Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the tag “Archimedes”

Infinitesimal Calculus 1: The Numbers Between Numbers

If I were to ask a person to name a number which comes between 1 and 3, everyone from a three-year-old child to a white-bearded great-grandfather is likely to respond by saying, “2.” If I rephrase the question to ask about a number between 1 and 2, then the young child might be confused, but a fourth-grader might be able to respond with 1\frac{1}{2}. We have to extend our understanding of what we mean by “number” to include some concepts which are not quite so intuitive. That is to say, in between the Integers, there are other numbers which are known as Rational numbers. In fact, given any Integer, n, there are an infinite number of Rational numbers which are greater than n and yet less than any other Integer which is greater than n.

There are numbers in between the Rational numbers, too. We can define some number, r, which is not equal to any Rational number. There are Rational numbers which are greater than r, and those which are less than r, but somehow our number r squeezes itself into a gap in between the Rational numbers. In order to find such a number, we need to further extend our understanding of “number” to include the Real numbers. This should all be very familiar to the average high-school student.

Now, what happens if we extend this idea one step further? Are there more numbers which are in between the Real numbers?

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Pi is Stupid

I teach Brazilian Jiu-Jitsu to people of all ages, from preschoolers to middle-aged parents. While BJJ, in itself, is not necessarily the most academic of pursuits, I also happen to be a huge nerd. So while teaching some of my 8 to 13 year-old students, it sometimes happens that I overhear them talking about their math classes, often to complain about ideas that they’re struggling to grasp. Being a huge nerd, and also a delighted teacher, I do my best to help them through these issues. If I can teach a kid how to find the length of the hypotenuse of a right triangle at the same time as teaching her how to finish a Triangle Choke, I become pretty much the proudest martial arts instructor you could hope to meet.

One of the things that my kids often use in their math classes, but almost never really understand, is the constant π (pi). They are taught π in class to help learn things like how to calculate the area of a circle, but they usually don’t really know what π actually is. They just think of it as some number that they have to memorize, never thinking about where the number comes from, or why it is what it is. Sometimes, I’ll tell the kids that they can earn their way out of doing push-ups if anyone can tell me what π is. Most often– after the jokes about desserts are made– I’ll hear someone say, “Coach, π is three-point-one-four!” Every now and again, one of the kids is clever enough to say, “Coach, π is three-point-one-four-on-into-infinity!” They get confused when I tell them that’s the value of π, but that is not what π actually is. It’s not their fault that they get confused by this; they were usually taught about π all wrong. I don’t even blame their math teachers, because most of the time, those math teachers were also taught about π in the wrong way. For a very long time, math classes have been teaching that π is a number, instead of teaching that π is the relationship between a circle’s circumference and its diameter. There is a reason it has been taught this way.

Ladies and gentlemen, π is just plain stupid.

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Alexandria: The Most Important City in History

In 332 BCE, Alexander the Great’s incredible military campaign advanced on Egypt. As his armies moved in, the people there regarded Alexander as a savior, hailing him as the Son of the Most High God, and declaring him Master of the Universe. The young conqueror quickly fell in love with the country, and in the following year, he founded a new capital city: Alexandria-by-Egypt. From its very inception, Alexandria was created to be one of the most important cities in the world. Its ports became a prominent trade destination, in the Mediterranean, and its culture flourished and prospered from a mix of disparate peoples, religions, and philosophies, even at its onset. The Lighthouse of Alexandria was an incredible and beautiful building, standing over 400 feet tall, regarded as one of the Seven Wonders of the Ancient World. But, without a doubt, the most incredible and amazingly important feature of Alexandria was the Museum.

The Museum of Alexandria became, almost immediately, the center of knowledge in the ancient world. It was not a museum, in the modern sense, but rather more like a modern research university. Students went there to learn all they could about the sciences of the day, while the teachers and academics received state salaries to simply do research and increase the knowledge of Mankind. The Museum boasted an incredible library, one which would quickly become the largest collection of books in the Ancient World. A tradition was developed, in the city, whereby foreign visitors would allow any books which they brought with them to be copied, so that the Library’s stocks would continue to increase. Vast amounts of knowledge were developed and stored in Alexandria.

The Museum was destined to make Alexandria-by-Egypt the most important city in the history of the world. An inordinately large amount of our modern knowledge of mathematics and science is owed directly to men educated or employed by this institution. What follows are brief descriptions of just 16 such scholars.

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