Monthly Archives: December 2014

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

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The History behind Differential Calculus

Calculus is one of the most important fields of Mathematics.  Calculus is a study of rates of change and motion, which we can see by the slope of a line or a curve. There are two major branches of calculus, Differential and Integral calculus, and they are inverses of each other. Integral calculus is used to find the areas under a curve, surface area or volume, and linear distance travel. Differential calculus (which concerns the derivative) mostly goes over the problem of finding the rate of change that is instantaneous, for example, the speed , velocity or an acceleration of an object. Differentiation is especially important in natural sciences, engineering and technology.

Image: Brandon Lim.

Image: Brandon Lim.

An example of differential calculus is if you wanted to find the velocity or the acceleration of an object, for example, a car. To find the velocity of a car, you would take the first derivative of a function (position at time t : dx/dt) and to find the acceleration you would take the second derivative of a function (dv/dt : change in velocity/change in time . This leads us to Newton’s law of motion, which is Force = Mass x Acceleration, where in this context, acceleration is the second derivative of a function.

Who was the person behind the development of calculus? Well, it wasn’t actually just one person. Sir Isaac Newton and Gottfried Wilhelm Leibniz were both credited with the development of calculus. Throughout their lives, they both argued on who came up with the idea first, both have accused each other of plagiarism. Those two weren’t the only ones who contributed to the discovery of Calculus. There have been many other known mathematician of that time that also helped with the development of calculus. For example, Rene Descartes indirectly helped create differential calculus by introducing variable magnitude.

Newton and Leibniz essentially created integral and differential calculus. They were both interested in objects that are in motion. However, they both looked at different aspects of this. Newton was more involved with the speed of a falling object and Leibniz with the slopes of curves to illustrate the rate of change. Although they both looked at different things, they both came up with the same results, hence the accusations of stealing the other’s ideas. However, combining both of their ideas, fundamental theorem of calculus was created, which links the concept integration to derivation.

It is hard to see the difference between the function and its derivative without having a visual presentation. In math, graphs are usually used to show what a function and its derivative look like. Any value of the first derivative at a given point is equal to the slope of the tangent to the graph of the function at that point. As we all know that in a graph, positive means increasing, so when the derivative is positive, the function must be increasing and when the derivative is negative, the function must be decreasing. When the value is zero at a point, the tangent is horizontal, and the function changes from increasing to decreasing, or from decreasing to increasing, depending on the value of the second derivative. The second derivative basically represents the curvature of the function. Since the first derivative shows the rate of change, the second derivative shows the rate of change of the rate of change. When the second derivative is positive, the function concave upwards and when the second derivative is negative, the function concave downwards.

To find a derivative of a function we have to make sure that the two x values are as close as possible so we can receive an accurate result. Derivative is defined by the limit of slope formulas as the x values become closer to each other. For example, we take a point which is on a curve, now we take another point that is closest to x, x+delta x. All we need to do now is plug this into the slope formula, one more thing, since we want the closest value to x, delta x has to be very small, so we find the derivative as delta x goes to 0; now we have the entire formula for derivative shown in the image.

IMG_20141220_102335

Differential Calculus helped evolve Math in many ways. It is used in many different fields of science, such as, physics, biology, and engineering.

Work Cited

http://www.edinformatics.com/inventions_inventors/calculus.htm

http://www.encyclopediaofmath.org/index.php/Differential_calculus

http://www.math10.com/en/maths-history/history5/origins-differential-integral2.html

http://science.jrank.org/pages/1134/Calculus-Differential-calculus.html

http://www.wyzant.com/resources/lessons/math/calculus/differentiation

 

Where are the Women?

Women have always been in the shadows of the technical fields. Throughout history, there have been a lucky and likely deserving few who have been formally acknowledged for their work and contributions in their fields. Today, women have such a great opportunity for the education and career option in these technical fields such as physics, engineering, computer science, and of course mathematics compared to those in the past. So why is it that we still find all of these fields significantly dominated by men?

There’s always the popular myth that women just aren’t as good at math and science as men are, though this is generally just a “gentler” way of claiming women are at the biological disadvantage of not being men. Even though once upon a time there were statistics to back up this theory, this is definitely no longer the case. In fact, many believe the primary reason for the statistics in the past is under representation. Because really, if only one girl in her grade takes a math course, and does remarkably average while the boys average out to above average, all the statistics are going to show is that the female population didn’t do nearly as well as the men. More recent studies (though still going back into the 1990s) have revealed a much different pattern: that the gap between genders in math and science abilities has become a myth itself.

