Today I was discussing the nature of number systems with a former seminary professor from Erskine Seminary. At first it seems strange that I would be discussing such a "mathematical" idea with a seminary professor, however, the idea is not as strange as it seems. What one will find through a discussion like this, that is, a discussion about number systems with a non-math person, is that number systems stretch far beyond mathematics. This is exactly what I discovered through my conversation with this former professor. The two main ideas that came out of this conversation were that: There is reasoning behind each number system, and, The uniqueness of each number system.
First, our conversation started with the origins of the current number system that we use here in the United States. Though it is up for some dispute, it is pretty generally accepted that our number system spawned from ancient Arabic numbering systems. Other than origins, is there a meaning behind the number system and, more importantly, what is the meaning? What came up in our conversation is that there is, in fact, a meaning behind our current number system. Although the numbers that most of us write daily do not display the following characteristics, they find their purpose in the traditional style of these numbers. There is a meaning! What is it? So we see, even in this common typeface that the number one, 1, is not just a line, but has a small tic or downslope at the top. It turns out that this is not simply the product of a sloppy writing, greasy teenager, but rather, is the remnant of the intentional meaning. Traditionally, the number 1 is written more like this ^, minus the superscript. We find actually that many countries still write the number 1 this way (Germany and other European countries). The traditional 2 is written more like Z, the traditional 3 is written more like a sideways W. Anyhow, what we notice about these numbers is vertices! Yes! As it was originally written, for the most part, the number of vertices or changes in direction signified which number it was. a 1 has one vertex, a 2 has
2 vertices, etc. This is something that I, and I assume, many other people, even mathematicians, have not thought about. What does this have to do with the role of a number system?
The answer to the previous question came, for me, in the furthering of our discussion about these unique properties in each number system. This will also serve to outline the role of our current number system. This discussion was not as in depth as you might think, in fact, it was pretty basic. Sometimes the most prolific understandings of an idea or concept. The main topic of our conversation revealed itself in Roman numerals. The professor outlined that in his current career as a superintendent, he works with young students teaching Greek from time to time and, in relation to that conversation, talked about Roman numerals with the students. On the one dollar bill, he explained, 1776 is printed on the bill in Roman numerals. He simply explained that the number is nearly 9 characters long. The students, naturally found this very confusing. We then talked about how incredibly difficult it would be to do any type of meaningful mathematics using that number system. It really brings into perspective, the incredible mathematical achievements by ancient mathematicians using number systems that were, basically horrible. What does this mean for the role of our number system? It means that it makes mathematics more accessible, meaningful, and more quickly progressing than ever before. Without our current, concise, and easily understood numbering system, mathematics would be slowwwww. It would be like doing internet research with dial up. Or eating rice with a single chopstick. Our current number system effectively eliminates the single chop stick rice eating of math.
Sunday, June 29, 2014
Monday, June 9, 2014
Journey Through Genius
The book that I read for our "book club," if you will, is Journey Through Genius by William Dunham. First of all, I would like to say that this is a great book. It is absolutely filled with information about not only some of the greatest proofs of all time, but also about the mathematicians themselves, the culture in which they lived, and the era in which they studied. That being said, I also want to add that this is not a book that you read for leisure in the evenings to relax. I would say that it is almost more of a book to read in spurts, not all at once. I say that because, though the information is extremely interesting, it would take an incredible amount of time to read through the entire book and actually have made sense of it all. I really loved the fact that the contents laid out the specific sections dedicated to particular proofs, theorems, and mathematicians. This allows the reader to skip around to the proofs that might be the most interesting to them at a particular time and truly dive in. That's the thing about this book. It's not about a joyful afternoon swim, but a rapids of facts and explanations that seem crazy to "dive into," but once you come out at the end of any of these rapids, you find yourself with a smile on your face. This is the beauty of mathematics at its finest. As described by the author, mathematics is not simply about practical application, memorization, or finite use. Rather, mathematics is like a Rembrandt a Picasso, a masterpiece. Something that is not only practical but visually and logically pleasing. Logic, in fact, is noted in this book as being one of the only prevailing pieces of history. Science changes and is replaced. Medical practices from centuries ago are naive attempts at best of something great, but the logic behind a tried, tested, and true proof of a theorem prevails over the centuries.
