Monthly Archives: March 2015

David Hilbert, Mathematical Genius

1912 Photograph of David Hilbert. Image: public domain, via Wikimedia Commons.

There have been many born who have had an impact on the field of mathematics; however, there are few who can be said to have discovered as many fundamental ideas as David Hilbert. He is known to many as one of the most influential mathematicians of both centuries he lived in. Not only did he solve many of the most pressing issues in mathematics at the time, he also created many of the tools used in the field to this day.

Born on the 23rd of January in the year of 1862, Hilbert began his seventy-nine year journey towards becoming renowned. He was the first child born and also the oldest. His father worked in the town of Konigsberg at the time while his mother tended the house. His mother was fascinated by prime numbers and astronomy. She taught him from home until he could enter his school, the Royal Friedrichskolleg. This was known as one of the best places to gain an education at the time, however, it didn’t stress science and mathematics as much as one would expect. Hilbert always demonstrated his love for the fields, however, and graduated with the top score in mathematics. He graduated in the fall of 1885 where he began his journey towards higher education.   His father disapproved of his choice towards higher education but this would not impact the soon to be brilliant mathematician.

The University of Königsberg was Hilbert’s next step in his educational journey. He turned his focus towards the field of mathematics, specifically calculus. Graduating with a doctorate degree focused on that of binary forms he accepted a job at a different university to work at the as a lecturer. He later traveled to the University of Gottingen as a lecturer for them as well. This new university was famous for producing many of the great minds of the time. Many of these were his colleagues and his classes were filled with brilliant young minds. A few years after this he became the editor of one of the most prevalent mathematical journals of the time. He soon became bored with this, however, and he made several trips to alleviate it. He planned to visit the twenty one greatest mathematicians of the world at the time, and after meeting a few he returned to work on solutions to some of the world’s greatest mathematical problems.

While working for his university in 1888 Hilbert demonstrated his first major contribution, which is that of the finiteness theorem. This famous theorem is the solution to Gordan’s Problem, which deals with how to represent infinitely many objects with finitely many building blocks. By taking a different approach than previously thought Hilbert backed his proof with the law of excluded middle to demonstrate that there could be a finite set of generators for binary forms. He went beyond what Gordan did by proving that the problem could be solved with more than two variables. Hilbert sent his results to be published in a journal where they would eventually be received. Although his result were dubbed to be one of the most important results that had happened to algebra it became very difficult for Hilbert to get them accepted. Due to the completely different approach he used Gordan did not admit that Hilbert’s method solved the problem on finite basis. This revolutionary idea itself became an issue to Hilbert as he had to go through many trials to get it known and accepted. Due to the fact that not all accepted this proof even after it was published, Hilbert was antagonized by several mathematicians. It took until another mathematician, Felix Klein, reviewed the proof many years later that it was accepted.

In the year of 1900 Hilbert created and proposed a list of 23 problems, those that were the most influential of the time. These problems included the Riemann hypothesis, solving a 7-th degree equation with algebra and proving that the arithmetic axioms are consistent. Due to the powerful change he made to geometry he now could gather the entire mathematical community to solve what was once unsolvable. He presented his problems to the International Congress of Mathematicians in Paris. The problems used were not all Hilbert’s creations as many were long known about and Hilbert had attempted to solve them yet failed. Some of these problems still remain today as challenges for our current mathematicians. This is due to the either precise or vague nature of each individual problem, as some leave much up to interpretation while others are extremely focused on one goal. Hilbert himself was very doubtful that these problems would be solved in the near future. The list of problems have been the focus for the mathematical community for years, and many people have tried to solve them. Any progress made would immediately be spread around and published into journals. However he believed that every problem had a solution.

Hilbert’s Fifth Problem, from the journal of Andrew Mattei Gleason. Image: Jean Berko Gleason via Wikimedia Commons.

What can be known as his greatest work is that on the field of equations. He stated that although there are an infinite number of equations they could be broken down to a finite number that could be used as building blocks for the others. He was unable to create this finite set, however, and this caused many to disregard this theory. Regardless of this a new field of mathematics was born. Hilbert’s vast knowledge of mathematics allowed him to accomplish many things, and as a result he has many different terms named after him such as Hilbert curves and Hilbert space. By furthering and discovering many principles in his field he became known as the most famous mathematician of his time.