So why then, despite women’s increased achievement levels in these subjects, are there so few making it to careers in math and science?

From a young age, girls most often rate their own mathematic ability lower than the boys do, despite there being no evidence of this fact reflected by grades or their teachers’ reporting. This very important finding could be the biggest clue we have to why we don’t find as many girls pursuing higher education or careers in these fields: poor perceptions of their own abilities. This is such an easy problem to help prevent if teachers and parents actually take the time to give informational, detailed feedback instead of a simple “this was wrong”. This praises the effort rather than the correctness of their result, encouraging them to understand the material rather than be discouraged by an incorrect conclusion.

Another problem come to light is the psychological pressure that stereotypes can put on a woman, doing just fine in her math and science classes, to leave the fields. This perceived threat most often occurs when a woman finds herself faced with the negative stereotypes surrounding female ability in the mathematic and scientific fields, despite there being no evidence of such differences. It is a thing she hears from the time she is young, that men are “just better” in many things, and these negative ideas appear on a day to day basis for most women in the fields. Another threat stereotypes bring along is an attack on a woman’s feminine identity. Women in math and science are often stereotypically portrayed in general media as being asexual, married to their work, physically unappealing, or just not real women. And while some women are very happy to fit any number of these ideas, which is completely fine of course, there are many that are so threatened by such ideas that it impacts their decision on whether or not to pursue the field, no matter their own ability.

Image: xkcd.com.

Then there’s always putting the representation of the entire gender on one girl’s shoulders. When one addresses a female student as being good or bad at something “for a girl”, they place the weight of upholding the reputation of all her female peers, past, present, and future, on her. There’s so much pressure on these girls not to fail, that many do not wish to even attempt. They’d rather step out of the field entirely than risk damaging such a fragile reputation.

Yes we have come a long way, and yes it’s such a wonderful improvement from how it used to be. Women do have professional and educational opportunities now that women even a few decades ago could never dream of having, and the woman that seize these opportunities and achieve so much are such inspirations to us all. So why don’t all women take advantage of the opportunity? Encouragement is a powerful determinant in anyone’s life, how can you blame so many women for falling out of the maths and sciences if they receive such an obvious lack of personal encouragement? Anyone can stand in front of a crowd and tell them women can do anything they set their mind to, but when faced one-on-one with someone who uses phrases like “you’re pretty good for a girl”, the effect is so much stronger. It’s not just the female perspective on women’s mathematical abilities that need improvement, but the male perspective as well.

References:

http://www.nytimes.com/2013/10/06/magazine/why-are-there-still-so-few-women-in-science.html?pagewanted=all&_r=1&

http://thesocietypages.org/socimages/2013/03/07/the-truth-about-gender-and-math/

http://www.apa.org/monitor/2010/07-08/gender-gap.aspx

http://secure.edc.org/WomensEquity/pubs/digests/digest-math.html

Infinite Series and the Riemann Hypothesis

I was fascinated by studying infinite series in calculus; the idea of adding up infinitely many elements and possibly having a finite number was not intuitive for me at first. These problems can help in bridging the gap between philosophical and practical aspects of mathematics. For instance, the famous Zeno’s paradox argues the impossibility of movement based on the infinite divisibility of space. With this paradox and assuming you have a certain distance to travel, you must first travel half that distance and then you must travel half the remaining distance, and then again half of the remaining distance and so on forever. So looking at this problem it seems as if we might not ever be able to get anywhere. However, if you were to stand a certain distance away from a wall and then attempt to do this, at some point you would undoubtedly walk yourself right into the wall. So is mathematics lying to us?

The answer no, this problem can be represented as an infinite series, 1/(2n) from n = 1 to n = infinity, which represents adding up the following elements (1/2 + 1/4 + 1/8 + 1/16 …). The problem is that we could never physically add up the infinitely many terms on a piece of paper, or even in all the books filling a library. What we can do is add up a portion of the sum, called a partial sum, to some nth element. Once we find the partial sum we can use calculus to take the limit as n grows increasing larger to infinity and see if we arrive, or converge, to a finite number. So for the series above, the partial sum can be represented as  1 – 1/(2n), and after taking the limit as n approaches infinity the second term goes to 0 and we are left with 1, a finite number. Even though you are adding up infinitely many elements, after traveling deep into your series, the elements that you are appending to the sum become so small that they are negligible to the total sum. These types of problems appear often in mathematics and especially integral calculus. In order to find the area under a curve in calculus we end up taking Riemann sums, drawing rectangles under the curve to approximate the area. Then we examine these sums when we take more and more smaller rectangles (infinitely many) that better approximate the area and eventually our our approximation turns into the actual solution. Infinite series appear all over in mathematics.