I would recommend this book to anyone who has an interest not just in the application of a theorem or how the proof is derived, but how these things are affected by the culture of the region, era of time, and specific personalities of the mathematicians who discovered them. If you are someone who is looking for a to the point, concise, and application geared explanation of some of these proofs, I would say this is not the book you are looking for. This book is not so much about learning a theorem or proof as it is discovering the theorems and proofs. So one might ask, "What is the difference between learning and discovering?" Well the answer to that is quite simple. There is a certain appreciation and fulfillment that comes from discovering something, while learning offers merely the satisfaction of a good grade, a praising note, or a gold star. The gratification in learning spawns from the reward, whereas the gratification in discovery spawns not only from the reward, but also the journey, hence the title of the book, Journey Through Genius. It is not titled, Genius Explained, How To Be a Genius, or Guess What This Theorem is Used For! And that is for a very good reason.
At the beginning of this blog I said that this book is not a leisurely read, but I have come to the conclusion that leisure, much like beauty, is in the eye of the beholder. For many mathematicians, like those mentioned in this book (Euclid, Archimedes, Euler, etc) these things may be one in the same.
I would recommend this book to anyone who has an interest not just in the application of a theorem or how the proof is derived, but how these things are affected by the culture of the region, era of time, and specific personalities of the mathematicians who discovered them. If you are someone who is looking for a to the point, concise, and application geared explanation of some of these proofs, I would say this is not the book you are looking for. This book is not so much about learning a theorem or proof as it is discovering the theorems and proofs. So one might ask, "What is the difference between learning and discovering?" Well the answer to that is quite simple. There is a certain appreciation and fulfillment that comes from discovering something, while learning offers merely the satisfaction of a good grade, a praising note, or a gold star. The gratification in learning spawns from the reward, whereas the gratification in discovery spawns not only from the reward, but also the journey, hence the title of the book, Journey Through Genius. It is not titled, Genius Explained, How To Be a Genius, or Guess What This Theorem is Used For! And that is for a very good reason.
At the beginning of this blog I said that this book is not a leisurely read, but I have come to the conclusion that leisure, much like beauty, is in the eye of the beholder. For many mathematicians, like those mentioned in this book (Euclid, Archimedes, Euler, etc) these things may be one in the same.
Sunday, June 8, 2014
The Role of The House of Wisdom
When examining the role of The House of Wisdom in mathematics I have found it to be very important to consider what the role of this place was in relation to anything. First, The House of Wisdom was founded by a man by the name of Caliph Harun al-Rashid. Now this is a guy who was pretty darn powerful in his day. He was intellectually, politically, and militarily resourceful, which is the exact way that many modern liberals would describe the late "Dubblya Bush." Further his surname translates to "the just," "the upright" or "the rightly-guided," which just further demonstrates the correlation made above. To get more to the point, and the value of the actual situation, this was a man who valued intellectuality and the discoveries that came along with it. He founded The House of Wisdom to be a place where intellectuals of many different disciplines could come together to research, explore, and translate ideas in the areas of science, mathematics, astronomy, medicine, alchemy and chemistry, zoology, geography and cartography. A good way to look at this place would be to think of it as a room where all of the best and brightest could come together to share their wisdom, learn from one another, and put their ideas into text of many languages, sort of like a coffee shop where people discuss the novels they are writing, but actually doing it.
Anyhow, how does this apply to the current state of mathematics? Given that the best and brightest were all gathered in this place, ideas could truly be explored and either proved or put to rest. This was a place where mathematicians could collaborate with other mathematicians and intellectuals to discuss problem points with whatever it was that they were exploring. I think that some of the greatest strides in mathematics came from collaboration and still do. How many people did it take to prove or disprove any point of discrete mathematics? Five line geometry? Any of Euclid's theorems? It takes often takes many people to work through a given math problem and The House of Wisdom was a place where people could do that. Not only did the work through the mathematics to come up with something that was valid and trustworthy, they translated those works into other languages so that the ideas could spread. Without this type of translation, many modern math strategies and formulas and theorems would not have been used, or maybe not discovered in certain cultures until years later. Maybe never. The sharing of ideas is incredibly important to the spreading of knowledge. The House of Wisdom is the hub of knowledge spread-ation, if you will. Like a raptor chases down its prey, displays every aspect of predation and devours that prey, so the spread-ation of mathematics hunts down the minds of skeptics and devours them into parabola shaped smile of discovery.