Later in his life Hilbert would publish a book called Grundlagen der Geometrie or the Foundations of Geometry. In this book he creates axioms to replace those made by Euclid and they lacked the flaws that Euclid’s had. This book also defines the once undefined terms used in prior books. This book is the overview of Hilbert’s views on geometry. Later on Hilbert sought to solve the foundational crisis of mathematics and in doing so he created a problem known as Hilbert’s problem. This problem was based around the foundational crisis of mathematics. It clarifies of the foundations of mathematics and in doing so it solves the problems based around their inconsistencies, these include paradoxes. It was, however, later proved to be impossible to solve Hilbert’s problem. It is also to be noted that at the same time as this book was published, another book based off of the same axioms was published by Robert L. Moore.

Hilbert was not exclusively a mathematician, as he was interested in the field of physics as well. One of his many trips he went to Bonn where he delved into the field. He investigated many problems while on this trip. His main focus was towards that of gasses and radiation, moving from the former to the latter over his career. His work was followed by Einstein’s work very closely and he even invited Einstein to his university to discuss it. Hilbert once again published a book based on the axioms he studied, this time on the field of physics. It allowed for steps to be made forward in the field of quantum mechanics as well as vastly impacting other famous mathematicians such as Schrodinger.

In the final years of his life Hilbert was the editor of one of the most prominent mathematical journals of the time, Mathematische Annalen. He retired from the university when he was 68 in 1933, due to the harsh rules imposed by the new Nazi regime that was beginning to take hold over the government. In the end Hilbert died due to health complications in 1943 only 10 years after his retirement. His funeral was attended by few people and his death was not known until many months later. Hilbert may have been forgotten by many at the time yet today he is remembered as one of the greatest mathematicians of his time.

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

http://www.britannica.com/EBchecked/topic/265698/David-Hilbert

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

http://www.storyofmathematics.com/20th_hilbert.html

http://www.sonoma.edu/Math/faculty/falbo/hilbert.html

http://www.thefamouspeople.com/profiles/david-hilbert-471.php

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Getting Something from Nothing

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

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

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

“Russel Peters-Invention of Zero”-Youtube

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

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

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

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

Further reading:

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

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

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

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

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

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

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

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

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

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

The Fascinating Plimpton 322

Mathematics is one of the most powerful tools in human existence. From basic counting and grocery shopping to advanced applications such as in computing and aerospace, math plays important roles and can be found everywhere. However, the math system was not built in one day. All the current have their past. Believe it or not, some archaeological findings in mathematics are still fascinating to us today, even though they were from hundreds or even thousands years ago. The clay tablet named Plimpton 322 is one of these valuable and interesting findings (Fig. 1).

Figure 1. The clay tablet Plimpton 322[1].

Plimpton 322 is a great presentation of how advanced Babylonian mathematics was. The clay tablet was written around 1800 BC. The math represented on the table is now known as Pythagorean triples. It was long before the development of Greek mathematics. The clay tablet was purchased by George Arthur Plimpton from an archaeological dealer in 1923. It now lies in the G.A Plipton Collection at Columbia University numbered 322[2]. This is also how the tablet was named Plimpton 322.

The table on the tablet was inscribed with four columns and fifteen rows. In the rightmost column, which is the 4th column, it includes the number of rows which is numbered from 1 to 15. Before we jump into the math, we need to know how numbers were recorded and how to translate these symbols to the math we understand. In the table, 1 is symbolized as which also called a “stroke”[1]. From 2 to 9, the numbers were inscribed by combining multiple of strokes. Figure 2 shows how numbers 2 through 9 were written in the table. The number 10 was written as.The same combination fashion works for numbers 10, 20, 30, 40, and 50 (Fig.3). How to write numbers between 10 and 20 or 20 and 30? the number system on the tablet still relys on combining “strokes”. 11 was written aswhich is an addition or combination result of 1 and 10. 0 is recorded as blank space. At this moment, you might think it is the same system as the system similar to the one we use today. However it is not quite. The math system we use now is based on 10. However, the system the Babylonian used was based on 60[1]. It means that if the number is written as, which tansclates to 1;2;1. It mathmatically represents 1*(60*60)+2*(60)+1=3721.

