Tag Archives: Hamilton

Imaginary Numbers: From Outcast to Respectability

Image: Matheepan Panchalingam, via Flickr.

Image: Matheepan Panchalingam, via Flickr.

Imaginary numbers, which are also known as complex numbers, have had a pretty bad reputation. When most people think of imaginary numbers, they probably break out in a cold sweat from the horrific memories of high school math class. They think that imaginary numbers are utterly incomprehensible and useless in the “real” world. “Imaginary numbers” sound very intimidating to people who are not familiar with them. They also sound highly theoretical with little or no use outside of pure mathematics. In fact, the exact opposite is true.

The most common imaginary number is i, which is formally defined as i = √-1. Since the act of squaring any real number always makes the number positive– whether it began as a negative number or not, it is impossible to find the square root of a negative number without using i. Thus, i made possible an entire class of math problems that were not possible before. For example, √-64 = 8i, cannot be done without using i, because √-64 does not exist in the real number line. Additionally, i can be easily changed from an “imaginary” number into a “real” number simply by squaring it: i² = -1.

The first known person to stumble upon the idea of using an imaginary number to take the square root of a negative number was the Greek mathematician Heron of Alexandria in 50 CE. He was trying to find the volume of a section of a pyramid using a formula that involved the slant height of the pyramid. However, certain values for the slant height would produce the square root of a negative number. Heron was very uncomfortable with this result, so in order to avoid using a negative number, he fudged his calculation by dropping the negative sign.

Girolamo Cardano was an Italian mathematician who was particularly interested in finding the solutions to cubic and quartic equations. In 1545, he published a book titled Ars Magna, which contained the solutions to cubic and quartic equations. One of the equations in his book gave the solution of 5 ± √-15. Commenting on this equation, Cardano wrote, “Dismissing mental tortures, and multiplying 5 + √ – 15 by 5 – √-15, we obtain 25 – (-15). Therefore the product is 40. …. and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless.”

Perhaps the first champion of imaginary numbers was Italian mathematician, Rafael Bombelli (1526-1572). Bombelli understood thattimes should equal -1, and that -i times should equal one. However, Bombelli could not find a practical use for this property, so he generally was not believed. Bombelli did have what people called a “wild idea” – that imaginary numbers could be used to get real answers.

Imaginary numbers continued to live in disgrace until the work of a series of mathematicians in the 18th and 19th centuries. Leonhard Euler helped clear up some of the problems with using imaginary numbers by developing the notation i to mean √-1. He also introduced the notation a+bi for complex numbers. Carl Friedrich Gauss  made imaginary numbers much more concrete and less “imaginary” when he graphed imaginary numbers as points on the complex plane in 1799. However, William Rowan Hamilton in 1833, delivered the coup de grace to imaginary numbers’ bad name when he advanced the idea that complex numbers could be expressed as a pair of real numbers. For example 4+3i could be written simply as (4,3). This made complex numbers much easier to understand and use.

Today, imaginary numbers are an essential part of the everyday calculations that make modern technology work. They are indispensable in the field of electrical engineering, particularly in the analysis of alternating current, like the electrical current that powers household appliances. Also, cell phones and air travel would not be possible without imaginary numbers because they are necessary in the computations involved in signal processing and radar. Imaginary numbers are even used by biologists when studying the firing events of neurons in the brain. Imaginary numbers have come a long way in the five hundred years since they were scoffed at for being absurd and totally useless.











Musings: The Poincaré Conjecture

Mathematics is no stranger to unsolved problems. Time and time again, equations, conjectures, and theorems have stumped mathematicians for generations. Perhaps the most famous of these problems was Fermat’s Last Theorem, which stated there is no solution for the equation xp+yp=zp, where x, y, and z are positive integers and p is an integer greater than 2. Pierre de Fermat proposed this theorem in 1637, and for over three hundred fifty years, it baffled mathematicians around the globe. It was not until 1994 that Andrew Wiles finally solved the centuries-old theorem.

Though the most famous, Fermat’s Last Theorem was by no means the only unsolved problem in mathematics. Many problems remain unsolved to this day, driving many institutions throughout the world to offer up prizes for the first person to present a working solution for any of the problems. Some few are general questions, such as “Are there infinitely many real quadratic number fields with unique factorization?” However, most of the problems are specific equations proposed by a single or multiple mathematicians and are generally named after their proposer(s), such as the Jacobian Conjecture or Hilbert’s Sixteenth Problem.

One such problem, proposed by Henri Poincaré in 1904 and thus named the Poincaré Conjecture, remained unsolved until 2002.  In order to encourage work on the conjecture, the Clay Mathematics Institute made it a part of the Millennium Problems, which included several of the most difficult mathematics problems without proofs. A proof to any of the problems, including the Poincaré Conjecture, came with a reward of one million US dollars. To this day, the Poincaré Conjecture remains the only problem solved.

