Chapter 2: The Calculus of Finite Differences
Many people feel like they know all of the math topics, however there are many more parts to know in the subject of mathematics. There are many topics in mathematics that many people do not know, or may know and not realize. In the Colossal Book of Mathematics the reader can see an example of such hidden topics in chapter two. This chapter is about the branch of mathematics called the calculus of finite differences that is not that well known to many people. Calculus of finite differences origin was in Methods Incrementorum which was a book published by an English mathematician. The chapter stresses that their are parts of this type of mathematics that can be extremely useful for many. The method of differences is also used in physics of natural world. Newton's formula and other sciences can be related to this branch of math. Chapter 2 uses the "Pancake problem" which is finding the maximum amount of pieces in which it can make "n" amounts of cuts and they can cross each other. Using the calculus of finite differences, you make a best guess for a formula and then try to prove those methods using deductive methods. It seems as though the challenge is finding the formula by which the problem can be solved and then being able to prove that such formula. For any finite series of numbers there is an infinite amount of formulas and functions that can generate those numbers. The chapter then refers to other problems that are about making "n" amount of cuts to create a maximum number of slices of something.
As my thoughts of the chapter go, I was very confused. This was very interesting in my opinion however that the sciences and Newton's law come into effect when solving a math problem that seems so simple. Reading the first paragraph of this chapter, hearing that this was a subject that not many knew, I expected to understand nothing. After reading the chapter however, I remember doing problems like this in high school and not even realizing it. The concept itself seems rather difficult and I had a hard time following with the chapter due to all the references to cutting items. My question regarding the chapter would be wondering if the calculus of finite differences only consists of these types of problems? By these problems I mean the cutting into n slices. Another confusion of mine came in when the chapter brought in Newton's Law, how does Newton's law actually help and when does this come into play with this topic?. To me the difficulty with the process described here is generating the formula. I do not understand how to generate the formula from the problem to best find a solution. A surprising aspect of the chapter is that a simple sounding problem such as the pancake problem could have such a deep way of figuring it out. I would have never guessed that such problems would be part of a topic of mathematics that I had never heard of. I also know that this topic would be something I would be interested to learn more about and learn the answers to all of my questions.
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ReplyDeleteIn chapter 2 it sounds as if it begins to explain exactly how much more complicated mathematics is than the small window that is exposed in grade school. Basically, there is a lot of depth in each portion of mathematics and there are infinite ways to reach “n”. Though the writer and myself found this chapter to be a bit confusing I believe some points mentioned were rather interesting. For example, the “Pancake Problem” intrigued me. Finding the most ways to cut a circle—or in this case a pancake— seemed a little bit more like a puzzle than a math problem. One question that I would as about the pancake problem is: How are there not an infinite amount of cuts that can be made? Though it is possible that I misunderstood the problem and there actually are an infinite amount of cuts that can be made on the circle or pancake. Not only is the pancake problem discussed, but the depth to which mathematics extends to is also mentioned, as I mentioned earlier. There really is no end to mathematics and I think that this chapter does a good job of demonstrating that. Overall, the writer did a good job or summarizing the chapter and her thoughts on the chapter.
ReplyDeleteChapter 2 was an interesting one. Finite calculus seems to be a very experimental sort of mathematics. Although it is used mainly for things like statistical and finding formulas, I think the most interesting place that finite calculus resides is in children’s math books, as a bridge of sorts between algebra and calculus. One of the questions that this reading made arise in me was this, “How can we apply things like the necklace problem to real life situations?” My understanding of the reading was that the only way to really develop and prove these formulas and ideas was to use the guess and check method. This doesn’t seem at all efficient or exact. If we are to apply finite calculus, won’t it take longer to come up with a reliable formula than it would to solve the problem a different way?
ReplyDeleteThe problems ranged from cutting a pancake to making necklaces to forming triangles. The necklace problem was the most intriguing to me. I felt like the amount of possible combinations was almost too high to calculate with certainty. As I continued to read, I found that my initial thinking was not far from the truth. Finite calculus was presented in the chapter like a hypothesis, something you could really only support, but not really prove definitively. I, like Tiffany was slightly confused with the application of these formulas because of this lack of certainty. Reading things like this reminds me that even though most people think of math as something that always has one right answer only, this isn’t always the case.
As I read through chapter 2, I found the idea of finite differences to be quite interesting. The fact that you can use this numerical technique to find an approximate solution to differential equations is fascinating. Before reading your blog and the chapter, I was unaware that you could approximately solve specific problems that would be easier to execute rather than trying to solve the differential equation directly. Using these methods in the chapter to find formulas seems like a pretty effective way to solving the specific problems. It gives you an approximation on which you can refine further but it also seems slightly inefficient as it seems it still would be difficult to find an exact solution. However, through reading, I gravitated more towards the pancake problem and necklace problem in the chapter. Those problems felt more appropriate for recreational math. Both the problems were more closely related to puzzles that put you into deep thought. For example, the pancake problem made me think about the finite or infinite slices on a pancake. Overall, I think you did a solid job on portraying the main ideas of this chapter. I too agree with the rest that it is hard to follow the high-level mathematics theory terminology without it being taught directly to us.
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