Figure 2. How the numbers from 2 to 9 were inscribed on Plimpton 322[1].

Figure 3. How the numbers 10, 20, 30, 40 were inscribed on Plimpton 322[1].

With the basic translation we just learned, now we can translate those symbols to the numbers we understand for the 2nd, 3rd and 4th columns. For the 4th column, it is listed 1 to 15 for each rows. The numbers here represent the row numbers (Fig. 4). Now we can also understand the 2nd column by using the addition base 60 number system (Fig. 5). You can go back to the earlier mathematical example of calculating 1;2;1 if you have any difficulties. When we have the basic understanding of how numbers are inscribed, we now need to know what these numbers mean or how the math works on the tablet. Archeologists translated the names on the top of each column which offers more information to the later discovery. On the 2nd column it has the word “width” and the 3rd it says “diagonal”. Mathematicians thought it might relate to Pythagorean triples. The results showed the guess was right if you ignore the error the tablet maker had (the corrected numbers are marked as red in Figure 5). For each row, the diagonal d, width w, and the unknown length of the last side of triangle l, make an equation w2+l2=d2. It is also true that all the ls are integers. So now the question remains on what does the 1st column do? It is actually the fraction of d^2/l^2. From Figure 1, every single row in the 1st column consists a series of numbers. The numbers can be translated into decimal numbers. For instance, in the first row where d=169 and l=120. Therefore, the fraction of d2/l2 should be about 1.9834. The first column has numbers 1;59;0;15. How is that 1.9834? On our normal 10 based system, for instance, 1.234 can be written as 1+2/10+3/(10*10)+ 4/(10*10*10). The same method also holds for the tablet decimal numbers. However, the only difference is that the numbers are 60 based. Therefore, 1:59:0:15, which is the decimal number inscribed on the fourth row of the 1st column can written as 1+59/(60)+0/(60*60)+15/(60*60*60)=1.9834.

Figure 4. The 4th column of Plimpton 322 (From the top to the bottom row, number of rows were inscribed which is 1 to 15)[1].

Figure 5. The 2nd and 3rd columns of Plimpton 322 (On each picture’s left side, the numbers on the left are the numbers decrypted from the inscribed symbols, the numbers in blue are the results after addition)[1].

Who made Plimpton 322 and what was the purpose of making it? Eleanor Robson, a Plimpton 322 researcher, believes that it was made by a male due to geographical reason. All the female scribes from ancient Mesopotamia lived much further north to where this tablet was made. Robson also does not agree that the tablet author was a professional mathematician because the professional academic disciplines is a phenomenon of the very recent past. There are two possibilities to the author’s identity which Robson brought up. First is that the author could have been a trainee scribe or a teacher. He might have taught simple math techniques in scribal schools. He also knew the document format used by the temple and palace administrators of Larsa [4]. Therefore, he was not a student. This leaves him highly likely to be a professional bureaucratic scribe. Robson also believes that the tablet served for teaching purposes. The tablet is similar to problem lists. This is likely to explain why the l numbers are missing on the tablet [4]. They are all integers and waiting to be solved from the known values. The fifteen rows with the same method also indicate that the teacher might offer this tablet for students to practice repeatedly on those problems.

Now the math puzzle of the ancient tablet Plimpton 322 is finally solved. However, the exact answers for how the tablet was created or who was the author still remain elusive. It is fascinating for us to know how incredibly advanced Babylonian math was back into the time around 1800 BC. They knew Pythagorean triples which today is still super commonly used in math and its fields of application. We can clearly see that the history of mathematics is also a part the history of human existence. Mathematics was created, developed and is still developing and helping human beings to reach new and more advanced eras in human history.

Reference:
[1] Casselman, Bill. “The Babylonian Tablet Plimpton 322.” The University of British Columbia. Web. 8 Feb. 2015.
[2] Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series 29, New Haven: American Oriental Society and the American Schools of Oriental Research, pp. 38–41.
[3] Joyce, David. “PLIMPTON 322.” Clark University. Web. 8 Feb. 2015.
[4] Eleanor Robson. Words and Pictures: New Light on Plimpton 322, American Mathematics Monthly 109, 2002.