Perhaps the most famous mathematical problem is the unsolved Riemann Hypothesis. This problem deals with finding the roots of the Riemann Zeta Function which itself is an infinite series. This function maps a complex number s to the series (1/1s + 1/2s + … + 1/ns) from n = 1 to n = infinity, or more formally written as:

RiemannZeta This function is defined for all complex numbers where the real part is greater than 1. This means that plugging in a complex number with real part greater than 1, the series will add up to, or converge to, a finite number. The mathematician for which the Zeta function was named after, Bernhard Riemann, was able to prove an analytic continuation of this function so that it is defined for all complex numbers s, except s = 1.

An analytic continuation is a technique used in complex analysis to extend the domain of an analytic function. The method is to find a new function which is defined over a larger domain. If the new function equals the old function on the intersection between the original function’s domain and the new function’s domain, then the new function is called an analytic continuation of the original function. So in the case of the Riemann Zeta function, an analytic continuation would be a new function which is defined exactly the same as the Zeta Function for complex values where the real part is greater than 1. However, the analytic continuation would also be defined on a larger domain, where complex values with real part less than one also have a value. For instance, plugging in the value s = -1 into the original Riemann Zeta Function would result in the series 1 + 2 + 3 + 4 + … off to infinity, which most of us will clearly recognize as being a divergent series. However, it is often said, especially in physics and string theory, that 1 + 2 + 3 + 4 + … = -1/12. This is because by use of Riemann’s analytic continuation, ζ(-1) = -1/12. Using this technique, all complex numbers besides s = 1, are defined for the Riemann Zeta Function. For s = 1 the function outputs the harmonic series, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … This series is not always clear to those who are first learning about convergence and divergence. The function 1/n tends to zero as it grows, but using methods in calculus we are able to prove that it is actually a divergent series and that it sums to infinity. Riemann was not able to account for this series in his analytic continuation, so the value s = 1 is a single pole, or singularity, where the function is undefined.

Some of the roots of the Zeta function are obvious to mathematicians, such as all of the negative even integers. Riemann conjectured that all other non-trivial roots would have a real part of 1/2, meaning on the complex plane all nontrivial roots would fall on the line Re[1/2]. However, while we have found many roots on this critical line (millions) this conjecture has yet to be proven. This is one of the Millennium Problems, so whoever does end up proving or disproving this will receive a prize of one million dollars awarded from the Clay Mathematics Institute. This problem is not simply about finding where a function equals, but has deep connections with number theory and specifically with the distribution of prime numbers where other proofs are yet to be complete without it.

Convergent infinite series are really interesting and not necessarily intuitive upon a first learning about them. Sometimes you can worry yourself thinking about the notion of infinity or the infinitesimal, it is hard to relate to something so large or small. The idea of zero is a little easier because we are more comfortable with the idea of nothing, but “a number smaller than any other number” is hard to wrap your mind around. So adding up these numbers that are practically nothing, but still something, seems like such an incredible mathematical achievement to me.

References:

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

Davis, Philip J., and Reuben Hersh. The Mathematical Experience. Birkhauser Boston, 1981. Print

Sondow, Jonathan and Weisstein, Eric W. “Riemann Zeta Function.” http://mathworld.wolfram.com/RiemannZetaFunction.html

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

Why Should You Learn Math?

Martin Gardner. Image: Konrad Jacobs, Erlangen, via Wikimedia Commons.

Martin Gardner. Image: Konrad Jacobs, Erlangen, via Wikimedia Commons.

People often ask why they should learn maths, what is it good for, or what is its practical purpose.  Many seem to think of literature, film, or history differently.  People see these things enhancing their lives everyday when they get a reference, or can recite an interesting fact to friends.  They get a bit of delight when they understand the phrase tilting at windmills, when they can recite some shakespeare to a loved one, or when they make friends laugh at an anecdote about Emperor Norton I.  They don’t realize that knowledge about maths can enrich their lives in similar ways.  Maths can be enjoyed in many aspects of life, such as understanding the jokes in the simpsons, understanding the origins of some idioms, recognizing absurd laws, and enjoying different kinds of puzzles and games.