Anyhow, how does this apply to the current state of mathematics? Given that the best and brightest were all gathered in this place, ideas could truly be explored and either proved or put to rest. This was a place where mathematicians could collaborate with other mathematicians and intellectuals to discuss problem points with whatever it was that they were exploring. I think that some of the greatest strides in mathematics came from collaboration and still do. How many people did it take to prove or disprove any point of discrete mathematics? Five line geometry? Any of Euclid's theorems? It takes often takes many people to work through a given math problem and The House of Wisdom was a place where people could do that. Not only did the work through the mathematics to come up with something that was valid and trustworthy, they translated those works into other languages so that the ideas could spread. Without this type of translation, many modern math strategies and formulas and theorems would not have been used, or maybe not discovered in certain cultures until years later. Maybe never. The sharing of ideas is incredibly important to the spreading of knowledge. The House of Wisdom is the hub of knowledge spread-ation, if you will. Like a raptor chases down its prey, displays every aspect of predation and devours that prey, so the spread-ation of mathematics hunts down the minds of skeptics and devours them into parabola shaped smile of discovery.
Multiplication In Other Countries
One thing that I have found to be extraordinarily interesting is the topic of multiplication. I got to thinking about what it really is and why we do it the way we do. It seems like everyone that I have every met has learned to do it the same way. There are, of course differences in terminology and sometime the order that things are done, but for the most part it's the same. You take your numbers and stack them up like a cake and then slowly work your way across, devouring each place value until you you get the very end (or beginning). What you find yourself with then is an empty plate...or a number, depending on what we are still talking about. Anyhow, I got to wondering if we all do multiplication that way because that's the only way to do it...that didn't seem logical, so there must be another way. Then it came to me! A calculator...that's how most people do multiplication these days. I include myself in that blanket statement. It honestly wasn't until my teacher assisting in a second grade class this past semester that I really had to retrain my brain to do multiplication like I learned at that age.
I couldn't possibly be building up this whole "how to do multiplication" thing, just to say that people use calculators, surely not. So what then? Well it turns out that there is another way, a pretty cool way, in fact, that doesn't involve writing numbers at all. At least in the way that we are used to. From my understanding, this multiplication has it's origins in China. So how do you do it?
Well basically it's like writing tallies. Lets say you wanted to multiply 14 and 12, you would start by writing one tally and then four tallies, representing 14. Then you would do the same thing for the 12 except crossing the lines horizontally over the 14 like this:
I couldn't possibly be building up this whole "how to do multiplication" thing, just to say that people use calculators, surely not. So what then? Well it turns out that there is another way, a pretty cool way, in fact, that doesn't involve writing numbers at all. At least in the way that we are used to. From my understanding, this multiplication has it's origins in China. So how do you do it?
Well basically it's like writing tallies. Lets say you wanted to multiply 14 and 12, you would start by writing one tally and then four tallies, representing 14. Then you would do the same thing for the 12 except crossing the lines horizontally over the 14 like this:
You then count up the intersections like the illustration below. The red indicates the hundreds place, the blue added up represents the tens place and the green represents the ones place. Of course you can do this same process with larger numbers, but then there is "carrying" involved. But! I thought this was a really cool, visual way to teach students how to do multiplication differently.
Wednesday, May 14, 2014
Nature of Mathematics
There are two questions that I aim to clarify, in light of my own understanding, through this post. The first question is, what is an axiom? An axiom is essentially to math, what a universal truth is to life. For instance, one of these universal truths, though not often thought about or commonly used, is that the truth is always the truth, even if no one believes it. So what this is saying is that, in life, if something happened and is true, even to the knowledge of only a single person, that true thing is, in fact, what actually happened. This holds true whether everyone or no one else thinks that this "true thing" is true. It is something that can be applied to any situation and must be accepted as real or fact by the person to whom it is true. This expands to all of life, in fact.
What this looks like in mathematics is an axiom. So an axiom is a truth about math, that we hold to be true regardless of the situation. In mathematical terms, this truth would in actuality be reminiscent of a proof. Though a proof must be proven to be accepted as true, an axiom (a statement about an aspect of math) is accepted as true without having to prove it....what the heck math!? Anyhow, I'm not sure who, but somewhere along the line some mathematician, who definitely was not bizarre in at least one way, decided that these statements didn't need to be proven. Who'd have thought?