How I Learned to Build Programs and Love Mathematics

A photo of an old computer. Image: Arkadiusz Sikorski via flickr.

A photo of an old computer. Image: Arkadiusz Sikorski via flickr.

Now more than ever, our society’s use of computers is far greater than it was before. From creating documents to playing video games, computers are to thank for the simplicity we have when doing these things. In addition, as time has gone by we have been able to create more “powerful” computers that can do things that we were never able to do before. But what actually is behind this “power”? One of the most obvious things people will tell you is the fact that current computers have better processors that allow more to be done in less time. While this is definitely a big reason why, another major reason why we have more “power” is because of Mathematics.

Everything that a computer does can be boiled down to numbers; most likely binary numbers but numbers nonetheless. The only thing the processor within a computer does is manipulate these numbers in specific ways. In most of these cases, the manipulations will be based around simple mathematics. One large part of mathematics is trying to create proofs. If we have a proof that makes a certain statement, then we don’t need to worry about it when making other proofs or functions. The Pythagorean Theorem is an excellent example of a simple mathematical proof that can easily be translated into a few number manipulations by the processor.

But what if we didn’t have the Pythagorean Theorem? Even if we didn’t have this proof, we could still figure out the length of a hypotenuse in a right triangle. One possible way of solving this problem would be by recursively going through every possible length of the hypotenuse and see if it can connect to the right angle produced by the other two sides. While this looks like a ludicrous way for a human to solve this problem, a computer could do this kind of computation relatively quickly. The major thing though is that, just like you, a computer can solve this problem much more quickly if it can use the Pythagorean Theorem which, in turn, will make the computer more “powerful”. While this example focused on a mathematical proof that almost everyone knows, there are many proofs out there which are far more obscure and may seem pointless for a human to use but actually make certain mathematical computations for a computer faster.

One of the most useful areas of mathematics for a computer would be linear algebra. Linear algebra spends a large amount of time dealing with number values within different dimensional vectors. The reason this is so useful for computer is because the primary way that numbers are stored on a computer is in different dimensional arrays which work just like vectors. This means that any proofs that we have that make certain types of vector manipulation easier will also make it easier for computers. A good example of this is how we can deal with linear equations in linear algebra using the Gaussian elimination algorithm. Unlike when we were in high school and the main way of solving these types of equations was by figuring out which exactly which variables we wanted to get rid of in which order, the Gaussian-elimination algorithm is much more straight forward by treating each equation as a vector. While this doesn’t really cut out time in solving the problem, it does make it much easier for programmers to implement linear equations in their code without accidentally doing any unnecessary work. This leads to a cleaner implementation of linear equations with less likelihood of programmers accidentally creating bugs in the code.

The best comparison of processor speed vs mathematics in a computer would have to be from the 1940s to 1990s when comparing America to the Soviet Union. In the 1940s World War 2 had started and, once this happened, both the Americans and Soviets started working on producing computers to help them crack codes and do other important tasks. The big difference, however, between America and the Soviet Union was that America had much more access to the components they needed to build computers, which meant they could build more and run faster. The Soviet Union, on the other hand, did not have access to nearly as many resources due to their extreme isolation from the rest of the world [1]. This meant that they had to build their computers with the resources and knowledge they had alone or copy designs from the western models. This meant that computers were a much more scarce resource and had to be used in the best way possible. If the computer they had didn’t have the speed necessary to run the program they created, then they had to build a better program.

When the Soviet Union fell, many of the Soviet Union’s mathematicians immigrated to America. Once they were in America the amount of papers that were produced by Americans working in the same area as a Soviet Union immigrant dropped significantly [2].  This, I believe, is mostly because of the fact that the only way the Soviet Union was able to compete with America in computational power was by having a better mathematical base in the programs they used. This, in turn, meant that the Soviet Union put a large emphasis on mathematics.