The Simpsons is chock-full of maths jokes and references.  In the episode “The Wizard of Evergreen Terrace” Homer Simpson models himself after Thomas Edison.  On a blackboard seen in the episode is the equation 398712 + 436512 = 447212 (“The Wizard of Evergreen Terrace”).  This is a joke about Fermat’s Last Theorem.  Fermat’s Last Theorem says that there are no integer solutions to the equation an + bn = cn when n is an integer greater than 2.  The equation on the board seems to contradict this if you plug it into a calculator.  This is because your calculator may not remember all the digits of the numbers as it calculates them.  The numbers on the board were picked by the writers as a joke to look like Homer had found a counterexample to FLT.  In the episode “Treehouse of Horrors VI”, Homer is pulled into the third dimension and runs into several maths references.  These include another false counterexample to Fermat’s Last Theorem, the statement P=NP, and euler’s identity, eπi= -1 (“Treehouse of Horrors VI”).  There are many other references in The Simpsons and its sister show Futurama.  These jokes and references come from the mathematicians who are working as writers on these shows.  The writers enjoy maths, and they can see that it can also be funny.

Mathematics sometimes even enters everyday speech.  If you have ever heard someone say something like “Your expectations do not square with reality” or “That is as hard as squaring the circle”  you know some idioms based on mathematics.  These phrases come from the ancient problem of squaring the circle.  The problem is trying to construct a square with the same area of a circle using only a straightedge and compass.  The first reference of the problem is from plutarch.  He claimed that when Anaxagoras was imprisoned he spent his time trying to construct a square with the same area as a circle (Boyer 57).  This problem persisted for thousands of years until, in 1882, it was proved to be impossible.

The impossibility to square the circle was not known to Edwin J. Goodwin, a physician from a small Indiana town.  Goodwin thought that he came up with a proof that he could square the circle, but it depended on the ratio of the circumference of a circle to its diameter to be 3.2 (Singh 24).  You might be aware (and if you are reading this blog you probably are aware) that the ratio of the circumference of a circle to its diameter is equal to π, an incommensurable number.  In 1897, Goodwin proposed a bill to the Indiana General Assembly to legislate the value of π.  He offered to let the schools in Indiana use his “Discovery” for free, and he offered to split the royalties from it with the state.  Sadly, the Indiana House of Representatives members did not understand the mathematics in the bill and passed it (Singh 25).  Luckily, when it got to the state senate there was a mathematician in the building.  The head of the Purdue University Mathematics Department, C. A. Waldo, saw the bill and explained the absurdity of it to the legislators.  After their brief lesson from Waldo, the state senators mocked the bill and did not pass it (Singh 25).

Mathematics can also provide a great deal of casual fun in the form of games and puzzles.  Martin Gardner’s column, Mathematical Games, in Scientific American introduced people to interesting things based on mathematics.  The column contained information about a variety of topics, including Graham’s Number, Hexaflexagons, and on April Fools in 1975 it contained a false counterexample to the Four Color Theorem.  Today, many people play video and board games that are complicated enough that knowing some mathematics will allow you to enjoy the game on more levels.  Minecraft is well known for allowing players to express their creativity and build whatever they would like.  In the game, the item redstone works like a circuit.  Redstone has 2 states, on and off, and can be used to make logic gates.  This lets players learn about boolean logic while they are trying to construct their crazy contraptions.  In strategy games, like Twilight Imperium or Risk, understanding the basics of probability is vital.  You need to understand how the dice are going to behave to know when you have the advantage or your opponent has it.  Should you attack twice with each attack having a low chance to succeed, or should you attack only once with a much better chance to succeed?  Should you buy the technology that increases damage with each hit or the one that gives a better chance to hit?  These kinds of questions come up all the time and can be answered by analyzing them mathematically.

Everyone knows that mathematics is useful for practical purposes.  We know that the sciences rely heavily on mathematics, that we should understand how interest works if we take out a loan, and that there are many other places that require maths in a practical way.  However, many people do not see that a little understanding of mathematics can expand their world in countless ways.  Mathematics is far more than just playing with numbers or a tool to help physicists.  Mathematics can be a way to understand a joke, it can help you play a game, or it can help you understand those around you.

