Tag Archives: algebra

The path to Analytic Geometry (Or a few of the many geniuses it took to learn 5th grade math)

Analytic geometry is the study of geometry using a coordinate system. Basically it’s the idea of expressing geometric objects such a as a line or a plane as an algebraic equation, think y=mx+b or ax+by+cz=k. This may be done by use of the more familiar Cartesian coordinates, by something such as polar coordinates or by just about any system for defining coordinates in a Euclidean space. The Common Core has the concept of graphing introduced in 5th grade, and graphing simple functions in the 8th grade. It’s quite interesting that something which took brilliant men so long to develop is now introduced to ten year olds.

The earliest evidence of anything resembling analytic geometry was by the Geek mathematician Menaechmus (380–320 BC), who was a student of Eudoxus and a tutor of Alexander the Great. Proclus and Eutocius both report that Menaechmus discovered the ellipse, hyperbola and parabola and that these were initially called the “Menaechmian triad”. These were used along with something resembling analytic geometry to solve the Delian problem, which is to, given the edge of a cube to construct the edge of a cube with double the volume. Though most of what we know of Menaechmus and his exact solution is second hand as his original work was lost, it appears as though he argued his solution for doubling the cube with proportions of a side length to the area of a side which fairly quickly leads to conics.

Another early manifestation of analytic geometry was by Omar Khayyám, whom we have mentioned in class. He drew a connection between algebra and geometry in his solution of general cubic equations. His idea to do this was to create a geometrical construction of a cubic equation by considering the variable to be the edge of a cube and constructing a set of curves from which a solution could be discerned. While it might seem far flung from Cartesian coordinates it was a significant leap in connecting the separate concepts of algebra and geometry.

Analytic geometry was more or less formalized in the early 17th century independently by René Descartes and Pierre de Fermat. Descartes published first and so he is commonly credited as the sole creator which leads to analytic geometry often being call Cartesian geometry. As Fermat has already been much discussed, I’ll skip his background and instead jump to Descartes. René Descartes was a French mathematician and philosopher who is most well known as the (co-)creator of analytic geometry and as the father of modern philosophy. He is the origin of the well-known quote “Je pense, donc je suis” or “I think, therefore I am” which appeared in in Discours de la methode (Discourse on the Method).

While the Fermat and Descartes constructions are equivalent, they did differ in several ways which primarily stem from which direction their creator worked. Fermat started with the algebraic equation and described the analogous geometric curve while Descartes worked in reverse, starting with the curve and finding the equation. To contrast the methods, the way most of us learn analytic geometry is much more similar to Fermat than to Descartes, where we learn to recognize that a degree 1 polynomial will represent a straight line then we learn how to find that line, next that quadratic function represents a parabola and so on. Whereas if we were to learn as Descartes’ work, we would take a straight line then learn that it represented a degree 1 polynomial which is similar to Fermat.  But then working further in this direction, it doesn’t make sense to jump to parabolas and instead to talk about conics and all degree 2 polynomials with no reason to talk specifically about parabolas.

In 1637, Descartes published his method of connecting arithmetic, algebra, and geometry in the appendix La géométrie (The Geometry) of Discourse on the Method. However, given Descartes’s opaque writing style (to discourage “dabblers”) as well as The Geometry being written in French rather than in the more common (for academic purposes) Latin, the book was not very well received until it was translated into Latin in 1649, by Frans van Schooten, with the addition of commentary clarifying certain arguments. Interestingly, though Descartes is credited with the invention of the coordinate plane, since he describes all necessary concepts, no equations are in fact graphed in The Geometry and his examples used only one axis. It was not until its translation into Latin that the concept of 2 axes was introduced in Schooten’s commentary.

One of the most important early uses for analytic geometry was to help prove the validity of the heliocentric theory of planetary motion, the (then) theory that the planets orbited around the Sun. As analytic geometry was one of the first methods one could use to actually make computations about curves, it was used to model elliptical orbits so as to demonstrate the correctness of this theory. Analytical geometry, and particularly Cartesian coordinates, were instrumental in the creation of calculus. Just consider how you might calculate something like the “area under the curve” without the concept of the curve being described by some algebraic equation. Similarly, the idea of rate of change of as function of time at a particular time becomes much clearer when thought of as the slope of the tangent line, but to do this, we need to think of the function as having some representation in the plane for which we need analytic geometry.

