 has six numbers less than or equal to it that are coprime with it: 1, 3, 5, 9, 11, and 13. Notice that if p is prime, φ(p) = p – 1 because every number less than p is coprime with p.

Now the Euler-Fermat theorem states that aφ(n) ≡ 1 mod n, which looks similar to ap -1 ≡ 1 mod p for a prime p. In fact, if = φ(p) = p – 1 for a prime p, the theorem reduces to Fermat’s little theorem.

Fermat’s little theorem has another generalization, Lagrange’s theorem. Joseph-Louis Lagrange was Euler’s student. Lagrange’s theorem generalizes both the previous theorems and doesn’t even require numbers. But due to the required background in group theory, I won’t go over the theorem. You can find links to more information on Lagrange’s theorem below.

Remember, a math whiz doesn’t need props like a magician does. Hook your audience with some modular arithmetic, and reel the people in with Fermat’s little theorem. If you want to get complicated, the most you’ll need is a pen and some paper.

Sources and cool stuff:

Math tricks: http://www.businessinsider.com/x-math-party-tricks-that-will-make-you-a-rockstar-2013-6?op=1

Modular arithmetic: http://nrich.maths.org/4350

Proof of Fermat’s little theorem: https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html

Euler-Fermat theorem and its proof: http://www.artofproblemsolving.com/Wiki/index.php/Euler%27s_Totient_Theorem

Lagrange’s theorem (only for the brave): http://cims.nyu.edu/~kiryl/teaching/aa/les102903.pdf

Fermat’s bio: http://www-history.mcs.st-and.ac.uk/Biographies/Fermat.html

Euler info: https://3010tangents.wordpress.com/2014/10/05/leonhard-euler-eulers-identity-fermats-last-theorem-and-the-basel-problem/

Lagrange’s bio: http://www-history.mcs.st-and.ac.uk/Biographies/Lagrange.html

Number theory textbooks: Gordan Savin’s Numbers, Groups, and Cryptography and George E. Andrews’s Number Theory

Interesting sources of math tricks and problems: Paul Zeitz’s The Art, Craft of Problem Solving and The USSR Olympiad Problem Book, and What is Mathematics by Richard Courant and Herbert Robbins

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

The Millennium Prize Problems

Although the Millennium Prize Problems carry a total of 7 million dollars in prize money, very few people know about them. In my opinion, this should be propagated more instead of the latest viral videos on YouTube, or the latest celebrity gossip. This was exactly the thinking of Landon Clay, who founded the Clay Mathematics Institute in 1998. Clay, a Harvard alumnus and successful businessman, is a firm believer and supporter of math and science as beneficial to all mankind, and he has set up this institute as well as the prize money.

I will give a very brief summary of each problem except the Poincaré Conjecture, which has been solved by Grigori Perelman. Keep in mind that these descriptions are not exhaustive as the areas of study of these fields are highly specialized. It cannot be possible to explain in a paragraph what mathematicians have been studying for decades or even centuries, so these descriptions will be very loose.

P vs NP

This is a problem in theoretical computer science. It involves two characters, P and NP. P represents problems that are easily solved by a computer. NP represents problems that are not necessarily easily solved by a computer but whose solutions can easily be checked if they are provided. By default, every P problems is an NP problem because if the computer can solve it, then it can check it. However, there are many examples of NP problems that aren’t known to be P (problems for which a computer can easily check a solution but for which we don’t yet know a fast algorithm for a computer to solve). Currently, many people believe that the evidence points to P not being equal to NP. Solving P vs NP would allow us to solve problems involving trillions of combinations without trying each one, and would greatly propel computing.

Hodge Conjecture

When Descartes married algebra and geometry by drawing a graph represented by a function, this revolutionized mathematics and it was never the same again. This enabled us to visualize and solve problems both geometrically or algebraically. The Hodge Conjecture implies a similar relationship exists between topology and algebra. Mathematicians soon found ways to describe more complicated shapes that were hard to imagine and only accessible via complicated equations. These shapes are known as “manifolds”. Depending on their properties, these manifolds have different “homology classes”. One example of manifolds with different homology classes is the sphere and the torus (AKA donut). In the case of the sphere, there is only one homology class: all shapes that are drawn on a sphere are homologically equivalent. In the case of a donut, there are multiple distinct homology classes. The Hodge Conjecture basically says that if you drew a random shape on a manifold, there is a rule you can apply to guarantee it can be described algebraically. Easy to describe in words but tricky to describe mathematically.