Sources

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

Singh, Simon.  The Simpsons and Their Mathematical Secrets. New York: Bloomsbury Publishing  Plc, 2013. Print.

“The Wizard of Evergreen Terrace” The Simpsons. Fox. KVVU-TV, Henderson. 20 Sep. 1998. Television.

““Treehouse of Horrors VI”” The Simpsons. Fox. KVVU-TV, Henderson. 29 Oct. 1995. Television.

Three Centers of a Triangle

There’s far too little geometry—excluding topology and non-Euclidean stuff—on this blog, so let’s add a little.

Euler Line

Euler line HU. Points H, U, and S are
respectively the circumcenter, centroid,
and orthocenter. Image: Rene Grothmann at the German Language Wikipedia.

Our goal is to get to the Euler line, a line that passes through a triangle’s circumcenter, centroid, and orthocenter. The line is only determined for non-equilateral triangles; the points coincide in the equilateral case. We’ll look at the three points above.

The circumcenter, centroid, and orthocenter are all “centers” of triangle. But what is a center of a triangle? Surely, it’s not a point equidistant to all points on the triangle. Our triangle would be a circle in that case.

The circumcenter of a triangle ABC is the center O of the circle K that triangle ABC is inscribed in.

Circumcenter

Circumcenter O of triangle ABC. Image drawn by me.

The circumcenter is actually the intersection of the three perpendicular bisectors of the triangle: FE, IG, and DH. To see this, first suppose that triangle ABC has a circumscribed circle K with center O. Draw radii AO, BO, and CO to each of the triangles vertices. This creates three smaller triangles AOBBOC, and AOC. In each of these smaller triangles, drop an altitude from O. For example, in triangle AOBaltitude OD would be dropped. This splits AOB into two smaller triangles that are congruent by SAS, Line OD is perpendicular to AB by construction, and AD = DB. Hence OD is indeed a perpendicular bisector of side AB. Repeating this for other sides shows that the center of the circumscribed circle is the intersection of ABC‘s perpendicular bisectors.

Moreover, the intersection of any to perpendicular bisectors is equidistant from each of the triangle’s vertices. The reader can see this by considering triangle AOC. Perpendicular bisector IG splits AOC into triangles that are congruent by SAS. It follows that lengths AO and OC are equal. Repeat for the other sides. We then see that the intersection of the perpendicular bisectors is equidistant from the triangle’s vertices. Thus the perpendicular bisectors of a triangle uniquely determine its circumcenter.

The centroid is the intersection of a triangle’s three medians, lines drawn from a vertex that bisect the opposite side. As said in class, the centroid is the center of mass for a thin, triangular solid with uniformly distributed mass.

Centroid

Centroid O of triangle ABC. Drawn by me.

The reader may suspect whether the three medians of a triangle intersect. Clearly two of the medians intersect; otherwise our triangle ABC would be a line. But the full proof is a little tedious. The proof involves assuming that two medians AF and CE intersect and drawing a parallelogram using the midpoints of the medians. We link to some proofs: http://jwilson.coe.uga.edu/EMAT6680Fa06/Chitsonga/MEDIAN/THE%20MEDIANS%20OF%20A%20TRIANGLE.htm uses classical geometry and http://math.stackexchange.com/questions/432143/prove-analytically-the-medians-of-a-triangle-intersect-in-a-trisection-point-of uses vectors.

Triangles

Four congruent triangles using midpoints. Image drawn by me.

Interestingly, the midpoints of the sides of triangle ABC—the ends of the medians—cut the triangle into four congruent triangles. We will prove this in a roundabout way. Let E be the midpoint of AB. Draw a line EF parallel to AC where F intersects BC. Similarly draw FD parallel to AB. By construction, EFDA and EFCD are parallelograms. Then AD = EF = DC, so D is the midpoint of AC. Similarly, F is the midpoint of BC. The reader can see that the triangles are congruent by repeatedly applying SAS.

Our final center is the orthocenter, the intersection of the three altitudes of a triangle. An altitude is a segment drawn from a vertex that is perpendicular to the opposite side. As with the two previous centers, the intersection of the altitudes at a single point isn’t immediately obvious.

Orthocenter

Orthocenter O of triangle ABC. Drawn by me.