Sources :

https://www1.maths.leeds.ac.uk/~sta6kvm/omar.pdf

http://www.math.wichita.edu/history/Men/descartes.html

http://www.encyclopedia.com/topic/Pierre_de_Fermat.aspx

Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics) Jul 7, 1999

by A. D. Aleksandrov and A. N. Kolmogorov

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

http://academic.sun.ac.za/mathed/Shoma/MATUNIT12_02.htm

http://www.cut-the-knot.org/WhatIs/WhatIsAnalyticGeometry.shtml

http://www.cut-the-knot.org/WhatIs/WhatIsAnalyticGeometry.shtml

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

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

http://en.wikipedia.org/wiki/La_G%C3%A9om%C3%A9trie

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Ancient India’s Mathematical Impact On The World

I’ve always wanted to travel to India, and I’m finally getting a chance to visit Chennai (along with some other places) this winter break.  I’ll be teaching my company’s Chennai, India team about service oriented architecture automation – aka boring computer stuff. However, I’ve also set some time aside to go sightseeing on the company’s dime!  We always seem to bring up India-birthed math topics, or mathematicians in class, so I thought it would be very fitting to blog about how India has impacted us!   Make sure you get your Tetanus, Diphtheriaand Typhoid booster shots, this journey may get a little out of hand!

*Spoiler alert: You can’t contract any foreign diseases from a blog post.

When I think of India, computer software, call centers, spicy food, and the Taj Mahal come to mind.  After making my way past these generalizations, I started to see how crucial this South Asian country’s mathematical contributions have been to mankind. India has been credited with giving the world many important mathematical discoveries and breakthroughs – place-value notation, zero, Verdic mathematics, and trigonometry are some of India’s more noteworthy contributions. This country has bred many game-changing mathematicians and astrologists. Over the course of my research I identified the “big three” mathematicians. The first, and arguably most important mathematician and astronomer (Ancient astronomers are similar to modern day astrologist!)  in India’s history, was Aryabhata.  Soon after Aryabhata, came Brahmagupta.  Brahmagupta followed in Aryabhata’s footsteps and built upon some of his more groundbreaking theories. Nearly 500 years later Bhaskara II (Not to be confused with Bhaskara I.) was born. While building upon the mathematical and astronomical work of his forefathers, Bhaskara II also paved his own way to become one of the “greats”. The “big three’s” findings, laid down some of the most vital building blocks in the history of mathematics, but how has that impacted us?

Aryabhata

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

Aryabhata

We will start off on this journey with Aryabhata (sometimes referred to as Arjehir), a well-known astrologist and mathematician, born in the Indian city of Taregana sometime between 476-550 AD. He lived during a time period we now refer to as “India’s mathematical golden age” (400-600 AD), and it is of no surprise why historians recognize this time period; Aryabhata’s achievements really were golden. He is most noted for dramatically changing the course of mathematics and astronomy through many avenues, which he recorded in a variety of texts.

Sanskrit writing. Image: Diggleburnz, via Flickr.

Sanskrit writing. Image: Diggleburnz, via Flickr.

Over the course of many wars and centuries, only one of Arybhata’s works survived. Aryabhatiya, which was written in Sanskrit at the age of 23, recorded the majority of his breakthroughs. Oddly enough, he only referenced himself 3 times throughout his workWithin this text, Aryabhata formulated accurate theories about our solar system and planets, all without a modern-day telescope. He recognized that there were 365 days in a year. He developed simplified rules for solving quadratic equations, and birthed trigonometry. Aryabhata’s original trigonometric signs were recorded as “jya, kojya, utkrama-jya and otkram jya” or sine, cosine, versine (equivalent to 1-cos(θ) ). He worked out the value of as well as the area of a triangle. Directly from Aryabhatiya he says: “ribhujasya phalashariram samadalakoti bhujardhasamvargah”. This translates to: “for a triangle, the result of a perpendicular with the half side is the area”. Most importantly, in my opinion, he created a place value system for numbers. Although in his time, he relied on the Sanskritic tradition of using letters of the alphabet to represent numbers. Aryabhata did not explicitly use a symbol for zero however. It kind of hard to conceptualize, but none of these things had ever been done, at least to this extent, before.

