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Wife of Pi

curiousCharacter — Tue, 23 Apr 2013 17:48:57 +0000

Wife of Pi cartoon by The Argyle Sweater

Book Review: The Feynman Lectures on Physics

curiousCharacter — Thu, 04 Apr 2013 19:30:15 +0000

The mathematician Niels Abel (1802 – 1829) said that his knowledge of mathematics came from reading papers by Euler, Gauss, and other great mathematicians. As he put it, “I learned from the masters, not their pupils”. If one wanted to use Abel’s philosophy for physics, the logical place to start would be The Feynman Lectures on Physics.

Zeno and His Arrow

curiousCharacter — Sat, 05 Jan 2013 18:27:57 +0000

As many readers will know, the comic above refers to the Greek philosopher Zeno, from about 490 BCE. He is famous for his "argument against motion" paradox. To get to a place, said Zeno, we must first go half way there. Once there, we must go half the remaining distance, or one fourth. Then we must move half the remaining distance, or one eighth, etc. So, moving from one place to another requires an infinite number of motions, which is impossible.

Escape Velocity – Part 2

curiousCharacter — Fri, 16 Nov 2012 19:05:53 +0000

In the previous post, I derived the formula for finding the escape velocity from a spherical mass, e.g., the Earth. In this post, I will go over some implications of this important formula, including the original prediction that black holes should exist.

Escape Velocity – Part 1

curiousCharacter — Tue, 30 Oct 2012 18:43:20 +0000

As the reader probably knows, any planet or moon has an Escape Velocity. A mass at the surface travelling at this speed or greater will go infinitely far – gravity will never be able to pull it back. This post is about escape velocity, starting with a derivation of the formula for computing it.

Euler Conquers the Cubic Equation

curiousCharacter — Mon, 15 Oct 2012 04:51:27 +0000

Everyone who knows algebra is familiar with the quadratic formula for solving second degree equations. It has been known since at least the 9th century. The logical next step in algebra was to find the corresponding formula for the 3rd degree (cubic) equation. Starting about 1520, a group of Italian mathematicians competed to find the fabled cubic formula, and in 1545, one of them, Girolamo Cardano (1501 – 1576), published the solution. As we will see, the formula is complex, and of limited practical use. However, the cubic problem has great historical importance. Not only was the art of algebra was advanced by the effort to find it, but mathematicians eventually realized that the formula would only worked correctly if they faced up to the existence of complex numbers. In this post, I will show a derivation of the formula, then show how it is used.

How to Solve IT, by George Polya

curiousCharacter — Sat, 06 Oct 2012 21:16:49 +0000

This post is the first of a series; in each one, I will describe a book on science or mathematics that I have found to be particularly interesting and informative, and which the reader might want to have a look at. Since I will choose only "good" books, each of my reviews will be generally positive. In a conventional book review, a book that does not seem to be a good one gets criticized. Here, if a book that I have read seems flawed, I will simply ignore it. How to Solve It, by George Polya In How to Solve It, Polya sets out to teach the reader to improve his/her ability to come up with solutions to problems that require some insight or creativity. His techniques and examples are mostly taken from mathematics (especially geometry), but he believes that these habits of thought apply also to physics, or any other area where the problems are quantitative in nature. Before he gets specific, Polya makes some general observations: 1) People often enjoy expending mental effort, as the devotion to crossword puzzles, Scrabble, and other activities shows. 2) This willingness to think does not often include mathematical things. Indeed, says Polya, mathematics is the most disliked subject in the schools. Then, many of those people who come to dislike math become elementary school teachers, where they induce a new generation to dislike it also. 3) One reason for the unpopularity of mathematics may be the schools' emphasis on rote learning and techniques. Students, Polya believes, should be given more practice work with problems requiring at least some creativity. Then, with some guidance from a teacher or tutor, the student can feel the reward of thinking through a problem to its solution. 4) The mental discipline of thinking about a problem, trying different things, and finally solving it, is an important life skill for any field of work,. Polya thinks that solving mathematics problems successfully may help to develop that discipline. After this introduction, Polya immediately list his four phases in solving a problem: 1) Understand the problem. Draw diagrams, pictures, or graphs. Define the variables, including those given and the one we are required to find. 2) Devise a plan. Have you seen something similar? Is there a formula that connects the problem's variables? Is there a simpler problem whose solution might help with this one? etc., etc. 3) Carry out your plan. 4) Look at the solution. Does the answer seem reasonable? Can I check the answer for correctness? The remainder of the book is mostly elaboration on these points, using many example problems. These examples are at the high school level, except for two or three that involve a little calculus. To give a sense of the difficulty level, here are two sample problems: 1) Bob has 10 pockets, and he has 44 coins, all alike. Can he place the coins in his pockets in such a way that each pocket has a different number of coins? 2) The perimeter of a right triangle is 60 centimeters. The altitude perpendicular to the hypotenuse is 12 centimeters. What are the lengths of the triangles sides? How to Solve It has been continuously in print since 1945, so that it is a sort of minor classic. From my experience as a tutor, the book and its techniques would be lost on a recalcitrant math hater. Those with more interest and motivation will find some excellent hints here.

Great Formulas: the Binomial Series, part 2

curiousCharacter — Sun, 22 Jul 2012 17:41:29 +0000

In the last post, I described Newton's Binomial Series formula, one of the most important results in mathematics. In this post, I will go through three practical examples of how the binomial series can be used.

Great Formulas: the Binomial Series

curiousCharacter — Tue, 10 Jul 2012 18:36:05 +0000

Algebra classes always teach the Binomial Theorem - writing out the expanded form of a binomial raised to a power. What they don't do is teach Newton's brilliant generalization of the Binomial Theorem, usually called the Binomial Series, in which the exponent can be anything, not just an integer. The Binomial Series is much more important and interesting, and in this post I will describe it, including how Newton figured it out.

Great Formulas: V Squared Over r

curiousCharacter — Fri, 15 Jun 2012 01:51:59 +0000

All physics classes cover the formula for acceleration when the motion is in a circle. In an example usually given, the formula lets us compute the tension in a string when a mass at the end of the string is being spun in a circle. Although the textbooks do not make a fuss over the formula, its discovery by Isaac Newton in 1655 was a landmark in the history of science. Deriving the formula required a great deal of insight into the nature of motion, and it was the starting point for Newton's work on gravity and the motion of planets. In this post, I will derive the formula using only basic mathematics, something physics courses do not usually do.