We show that the altitudes of triangle ABC intersect. Construct triangle DEF with triangle ABC inscribed in it by making sides DF, FE, and DE parallel respectively to BC, AB, and AC. Draw altitude BK where intersects DF. Since AC is parallel to DEBK is perpendicular to DE. Moreover, ADBC and BACE are parallelograms, so DB = AC = AE. Hence BK is a perpendicular bisector of DE. We repeat the argument for the other altitudes of triangle ABC. Then the altitudes of ABC intersect because the perpendicular bisectors of DEF intersect.

There are a few other centers of a triangle that are either irrelevant to the Euler line or take too long to construct (i.e. I’m tired of drawing diagrams). The incenter is the center of the circle inscribed within a triangle. The incenter also turns out to be the center of a triangle’s angle bisectors. The Euler line doesn’t pass through the incenter.

The nine-point circle is the circle that passes through the feet of the altitudes (the end that isn’t the vertex) of a triangle.

Nine-Point Circle

Nine-point circle of ABC. Image: Maksim, via Wikimedia Commons.

Strangely, the circle also passes through the midpoints of the sides of its triangle. But that’s not all. The circle passes through the Euler points, the midpoints of the segments joining the triangle’s vertices to the triangle’s orthocenter. Thus the nine-point circle does indeed pass through nine special points of a triangle. The center of the nine-point circle lies on the Euler line.

After all this, we still haven’t proved that the circumcenter, centroid, and orthocenter lie on the same line. We won’t prove this. Here’s a video of the proof by Salman Khan: https://www.youtube.com/watch?v=t_EgAi574sM. The proof uses a few facts about the centers we haven’t discussed, but these facts aren’t too hard to show. Refer back to my four congruent triangles picture. Let O, K, and L respectively be the circumcenter, centroid, and orthocenter of triangle ABC. Then Khan proves that triangle DOK is similar to triangle BLK. This implies angles DKO and CKL are equal, which means O, K, and L lie on the same line.

Sources and cool stuff:

H.S.M. Coxeter and Samuel L. Greitzer’s Geometry Revisited

Paul Zeitz’s The Art and Craft of Problem Solving (Chapter 8 is called “Geometry for Americans”)

Wolfram on the nine-point circle: http://mathworld.wolfram.com/Nine-PointCircle.html

A fun way to play with the Euler line: http://www.mathopenref.com/eulerline.html

Khan’s Euler line video: https://www.youtube.com/watch?v=t_EgAi574sM

Wolfram on the Euler line: http://mathworld.wolfram.com/EulerLine.html

Classical median proof: http://jwilson.coe.uga.edu/EMAT6680Fa06/Chitsonga/MEDIAN/THE%20MEDIANS%20OF%20A%20TRIANGLE.htm

Vector median proof: http://math.stackexchange.com/questions/432143/prove-analytically-the-medians-of-a-triangle-intersect-in-a-trisection-point-of

Modular Arithmetic and how it works

As children, we grew up learning how to count to 10. Why 10? Well this could be easily justified using the fact that we as humans have 10 fingers and any whole number up to 10 could be easily represented by a quick show of fingers. But what happened when we, as children with this new found power of counting objects up to 10, encountered a number greater than 10? Did we take off our shoes and start counting with our toes? That might have solved the issue for numbers greater than 10 but less than 20 (assuming you aren’t polydactylic) but in all reality, we needed a way to transcend the idea of representing objects with our fingers and/or toes and represent any number, no matter how large.

How did we do this? By using a Place Value system with a base 10. “Place Value” means that using a limited number of symbols, we can represent any number by using these symbols in a variety of combinations. The value of each symbol is based on the position or “place” where the symbol is located in the sequence of symbols.

For example, pick a base. The very first column or “place” should be used for all the symbols preceding the base until the base itself is reached. This is called the “units” place or informally as the “ones” place. This place is usually the farthest left or right place in a sequence. For instructional purposes and for familiarity, we will place the units place on the far right of the sequence.

Once the base has been reached, a second place will be added to the left indicating how many “bases” have been reached. When the amount of “bases reached” has reached the base amount, then a new place is added, again to the left, indicating how many bases of bases have been reached and so on and so forth.

For example, our familiar base 10 system works as follows:

_____ . . . _____ _____ _____ _____

A comma is added after every 3 digits for practical purposes to easily differentiate places in more complex combinations of numbers.