Riemann Hypothesis

A graph of the Zeta function. Image: Public domain, via Wikimedia Commons.

A graph of the Zeta function. Image: Public domain, via Wikimedia Commons.

The Riemann Hypothesis has profound implications in number theory and tries to tackle perhaps the longest standing question in mathematics: Where are all the prime numbers, and how are they distributed? The hypothesis involves the “trivial” and “non-trivial” zeroes of the complex Zeta Function. A complex function is a function that takes in complex numbers of the form a + bi, and spits out a complex number. Trivial zeroes of the Zeta function occur at negative, even integers (-2,-4,-6,…). The conjecture is that all other zeroes (“non-trivial) have the form ½+bi. Although these zeroes have been computed to the trillion digits and held true, this does not constitute a proof and a general proof is being sought. The distribution of prime numbers is seemingly random and sporadic with no telling when or where the next one will pop up in the number line. This also leads to a deeper philosophical question. How can something as structured and ordered like mathematics have something as chaotic and random as prime numbers as one of its foundations?

Birch and Swinnerton-Dyer Conjecture

Examples of elliptic curves. Image: Chas zzz brown, via Wikimedia Commons.

Examples of elliptic curves. Image: Chas zzz brown, via Wikimedia Commons.

A Diophantine equation is a polynomial equation for which mathematicians are searching for integer or rational solutions. The study these equations, is known as arithmetic geometry. A well-known example of a Diophantine equation is the Pythagorean theorem. These equations are named in honor of the Greek mathematician, Diophantus, who studied these types of equations. An elliptic curve is a graphic representation of a Diophantine Equation of the form y2 = x3 + ax + b. This conjecture states that for an elliptic curve, E, the algebraic rank and geometric rank are the same. In other words, you can find the algebraic rank by finding the geometric rank and vice versa. The rank is essentially the number of rational solutions with a 0 rank meaning a finite number of solutions, and a rank greater than or equal to 1 having infinite solutions.

Navier-Stokes Equations

Image: UserA1, via Wikimedia Commons.

Image: UserA1, via Wikimedia Commons.

The Navier-Stokes equations of fluid flow are partial differential equations that physicists use to model ocean currents, weather patterns, and other phenomena. These equations, named after Claude-Louis Navier and George Gabriel Stokes, have been stumping mathematicians for about 150 years. The problem lies in that these equations are so complex, that one cannot tell whether these equations will suddenly have a spike or blow up as time goes on. It’s similar to the story of the cat in Dr. Seuss’s “The Cat in the Hat Comes Back”, where the cat makes a stain he cannot clean up. He calls on the help of a smaller version of himself called Little Cat A, who then calls on an even smaller cat called Little Cat B and so on until the microscopic Little Cat Z unleashes a VOOM on the stain and it disappears. The Navier-Stokes problem is asking whether we can predict where the VOOMs are. Fluids can be both viscous liquids and gases and solving these equations will impact areas such as meteorology and fluid dynamics.

Yang-Mills Theory

This theory is the basis of elementary particle theory, but relies on a very weird concept, the concept of infinitely small numbers to describe the weight of these “massless” particles. This is a contradiction since these particles travel at the speed of light and anything travelling that fast must have infinite mass. New foundations and approaches to physics is required to solve this theory. Solving the mass-gap problem would mean the existence of a rigorous Quantum Field Theory. The techniques used could also be applied to other results in Quantum Field Theory.

References:

P vs NP

http://danielmiessler.com/study/pvsnp/

Hodge Conjecture

http://www.theguardian.com/science/blog/2011/mar/01/million-dollars-maths-hodge-conjecture

Riemann Hypothesis

http://qntm.org/riemann

BSD

http://theconversation.com/millennium-prize-the-birch-and-swinnerton-dyer-conjecture-4242

Navier Stokes

http://www.cs.umd.edu/~mount/Indep/Steven_Dobek/dobek-stable-fluid-final-2012.pdf

Yang Mills

http://theconversation.com/millennium-prize-the-yang-mills-existence-and-mass-gap-problem-3848