For the longest time, the way America created more “powerful” computers was by increasing the speed of the processor. However, we have now reached a barrier when trying to produce faster processors. Instead of producing faster processors, now we see that computers will have multiple processors. While this can lead to more “powerful” computers, the power of a computer is now fully based on the programs that people make for them and the mathematical proofs they use in those programs. In conclusion, now more than ever the power of a computer fully relies on our expansion in the area of mathematics.

Sources:

[1] (http://en.wikipedia.org/wiki/History_of_computer_hardware_in_Soviet_Bloc_countries) Soviet Union didn’t have access to the same computers that Capitalist countries had. Either had to build their own or create copies of Western Models.

[2] (http://www3.nd.edu/~kdoran/Doran_Math.pdf) Soviet Union Immigrants started doing more Mathematics Papers than Americans after the fall of the Soviet Union (Better at Mathematics?)

Thinking in different bases

Very few people ever stop to think about why numbers are the way they are. Have you ever stopped to consider why you live in a society that uses a ten based (or decimal) number system? This is a system where you start at 1 and go all the way up to 9 before you reset with one followed by a zero (or 10). Well if you haven’t taken the time to ponder this deeply complex situation that you find yourself in, fear not, for I will explain why. For the most part, human beings find themselves in the possession of 10 fingers. So the leading hypothesis for why we use a base 10 system is because it was convenient since we could count to 10 on our hands. For that single and simple reason, society decided that its number system would have a base of 10. There you have it, one of the great mysteries of life has been cleared up for you.

Now that you have an open space where a mystery used to rest, allow me to fill that spot with a new mystery. Why don’t we use a differently based number system? No really, think about it! Throughout history, multiple societies chose not to use a decimal system. For example, Babylonians used a base 60 number system and the Mayans used a base 20 system. Likewise, there are quite a few key things in our lives today that don’t really rely on the conventional base 10 way of doing things.

binaryFor example, we are surrounded by computers, which utilize Binary, a base 2 number system. At every position there is either a 1 or a 0, so the number ten in binary looks like 1010.

Another example of something that isn’t really 10 based is a clock. Take a second to look at an old school clock with an hour and minute hand (gasp! Not digital!) . You’ll quickly notice that it goes from 1 to 12 instead of 1 to 10. Weeks are broken down into 7 days and minutes as well as hours are in chunks of 60. So as you can see, the case could be made to switch to a different based number system. Let’s take a look at another option that society could use in place of the current base 10 system.

The system that we’ll look at is one of my personal favorites (thanks to my Computer Science bias), the hexadecimal system. As you can probably guess from the name, the hexadecimal system adds six to the base 10 system, leaving us with a grand total of base 16! The system uses 0 to 9 to represent numbers 0 to 9 and then uses A to F to represent the numbers 10 to 15. Hexadecimal is a positional numeral system just like the decimal system. Just to give you a better idea of what hexadecimal is, lets learn how we can represent a hexadecimal number in decimal.

Let’s take the random hexadecimal number 3FB1. Since it is a positional system, meaning the position of the symbol is part of its value, we can just take the symbol and multiply the symbol’s value by its base to the power of its position (position numbering goes right to left and starts with 0 rather than 1). So we would take (3 x 163) + (15 x 162) + (11 x 161) + (1 x 160). Simplifying this further we get (3 x 4096) + (15 x 256) + (11 x 16) + (1 x 1). At the end of this we are left with 16,305. So right off the bat we can begin to see the some of the potential benefits that would come with using a base 16 system. Firstly, we notice that it takes fewer symbols to represent numbers. Where in decimal we had to use 5 symbols (6 if you count the comma) to represent 16305, in hexadecimal we only had to use 4. We can also note that because of this space bonus, we could potentially represent higher numbers in the same number of hexadecimal characters. Even though this all sounds great, there are some disadvantages that would come with using the hexadecimal system. For one, performing mathematical operations on base 16 numbers can get complicated quickly (a base 16 multiplication table has 256 instead of 100 elements). Try performing long division on two hexadecimal numbers! Also, I personally believe that it would be trickier to set up equations with variables due to the fact that the characters “A”, “B” and “C” could no longer be used (there goes the quadratic formula song). On a similar note, there are many people who think that we would be better off on a duodecimal system (base 12), but that is a conversation for another time.