So, what in the world do we use these axioms for? Why would we need something that is accepted as true, without having been proven? Axioms are used for proving other mathematical statements which could not otherwise be proven. That is, for many mathematical statements you come to a point in the proof of said statement where you have to accept at least something as true. If you didn't then you would have to continue breaking down the statement forever...but seriously it would get frustrating or impossible to prove some things if you didn't accept some truth as true. That truth is an axiom. They are essentially the most basic statement that you can use in a proof that doesn't have to be proven itself. An example would be that if we could not assume that all right angles were congruent, we could not logically apply the pythagorean theorem to any triangle. This would be in light of the fact that, if not all right angles are 90 degrees, then they could be other degrees. Smaller or larger. A change in angle measure means a change in segment length on a triangle, which would be impossible to calculate with the pythagorean theorem, because the pythagorean theorem only works if there is a right angle.
That seems confusing. But a simple way to think of an axiom is this. "I would not be at Grand Valley right now if I didn't apply, I wouldn't have applied if I didn't know it existed, I wouldn't have known it existed if...if...if....if...if....................axiom." An axiom is the last in the line of the "ifs"...if you will.
What this looks like in mathematics is an axiom. So an axiom is a truth about math, that we hold to be true regardless of the situation. In mathematical terms, this truth would in actuality be reminiscent of a proof. Though a proof must be proven to be accepted as true, an axiom (a statement about an aspect of math) is accepted as true without having to prove it....what the heck math!? Anyhow, I'm not sure who, but somewhere along the line some mathematician, who definitely was not bizarre in at least one way, decided that these statements didn't need to be proven. Who'd have thought?
So, what in the world do we use these axioms for? Why would we need something that is accepted as true, without having been proven? Axioms are used for proving other mathematical statements which could not otherwise be proven. That is, for many mathematical statements you come to a point in the proof of said statement where you have to accept at least something as true. If you didn't then you would have to continue breaking down the statement forever...but seriously it would get frustrating or impossible to prove some things if you didn't accept some truth as true. That truth is an axiom. They are essentially the most basic statement that you can use in a proof that doesn't have to be proven itself. An example would be that if we could not assume that all right angles were congruent, we could not logically apply the pythagorean theorem to any triangle. This would be in light of the fact that, if not all right angles are 90 degrees, then they could be other degrees. Smaller or larger. A change in angle measure means a change in segment length on a triangle, which would be impossible to calculate with the pythagorean theorem, because the pythagorean theorem only works if there is a right angle.
That seems confusing. But a simple way to think of an axiom is this. "I would not be at Grand Valley right now if I didn't apply, I wouldn't have applied if I didn't know it existed, I wouldn't have known it existed if...if...if....if...if....................axiom." An axiom is the last in the line of the "ifs"...if you will.
Wednesday, May 7, 2014
When I think of math, I naturally think of all of the math courses that I have taken at Grand Valley. Throughout all of these different math courses, I have seen that math can look vastly different and has just about an infinite amount of applications. Math, for me, has ranged from how to teach it to children using manipulatives all the way to proving that zero is, in fact, less than one. Throughout the duration of my journey from colored blocks to more abstract applications of mathematics I have found that the best way to describe math is that it is helpful. That's what math is. Given that there are so many different variations of math, I find that it would be unhelpful to list what math is in all of these circumstances. I do, however, find it helpful to talk about the fact that, in light of there being so many different mathematics applications, math is helpful. Why are there so many different types of math? Because there are so many different purposes for it. Math can help me figure out how many more apples I have than Jane, it can help me to calculate the number of years it will take me to pay off my students loans, or it can even help me to figure out the best way to invest my (hypothetical) lottery winnings. So the most basic and fulfilling ways to describe what math is, in my opinion, simply to say that it is helpful. It helps me, it helps you, it helps everyone.
As far as the biggest achievements in math go, I would say that they are:
1. The development of the pythagorean theorem. Well done Pythagoras. (What can't that be used for?!)
2. Calculus. What on earth would we do if we couldn't estimate the volume of objects by slicing them into tiny discs?
3. Matrices can literally be used for more things that I can think of. I'm glad that those are around.
4. Multiplication and division. How else would we divide 35 camels up for some strangers!?
5. Addition and subtraction. We have to pay for things, sometimes at the same time. Without addition and subtraction we would be left wondering how much the cone costs.
As far as the biggest achievements in math go, I would say that they are:
1. The development of the pythagorean theorem. Well done Pythagoras. (What can't that be used for?!)
2. Calculus. What on earth would we do if we couldn't estimate the volume of objects by slicing them into tiny discs?
3. Matrices can literally be used for more things that I can think of. I'm glad that those are around.
4. Multiplication and division. How else would we divide 35 camels up for some strangers!?
5. Addition and subtraction. We have to pay for things, sometimes at the same time. Without addition and subtraction we would be left wondering how much the cone costs.
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