Brahmagupta

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

Brahmagupta. Image: public domain, via Wikimedia Commons.

Brahmagupta was born in Bhinmal, India presumably a short time after Aryabhata’s death in 598 AD. He wrote 4 books growing up, and his first widely accepted mathematical text was written in 624 when he was only 26 years old! I find it funny that most of the chapters in his texts were dedicated to disproving rival mathematicians’ theories. Brahmagupta’s most notable accomplishments were laying down the basic rules of arithmetic, specifically multiplication of positive, negative, and zero values. In chapter 7 of his book, Brahmasphutasiddhanta (Meaning – The Opening of the Universe), he outlines his groundbreaking arithmetical rules. In the context below, fortunes represent positive numbers, and debts represent negative numbers:

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.

However it seems Brahmagupta made some mistakes when explaining the rules of zero division:

Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.

Since our early teens we’ve know anything divided by zero is not zero. When zero is the denominator, the fraction will always “fall over” – that’s how I learned it as a youngin! However, we still have to give Brahmagupta credit, he was so close to getting it all right.

Bhaskara II

Bhaskara II is similar to the other mathematicians we’ve discussed in this post.  He was born in 1114 AD, in modern day Karnataka, India.  He is known as one of the leading mathematicians of India’s 12th century.  He blessed the world with many texts but Siddhanta Shiromani, and Bijaganita (translates to “Algebra”) are the ones that have shined through the centuries.  These specific texts documented some of his more important discoveries. In Bijaganita, Bhaskara demonstrated a proof of the Pythagorean theorem, and introduced a cyclic chakravala method for solving indeterminate quadratic equations:

y = ax2 + bx + c

Coincidentally, William Brouncker was credited for deriving a similar method to solve these equations in 1657, however his solution is more complex. From Siddhanta Shiromani, Bhaskara gave us these trigonometric identities:

 sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
sin(a – b) = sin(a) cos(b) – cos(a) sin(b)

If I had a dollar for every time I relied on these identities, or any of their variations throughout my mathematical career, I’d probably have enough money for a new laptop! Although Newton and Leibniz are credited for “inventing” calculus, Bhaskara had actually discovered differential calculus principles and some of their applications.

A World Without Aryabhata, Brahmagupta and Bhaskara II

I know this is a long shot, but let’s entertain the idea of a world without any of Aryabhata’s, Brahmagupta’s, or Bhaskara’s work.  Granted, future mathematicians would have undoubtedly discovered a portion of the “big three’s” breakthroughs, at least in one way or another. While it’s pretty obvious someone else would’ve invented a number system with a placeholder, or a zero equivalent, it’s not as clear with more complex things such as trigonometry. The foundation built by the “big three” could’ve altered slightly. This alteration could’ve given us a Leaning Tower of Pisa rather than an Eiffel tower – metaphorically speaking, that is. The main point you have to realize is: without the “big three” the progression of mathematics would have been slowed in one way or another, thus effecting our world today. If the “big three” didn’t exist there’s no telling how far back it could’ve set humanity.

That being said, these mathematicians’ theories, methods, and proofs served as building blocks for other mathematicians (globally). If you want to build out a brilliant theorem or proof, you have to start with, or at least incorporate the basics, at some point. Without these basics, the world would have been set back, at least in the realm trigonometry and algebra. It’s hard to imagine using any other number system than what we use today, especially without a numerical placeholder! Young children would be less eager to learn math because writing down large numbers would be a tedious process.  What would we have used in place of zero? What about  math with negative numbers?

Trigonometry electrifies our lives and rings in our ears.  I think it is the biggest part of Aryabhata’s work that we take for granted. Without his trigonometric discoveries we wouldn’t have useful conventional electricity. The natural flow of alternating current, or AC current, is represented by the sine function. Electrical engineers and scientists use this function to model voltage and build the electronics we use every day. Alternating current primarily comes from power outlets, but it can also be synthesized in our electronic devices. Trigonometry is also extremely relevant today in music. Sine and cosine functions are used to visualize sound waves. This is especially important in music theory and sound production. A musical note or chord can be modeled with one or many sine waves. This allows sound engineers to morph voices and instruments into perfect harmony. However, Aryabhata is to blame for all that auto-tuned, T-Pain nonsense we hear on the radio!  Lastly, trigonometry has a strong presence in modern day architecture. It’s a necessity when building complex structures and designs. We’d have to say goodbye to beautiful architecture and reliable suspension bridges if it weren’t for Aryabhata.