So next time when you are counting on your fingers, take some time to think about the effects of you simply having 10 fingers!

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

http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system

http://en.wikipedia.org/wiki/Hexadecimal#Conversion

http://mathforum.org/library/drmath/view/63375.html

Applications of Imaginary Numbers

The concept of imaginary numbers has always been a fascinating one. The Greek mathematician Heron of Alexandria, born around 10 AD, is noted as being the first person to have come up with the idea of imaginary numbers. It wasn’t until the 1500’s, though, that rules for arithmetic and notation for complex numbers really came to fruition. Of course, at the time most people thought imaginary numbers were just stupid and pointless. Heck, today I’m pretty sure most people still think imaginary and complex numbers are stupid and pointless. Surely they can be used for more than just generating pretty looking fractals (like the Mandelbrot set), right?

Yes, because of imaginary numbers there is a solution to any type of polynomial equation… but there has to be more use to them than that, right? The topic I wish to present in this article is about some of the other applications of imaginary numbers. Imaginary numbers are really useful and they can be used to do all sorts of awesome things!

While presenting this information, I do not claim to list every single practical use of imaginary numbers. There are many useful applications that involve some crazy complicated mathematics and are admittedly beyond the scope of my understanding at the present time. Rather, I wish to share a few of my favorite applications of imaginary numbers. It is my hope that the reader will learn more about why mathematicians have studied so much about imaginary and complex numbers.

First, complex numbers have a remarkable application in triangular geometry. There is a fascinating theorem called “Marden’s theorem”. I read about this theorem in an article written by Dan Kalman, a doctor of mathematics who works in the Department of Mathematics and Statistics at American University. He claims that this theorem is “the Most Marvelous Theorem in Mathematics.”

caption

A visualization of a Steiner inellipse with its foci. The ellipse is based on the polynomial p(z)=z3-(9+9i)z2+(3+52i)z+(33-39i). The black dots are the zeros of p(z), and the red dots are the zeroes of p'(z) and the foci of the inellipse. Uploaded by User Kmhkmh for Wikipedia on 2/6/2010. Creative Commons license. Reuse permitted.

Basically, this theorem can help one find the foci of a Steiner inelipse. A Steiner inellipse is simply an ellipse that is inside of a triangle and is tangent to the midpoints of the three sides of the triangle. Such an ellipse is shown in the following diagram.

The foci of a Steiner inellipse can be found by using complex numbers! The triangle’s vertices can be written as points in the complex plane as follows: a = xA + yAi, b = xB + yBi, and c = xC + yCi. Marden’s theorem states that if you take the derivative of the cubic equation (x-a)(x-b)(x-c) = 0 and set it to zero, then the solutions of this equation will be the two foci of the Steiner inellipse in complex numbers. Isn’t that a really bizarre theorem? If you think about it, though, it makes some intuitive sense. When you take the derivative of an equation and set it equal to zero, the solutions of that equation give you the maximum and minimum values found on the arcs in the equation. A regular cubic equation could have up to two arcs, so it’s natural that there would be two max/min values. The fact that these two values are the two foci of the inellipse is really interesting.

As it turns out, using complex numbers here gives us a very amazing and useful geometric tool to use. There are also a few generalizations of this theorem that apply to different types of polynomials and other geometric shapes!

So it seems that first we have geometric applications for complex numbers. Now I would like to present a second category of applications. These are related to phasor calculus. A phasor is a complex number that represents a sinusoidal function. Thanks to the amazing Euler’s formula (e= cosx + sinx), sinusoidal functions can be rewritten as complex numbers. This allows for easier problem solving and analysis for many types of problems.

For instance, in electrical engineering alternating currents can be a pain to analyze sometimes. After all, they have voltages that exhibit sinusoidal behavior. With the use of phasors, one can analyze aspects of AC circuits more easily. Analysis of resistors, capacitors, and inductors can be combined into a single complex number, which is called the impedance. Phasors are comparatively easy to interpret, so it’s a lot easier to study AC circuits when studying them in the complex plane! In addition to AC circuits, complex numbers are similarly useful when studying electromagnetic fields, where the quantities of electric and magnetic field strength are combined into a single complex number.