Now let’s go back 4,000 years ago to Sumer, a region of Mesopotamia, (modern-day Iraq). There, children learned to count, but using 60 as a base. Why 60? Did the children of that time have 60 fingers and/or toes? Probably not. The reason for using this number as a base has not been explicitly recorded but there are two convincing hypothesis on why a base 60 number system developed.

One idea, is that instead of using their whole finger to represent a single number, the Babylonians actually counted the 12 knuckles of the four fingers on one hand, using the thumb as a “pointer” and the five fingers on the other as multiples of twelve. So on one hand they had 1-12 and on the other they had how many 12’s, for a total of 12 x 5 = 60.

The other idea is that the number 60 has many divisors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. In fact, 60 is the smallest number divisible by all integers from 1 to 6. This could prove very useful by being able to do division using more whole numbers and resulting in less fractions.

Base 60 is still used in many aspects of our lives today such as the 60 seconds in a minute and the 60 minutes in an hour. The circles is traditionally divided into which are also subdivided into 60 minutes of arc and further divided into 60 seconds of arc.

Modular Arithmetic

Image: Spindled, via Wikimedia Commons.

Image: Spindled, via Wikimedia Commons.

Now that we have a brief overview about bases, we can apply the power of Modular Arithmetic to change counting bases. Modular Arithmetic is a very handy and useful tool in mathematics invented by the famous Mathematician Carl Friedrich Gauss in 1801. We know that the number line is infinitely long but if we were to wrap this infinitely long number line around a circle of a given circumference n, we would notice that numbers would “line up” or over lap around the circle. This is the idea behind modular arithmetic. Keep in mind that we are dealing with integers here and not the real numbers.

So the number indicating how large the circle is n, is called the modulus. And we say that after one wrap around, any numbers that line up are congruent. In mathematical terms, when a number a, leaves the same remainder as a number b, we say a and b are congruent written

a ≡ b mod n

The “mod n” part is just notation letting us know that we are in mod n and is not actually part of the equation, per se. However, when the context is understood, it should be OK to omit writing this every time.

In general, any modulo n has n residue classes, one for each integer from 0 to n-1.

Let’s use the timer on your microwave as an example of a base. So we will have n residue classes from the integers 0 to 59.

0, 1, 2, 3, … 56, 57, 58 59

We call this modulo 60 or mod 60 for short. When we add 1 to 59, we return to 0. This is true for any modulus, even our own familiar base 10 (when we add 1 to 9, we return to 0) or even every day objects like traffic lights (Red, Green, Yellow, Red, …). The integers from 0 to 59 in our base 60 example are called Residue Classes.

Now for a quick example, when I was in the military, we would tell time using the 24-hour clock. This is different than the usual 12 hour clock where all 24 hours are represented twice and distinguished using A.M. or P.M.

So when I would get asked what time I would be ready to get picked on Friday for the weekend, I would reply 1600. Of course this did not make sense to most people because the face of a clock only has the numbers 1-12 listed on it. How could I explain correctly to them what time to pick me up so as to maximize our time together on the sunny beaches of San Diego? Using modular arithmetic of course!

Numbers are said to be congruent if their difference is divisible by the modulus. Or stated more succinctly, a is congruent to b if a-b is divisible by n shown algebraically

a ≡ b mod n if a-b / kn for some k

This basically means that the difference must be divisible by the base.

In our example, let’s show that 1600 is congruent to 4:00. For lingo purposes, just think of the colon as “hundred hours” to be in step with 1600. 1600 – 400 is 1200, a multiple of 12. Written

1600 ≡ 400 mod 1200

1600-400 /1200

“So 4:00 P.M. civilian. Don’t be late.”

Another cool example of things you can do with modular arithmetic is calculate the last digit or remainder of a huge number like . Try doing that by hand! Here is how we would do it mod 10.

1919 ≡ 919; (because 19 is congruent to 9 mod 10)

(92)9*9 ≡ (81)9*9 ;

(1)9*9 ≡ 9; (because 81 is congruent to 1 mod 10)

References:

Use of base 60 using hands

https://www.youtube.com/watch?v=_XBJBG3vKS8&list=PLMpRfUAf1yQGhrBCe6moyrW49mJlNgf0n

Base 60 as a base

http://www.storyofmathematics.com/sumerian.html

Sub-divisions of angles into minutes and seconds

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

Modular Arithmetic

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

https://brilliant.org/discussions/thread/modular-arithmetic-2/