References:

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

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

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

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

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

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

History of Mathematics – BBC:

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

Fermat prime and Mersenne prime

Fermat prime

https://i1.wp.com/upload.wikimedia.org/wikipedia/commons/f/f3/Pierre_de_Fermat.jpg

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form. Fn=22n+1, where n is a nonnegative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617. And he is the first to investigate numbers of the form 22n.

There is the Pépin test which gives sufficient and necessary condition for the primality of the Fermat prime and this can only be implemented by use of modern computers. Pépin’s test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth’s test. The test is named for a French mathematician, Théophile Pépin.Let Fn be a Fermat number. Fn is prime if and only if 3(Fn-1)/2 = -1 (mod Fn).Here 3 can be replaced by any positive integer k for which the Jacobi symbol (k|Fn) is -1. These include k=3, 5, and 10.If Fn is prime, this primality can be shown by Pepin’s test, but when Fn is composite, Pepin’s test does not tell us what the factors will be (only that it is composite). For example, Selfridge and Hurwitz showed that F14 was composite in 1963, but we still do not know any of its divisors. (Chris K. Caldwell)

Mersenne numbers, which take the form of 2n-1, were named after Marin Mersenne, a French monk from the early 17 century, who corresponded with Fermat. We are particularly interested in the case when Mersenne number are prime. It is not doubt that the first 17 primes of the form 2n-1 match the following n values: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281 (Zegarelli 287).

The first 12 Mersenne primes were known since 1914 and the 12th, 2127-1, was established by Anatole Lucas in 1876. It was one of the largest-known prime number for over 75 years. The next 5 Mersenne primes (p=13 to 17)were discovered in 1952. It was in 1952 when the testing program for Mersenne numbers was began. This led to the establishment of three other primes. More testing has been carried out using modern day computers and the smallest Mersenne number that is untested is 22309-1 (~2013), and this has not been a case of great interest. There is a conjecture that 2n-1 is always prime when n is a Mersenne number. And the more interesting case is the 28191-1 because the 8191 also is the Mersenne number. (Křížek, Florian and Lawrence 214).

Works Cited

Chris K. Caldwell (2014). Pepin’s test . [ONLINE] Available at: http://primes.utm.edu/glossary/page.php?sort=PepinsTest.

Crandall, Richard E, and Carl Pomerance. Prime Numbers: A Computational Perspective. New York: Springer, 2005. Internet resource.

Zegarelli, Mark. Basic Math & Pre-Algebra Workbook for Dummies. Hoboken, N.J: Wiley, 2008. Internet resource.

Křížek, Michal, Florian Luca, and Lawrence Somer. 17 Lectures on Fermat Numbers:    From Number Theory to Geometry. New York, NY [u.a.: Springer, 2001. Print.

The Father of Algebra: Al-Khwarizmi or Diophantus?

The cover of Diophantus' book Arithmetica. Image: Public domain, via Wikimedia Commons.

The cover of Diophantus’ book Arithmetica. Image: Public domain, via Wikimedia Commons.

The history of algebra is very intriguing because of the many cultures that contributed to its origins. Although there were many ancient civilizations that studied algebra, there are two men that are best know for bringing algebra to our modern day: Al-Khwarizmi and Diophantus. The debate as to who is the “father” of our modern day algebra is still a subject of interest to which I hope to bring some light. I would like to share with you the lives of both of these mathematicians, their works and their legacy.