The last application I wish to bring up involves the usage of imaginary numbers to solve integration problems. As it turns out, we can use the aforementioned Euler’s formula to simplify real integration problems and help us find real answers. This is done by using a base integral that has a complex solution. An example of a base integral would be∫ e(1+i)xdx. Using simple u substitution, we can find the answer to this integral, which is ((1-i)/2)e(1+i)x + c1 + ic2. With this known imaginary answer, we can compute the answer to a real integral.

Consider, for example,∫ excosxdx. First, we rewrite the previously mentioned base integral as: ∫ exeixdx. Then we can use Euler’s formula to alter this integral further:∫ exeixdx = ∫ ex(cosx + isinx)dx. This will further simplify to∫ excosxdx + i∫ exsinxdx. We can set the known solution of the base integral equal to this complex integral and solve for ∫ excosxdx , which is the real integral we are trying to compute. We will see that the imaginary parts must be equal and the real parts must also be equal. Solving in this manner will show us that ∫ excosxdx = .5ex(cosx + sinx) + c. Hopefully I don’t have to explain how useful integrals are! The fact that complex numbers can help us solve integrals alone means they are really useful.

I think in general it seems that whenever there’s an oscillatory phenomenon of any kind then complex numbers are naturally helpful in describing said phenomenon. Complex numbers have multiple substantial applications in a multitude of scientific problems. In addition to the few I’ve mentioned, complex numbers are also used in: quantum mechanics, control theory, signal processing, vibration studies, cartography, and fluid dynamics. Dang. Since a long time ago complex numbers have been thought of as trivial and inconsequential. Descartes himself (who coined the term “imaginary”) called these types of numbers imaginary because he meant for this to be derogatory. However, as we have learned more about math throughout the ages we have found many a useful application for imaginary numbers.

The aforementioned Mandelbrot set. This is a fractal involving a set of complex numbers. Uploaded by User Localhost00 on 10/13/2013. Creative Commons license. Reuse permitted.

Sources:

-Imaginary Number. (n.d.). Retrieved February 24, 2015 from Wikipedia: http://en.wikipedia.org/wiki/Imaginary_number#cite_ref-1

-Complex Number. (n.d.). Retrieved February 24, 2015 from Wikipedia:

http://en.wikipedia.org/wiki/Complex_number#Applications

-Dan Kalman, “The Most Marvelous Theorem in Mathematics,” Loci (March 2008)

– P. Ceperley. 8/28/2007. Phasors. Retrieved from:

http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html

-Integration Involving Complex Numbers. Retrieved February 24, 2015 from:

http://www.mathnotes.org/?pid=109#?pid=109

Googols

During class, we spoke briefly about Graham’s number.  It’s a number so vast that it can’t be written in our conventional decimal number system.  There are other very large numbers as well, such as the googolplex.  We run into a problem when talking about numbers on this scale, because we rarely encounter really large numbers in day to day life.  In fact, very few people have a frame of reference for how large numbers like a trillion really are.  One of my favorite numbers is the googol.  It’s the number that nerdy kids throw around when they want to sound smart when describing innumerable quantities.  The definition of a googol is pretty simple, 10100.  The one question that remains:  What would it take to have a googol?

An easy way to relate large numbers is to compare them with money.  Consider a $100 bill.  It’s a rather light, thin piece of paper, but it holds real value to a lot of people.  Its dimensions are 6.14 in. x 2.61 in. [1], and it is .0043 in. thick. [2]

Assembling a million dollars in one location using $100 bills is easy (if you have the money, of course).  If you were to stack them neatly into a cube, the cube would be about a quarter of a foot long, a quarter of a foot wide, and a quarter of a foot high.  This is about the size of a basketball.

This may be all well and good, but what about a billion?  Each side of the cube now is 7.36 feet long. This cube would also weigh 11 tons. This is about the weight of 3.5 Subaru Foresters.

