Author Archives: heathfrench

Drop the Base

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A demonstration of exactly how a shift of base can change our perception of time. Image: Jeremy Keith via flickr.

As a kid, when we were first introduced to numbers, they were just something we memorized, learned to accept, and started using on a regular basis. While this seems almost second nature to most of us, there was a time where the idea of a number system was a new thing and, like all new things, it was discovered multiple times by different people who had different setups. One of the more interesting areas of variation between different number systems would be the base that different number systems used.

Before going into detail about what a base is, it is important to understand that base systems are primarily used by number systems which also use position to determine how large a number is. For example, the Arabic numeral system is positional because I can use the same symbol in a different position to change the value of the number. While 01 is only the value one in this system, by switching these numbers around to 10, I have changed the value now to ten. This is different from something like the Roman numeral system which, for the most part, wouldn’t be considered a positional system because in two different numbers, like X and XIII, the value of the symbol X doesn’t change.

Now, what does this have to do with the base of a number system? The thing is, the base of a positional number system is the number of different symbols you can have in any single position. For example, the Arabic numeral system is base 10 because we can have ten different symbols in a single position (1, 2, 3, 4, 5, 6, 7, 8, 9, 0). In addition to defining how many different symbols you can have in any one position, the value of the base will also affect how much of a change in value a symbol will have based on its position. As I had mentioned earlier, different number system have different bases. The primary reason why would most likely be just because they may have had a different system for counting which lead to that decision. Having a base 10 system is the more common one and a lot of people give credit to that due to the fact that the average number of fingers we have on our hands combined is ten and people like to count using their fingers. On the other hand, the Mayans had a numeral system which consisted of base 20. Unlike most people from Europe, the Mayans wouldn’t wear shoes which meant they could count using ten fingers and ten toes. Even the Babylonian’s had a numeral system with base 60. I honestly couldn’t say why but I am sure they had a good reason for doing so.

Even current day computers use a different base than 10. Instead, computers count using base 2 which means they can only have a 0 or 1 in any position. How can something like this work? The reason why different numeral systems can have different bases is because all positional systems use mathematics in combination with the base size to determine how important a certain symbol is based on its position. This means that it is easy to convert from any base system into a different one. For example, if I want to convert the binary number (100010) into a base 10 number, all I need to do is figure out the base 10 value at each position and add them together. Since this is base 2, every position will be multiplied by 2i with i being the current position. This means:

100010 -> 1*25 + 0*24 + 0*23 + 0*22 + 1*21 + 0*20 -> 1*32 + 0*16 + 0*8 + 0*4 + 1*2 + 0*1 -> 32 + 2 = 34

To make things even more bizarre, a base can be found in more than just numeral systems. Another great example of a system which has a base is the alphabet. In our case, the Roman alphabet has a 26 base system or 52 (if you include capitalization). In addition, a lot of the different measuring units we have also have their own setups for bases. There are 12 inches in a foot, 16 ounces in a pound, 60 seconds in a minuet, and even 12 months in a year. And yet, for all of these we use a base 10 counting system instead of creating our own symbols for each measuring units. Then again, imagine how confusing that would be. In most places, people realize how difficult it can be constantly converting from one base system to another which is why certain measuring systems like the metric system uses a constant base of 10 between unit sizes to make things easier.

In the end, the point is that different bases are used everywhere. Whether you are dealing with numbers or some other system entirely, you will usually be able to find a base of some kind connected to the system. While it may be difficult to have to constantly deal with different kinds of bases, bases are necessary for people to be able to have such a large variety with such a limited number of symbols. Bases are here and they are here to stay.

Sources:

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html http://mathcentral.uregina.ca/RR/database/RR.09.00/hubbard1/MayanNumerals.html

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?)

The Beauty of The Elements

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A stone statue by Joseph Durham depicting the famous mathematician Euclid. Image: Garrett Coakley via flickr.

When I was in high school, I eventually learned about the mathematical subject known as geometry. Unlike most schools though, instead of our teacher having us sit down and listen to them talk about the subject, our teacher had each and every one of us go to the library and rent a copy of Euclid’s Elements (Book 1). From that point on till the end of the first semester, each day we would separately read from Euclid’s Elements and then try to prove to our teacher each and every postulate using Euclid’s methods. It wasn’t until recently that I discovered that most children do not learn about geometry in this fashion and how unique of an experience I had. While I can see some of the possible advantages behind the new ways people learn about geometry, I still believe that Euclid’s The Elements has its own advantages that some of these other sources don’t.

One of the most noticeable things about The Elements is that each and every one of Euclid’s postulates build exceptionally well off of each other. While I see proofs building off of each other in most other texts books, there is just something about the way it is done in The Elements that feels much smoother. Perhaps the big advantage with a book like The Elements is that it was never meant to be a “text” book but rather a book for people who are interested in learning about geometry. Because of this, it doesn’t have to continually throw out real world examples or ask the reader to try to use this proof in specific scenarios. Instead, The Elements will just make a statements, go about proving that statement, and then go straight into making another statement and most likely prove it using the previously proven statement.

Another difference between The Elements and other geometry books which I believe makes it far superior is the general way in which it goes about solving proofs. Nowadays, most geometry books will use a popular form of algebra and a number system to solve equations. However, Euclid’s Elements is fully self-contained and takes nothing for granted. Because this book was created in a time where people didn’t necessarily have access to other sources, everything that is necessary to understand what is being stated in this book is there; including its own algebraic system. This self-contained version of algebra within The Elements uses simple comparisons between lines and shapes to each other which replaces constants and variables found in other forms of algebra (which is also explained in the book) to prove that the different statements that are being made are true. These comparisons in combination with previously proven statements allows The Elements to create proofs of all different kinds. While the algebraic like system Euclid’s Elements uses to solve equations may be a little difficult to get one’s mind around sometimes it makes the proofs within its pages much more difficult to refute than other geometry books.

So, why do we not use this book to teach students about geometry today? Perhaps the biggest reason and most obvious is that The Elements is a difficult book to read. Unlike most textbooks today, it doesn’t use numbers and doesn’t give examples. However, just because current day geometry books are easier to teach with and easier for students to understand does not mean that they are better books. Perhaps the final reason that I believe The Elements is such a great geometry book compared to others is that the reader must want to learn about geometry if they wish to get anywhere in Euclid’s Elements.  But, if they are able to get through Euclid’s Elements, they will have a much stronger fundamental idea of geometry than from other textbooks. While it is easy to state the fact that someone who survives being stranded in the wilderness will have a better idea of how to survive in the wild than someone who hasn’t, it doesn’t change the fact that it is true.

Going back to my classroom experience, I thoroughly enjoyed going through the proofs in The Elements and I would spend most of my lunch time going to my teacher and proofing more of Euclid’s Statements. After about 2 weeks of starting the book, I had finished it. After that point, I spent the rest of time in class helping other students understand The Elements. Unfortunately, most of the other students had a hard time getting through that semester and only a few other students were able to understand it in a similar fashion as myself. However, those among us who did understand Euclid’s The Elements had no troubles passing the second semester of class which was going back to the more common form of geometry. In conclusion, I believe that Euclid’s Elements is a fantastic book that does more for geometry than any other book out there and, if someone is really interested in geometry, they should do their best to read through and understand The Elements if they want the best foundation in geometry they can have.

Sources:

(http://en.wikipedia.org/wiki/Euclid%27s_Elements)

(https://archive.org/details/bub_gb_UhgPAAAAIAAJ)