Diophantus:

Much of the life of the Greek mathematician Diophantus is unknown, but we do know that he lived in Egypt sometime after 150 BCE and before 350 CE. From what we have found it seems most likely that he lived during the 3rd century CE. We also have knowledge of his works that were popularized in the 17th and 18th centuries. Arithmetica, one of his greatest works, consists of 13 books of 130 algebraic problems. Out of these books, six were thought to be the only ones to have survived. However, in what’s known as Astan-i Quds, an Arabic manuscript, are thought to be the remaining books of Arithmetica. Even though these manuscripts have been found, some are not convinced of its veracity. The problems found in Arithemtica are known as Diophantine equations. These equations included polynomial equations, linear Diophantine equations, and Diophantine approximations among other Diophantine problems. Other works include Porisms, a collection of lemmas, and many works on polygonal and geometric, all of which helped expand mathematics.

Diophantine polynomial equations are polynomials with a number of unknowns for which only a rational solution is found. These equations usually had many solutions because of their many unknowns. Diophantus generally would only solve for one solution, instead of solving for all or them. A linear Diophantine equation is two sums of monomials of degree zero or more. To solve these equations one would have to use what is called Diophantine analysis. A Diophantine analysis would ask a series of questions, which would help find the solution.

Now the question is, what mark did this make on history? Although it is hard to know exactly who was influenced by Diophantus, we do have knowledge of many mathematicians who were influenced by his work. I would say the most famous work to have come from studying Diophantine equations was from Pierre de Fermat. De Fermat was studying Arithmetica when he scribbled “x^n+y^n=x^n where x, y, z, and n are non-zero integers, has no solution with n greater than 2.” This scribble is better known as Fermat’s Last Theorem, which later inspired algebraic number theorem. Other mathematicians that were inspired by his work are Andrew Wiles, who found proof of Fermats’ theorem, John Chortasmenos, a monk and mathematician, and Wilbur Knorr, a math historian. Above all else he was one of the first people to use symbols in mathematics. This is something we are all used to today in our mathematics from a young age.

Al-Khwarizmi:

A statue of al-Khwarizmi in Uzbekistan. Image: Jori Avlis, via Flickr.

A statue of al-Khwarizmi in Uzbekistan. Image: Jori Avlis, via Flickr.

Abu Abdallah Muhammad ibn Musa al-Khwarizmi, better known as al-Khwarizmi, was a Persian mathematician born in the latter part of the 8th century CE. One of his greatest works was Compendious Book on Calculation by Completion and Balancing. He also had books on arithmetic, astronomy, trigonometry, and geography to name a few. He also helped make the Indian numeric system part of western culture.

His most famous book, as mentioned earlier, is where we get the name algebra. The Arabic name of his book, Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala, has the word Al-jabr, which means restoration. Al-jabr was the beginnings of the word algebra. When this book was translated into Latin it was called Liber Algebrae et Almucabola, which indicates clearly the source of algebra. This book expounded on the knowledge of quadratic functions among others. The book has hints of influence from past mathematicians, but the ties with Indian mathematics is most evident. One loss from the Indian mathematics was that of negative numbers. Because negative numbers were not used, equations with negative solutions were not studied. His book used squares, roots, and numbers to describe the equations. It also introduced the forcing of one side to be equal the other, which is what we would use today. This was the completing part. Balancing was done by subtracting the same amount from both sides of the equation. He also dealt with measuring areas and volumes. His work also included the concept of Algorithm, which is used in our everyday lives.

We now know what he taught, but, again, who or what was influenced by his works? In the 12th century, when his book was translated into Latin, Europe began to become familiar with his work. After a few centuries his work helped get Europe out of the dark ages.

Although I believe that both Diophantus and al-Khwarizmi contributed greatly to the math world, I think that al-Khwarizmi should be considered the father of algebra. This is due to the fact that his work is much closer to the algebra that is used today. His work was used for so long and was never lost. His worked helped Europe out of the dark ages, Diophantus did great work but al-Khwarizmi pushed the mathematical world in a great direction.

References

http://www.britannica.com/EBchecked/topic/164347/Diophantus-of-Alexandria#ref704023

http://www.onislam.net/english/reading-islam/research-studies/islamic-history/454243-al-khwarizmi-the-father-of-algebra.html

http://gulfnews.com/life-style/people/the-father-of-algebra-abu-jaafar-mohammad-ibn-mousa-al-khwarizmi-1.1233076

http://www.famousscientists.org/muhammad-ibn-musa-al-khwarizmi/