The Poincaré Conjecture is a problem in geometry but concerns a concept that, for many, is difficult to comprehend and all but impossible to visualize. The best means to approach it is to imagine a sphere, perfectly smooth and perfectly proportioned. Now, imagine an infinitesimally-thin, perfectly flat sheet of cardboard cuts into the sphere. If you were to take a pen and draw on the cardboard where the sphere and the cardboard intersect, you would produce a circle. If you were to take the sheet of cardboard and move it up through the sphere, the circle where it and the sphere intersect would gradually shrink. Eventually, just as the cardboard is at the edge of the sphere, the circle will have shrunk to a single point.

Plane-sphere intersection. Image: Zephyris and Pbroks13, via Wikimedia Commons.

Note that in the field of topology, this visualization applies to any shape that is homeomorphic to a three dimensional sphere (referred to as a 2-sphere in topology since its surface locally looks like a two dimensional plane, much as how the Earth appears flat while standing on its surface). Homeomorphic refers to a concept in the field of topology concerning, what is essentially, the distortion of a shape. For instance, one of the simplest examples in three dimensions is that a cube is homeomorphic to a sphere, since if you were to compress and mold the cube (much as you would your childhood PlayDoh), you could eventually shape it into a sphere. However, in topology, you are not allowed to create or close holes in a shape. This is why shapes such as a donut or a cinder-block are not homeomorphic to a sphere, due to the holes that go through them.

Poincaré proposed a concept concerning homeomorphism and the previously described visualization, and it is here where imagining the problem no longer becomes possible. We live in a three-dimensional world, where any position in space can be plotted based on relativity to three axes, all perpendicular to each other. To imagine a fourth spatial dimension perpendicular to those three is mentally impossible, as is any shape with higher dimensions, and yet many problems in geometry and physics relate to a fourth and even higher dimensions. The Poincaré Conjecture relates to these higher dimensional shapes, specifically closed 3-manifolds (shapes with a locally three dimensional surface). It states that, if a loop can be drawn on a closed 3-manifold and then be constricted to a single point, much like the intersection of the cardboard plane and the sphere in the aforementioned example, then the closed 3-manifold is homeomorphic to a 3-sphere, the set of points equidistant from a central point in four dimensions (Morgan).

If the concept of the Poincaré Conjecture is difficult to conceive, its solution by Russian mathematician Grigori Perelman in 2002 is almost incomprehensible. Due to the number of variables involved, one could not simply set up a system of equations between a three-dimensional space and a 3-sphere. Instead, Perelman used a differential geometry concept called Ricci Flow, developed by American mathematician Richard Hamilton. In short, it is a system which automatically contracts to a point on any surface, and it proved to be the precise tool needed to prove the Poincaré Conjecture. (THIS video does a good job of explaining it in layman’s terms) (Numberphile)

An example of Ricci flow. Image: CBM, via Wikimedia Commons.

Interestingly, despite the immense difficulty of solving such an abstract problem as the Poincaré Conjecture, Perelman refused the prize awarded to him for his accomplishment. His solution to the problem was an exercise in his own enjoyment, and as he later stated upon being offered the Fields Medal (the mathematician equivalent of the Nobel) and the immense monetary prize,  “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” Later, he also argued that his contribution to the solution of the Poincaré Conjecture was “no greater than that of… Richard Hamilton,” and that he felt the organized mathematical community was “unjust.” (BBC News, Ritter)

To this day, the Poincaré Conjecture remains the only Millennium Problem solved. Its proof wound up leading to the solution of various other related geometrical problems and closed a century-old mystery. As the field of mathematics continues to grow and progress, it is only a matter of time until other unsolved problems come to resolution.

Works Cited

Morgan, John W. “RECENT PROGRESS ON THE POINCARÉ CONJECTURE AND THE CLASSIFICATION OF 3-MANIFOLDS.” The American Mathematical Society 42.1 (2004): 57-78. The American Mathematical Society. The American Mathematical Society, 29 Oct. 2004. Web. 9 Oct. 2014. http://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01045-6/S0273-0979-04-01045-6.pdf

Jaffe, Arthur M. “The Millennium Prize Problems.” The Clay Mathematics Institute. The Clay Mathematics Institute, 4 May 2000. Web. 09 Oct. 2014.

Numberphile. “Ricci Flow – Numberphile.” YouTube. YouTube, 23 Apr. 2014. Web. 09 Oct. 2014.

“Russian Maths Genius Perelman Urged to Take $1m Prize.” BBC News. BBC, 24 Mar. 2010. Web. 09 Oct. 2014.

Ritter, Malcom. “Russian Mathematician Rejects $1 Million Prize.” Russian Mathematician Rejects $1 Million Prize. The Associated Press, 1 July 2010. Web. 09 Oct. 2014.