How about the glorious trillion?  A trillion dollars would form a cube 73.6 feet long.  If stacked on a football field, the pile of money would be 7 feet deep.  The current national debt of the United States is around $17 trillion.  This money on a football field would be 117 feet deep.  It’s about the money required to half-fill Rice-Eccles stadium with $100 bills.

As the numbers grow, the money pile gets more and more ridiculous. Let’s consider how the short scale number system works.  A million is a one with six zeroes.  A billion is a one with nine zeroes.  Every ‘–illion’ thereafter has an additional three zeroes.  How do we determine what should prefix ‘-illion’?  We count in Latin.  Therefore, we get prefixes like ‘quad-‘, ‘quint-‘, and ‘sept-‘, just to name a few.  A googol in the short scale system is 10 duotrigintillion, or ’32-illion’.

A quintillion dollars (that’s $1,000,000,000,000,000,000) would form a cube 7,360 feet long, or about 1.4 miles on each side.  This is more money than the purchasing power of the Earth by a factor of 11,300. [3] In addition, if we divided up this money equally among the world’s population, each person gets $140,350,877.  This would also likely cause the world’s economy to completely fall apart.

A septillion dollars would form a cube measuring 139x139x139 miles.  This is rapidly getting harder to comprehend.  This cube of money would weigh 11 quadrillion tons.  It’s also enough to completely cover the Earth in money 83 feet deep.

A nonillion dollars would form a cube 13,940 miles across.  This is an enormous amount of money, and would in fact form a cube much larger than the Earth.

An undecillion dollars would create a cube 1.4 million miles across, which is larger than the sun.  Oddly enough, the bakery chain, Au Bon Pain, was once sued for $2 undecillion dollars. [4] While the lawsuit may have been filed, it was under spurious claims, and was subsequently dropped.

We’re starting to run out of comparable objects.  The sun is the largest object that people are most people are familiar with.  However, there are still some valid comparisons to be made.  1 AU (Astronomical Unit) is 92,955,806 miles across.[5]  It’s defined as the median distance from the Earth to the Sun.  One tredecillion dollars (for reference, that’s $1,000,000,000,000,000,000,000,000,000,000,000,000,000,000) would form a cube of $100 bills that is 1.5 AU across.  You could line up one corner of the cube on the center of the sun and have the other corner reach Mars.

By now, we’ve reached Mars from the Sun.  If we increase the value of our cube to $1 quindecillion, our cube has grown to the point that we could set a corner of it on the Sun, and have the other corner of the cube reach the Voyager 1 probe, the farthest man-made object from Earth.  We’re now at 150 AU.  How much further do we need to go to reach a googol?

Well, to get a cube that reaches from the Sun to the center of the Milky Way galaxy, we need $1.2 duovigintillion (That’s 22-illion).  This is around 1.7 billion AU.  Having exhausted our units again, we’ll need to switch to light-years.  There are 63,241 AU in one light-year.  One trevigintillion dollars would form a cube 237,000 ly across.

It takes roughly $1.1 quattorvigintillion to form a cube of money that stretches from our sun to the Andromeda Galaxy.  We’ve also long passed the point at which the cube of money would collapse to form a black hole.  The cube would now be 2,400,000 ly across.

We're going to need a lot of these. Image by Bureau of Engraving and Printing, via Wikimedia Commons.

We’re going to need a lot of these. Image by Bureau of Engraving and Printing, via Wikimedia Commons.

Once we reach $4 octovigintillion, we have a cube of money that is the size of the observable universe. Unfortunately, we’ve really run out of reference points.  The only thing that we can do now to reach a googol is to increase the value of our bills!

Finally, we’d need to increase the value of our bills to quintillion dollar notes, but at last, we are successful.  We filled the observable universe with bills that are worth thousands of times more than the world economy, but we’ve finally gotten a googol!  Congratulations!

Sources:

[1] http://en.wikipedia.org/wiki/United_States_one_hundred-dollar_bill

[2] http://www.ehd.org/science_technology_largenumbers.php

[3] CIA Factbook: https://www.cia.gov/library/publications/the-world-factbook/geos/xx.html

[4] http://www.loweringthebar.net/2014/05/2-undecillion-dollar-demand.html

[5] http://en.wikipedia.org/wiki/Astronomical_unit