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Finding Formulas by Guessing

curiousCharacter — Wed, 05 Oct 2011 06:31:39 +0000

In my high school library, there was a copy of the classic book "Men of Mathematics" by E.T. Bell. Each chapter is a short biography of a notable mathematician of the past (and yes, they are all men). In the chapter on Carl Friedrich Gauss (1777-1855), I first encountered the famous story of how, as a 10 year old student, Gauss baffled his teacher by instantly solving a problem that the teacher assumed would occupy his student's an hour or more. In this post, I will describe Gauss' insight, then show how a related problem can be solved by using some informed guesswork.

L2: What is it, and Where is it?

curiousCharacter — Wed, 24 Aug 2011 04:55:37 +0000

Instead of orbiting Earth, the James Webb Space Telescope will orbit the Sun in a special spot beyond the Earth that is a sort of gravitational island. The location is called “L2” (Lagrange Point 2). In this post, I will describe what L2 is, then show how to compute where it is relative to Earth. It's a terrific example of the power of basic algebra.

A Gem From Newton’s Principia

curiousCharacter — Tue, 26 Apr 2011 20:02:59 +0000

Isaac Newton's Mathematica Principia (1687) has been described as the most important, but also the least read, scientific book ever written. It has been little read mostly because it has been little comprehended. The book is filled with complex geometric diagrams, and Newton's explanations are brief, the assumption being that the reader's mathematical knowledge and ability is very high. However, there is at least one result that Newton derived in the Principia that is fairly easy to understand, and I will describe it in this post. It also happens to be one of the important theorems in the Principia: a proof that Kepler's Second Law of planetary motion isa consequence of mechanics.

Word Processing and Mathematics

curiousCharacter — Sun, 06 Mar 2011 06:28:14 +0000

This post is about creating, displaying, and publishing documents that have mathematical content. This is a troublesome thing for mathematics and science workers, because most word processing systems treat math as an add-on or an afterthought, if they have any provision for it at all. Even when mathematics is supported, it may be difficult or impossible to do much more than print the document on paper.

Dimensional Analysis

curiousCharacter — Sat, 05 Feb 2011 09:01:35 +0000

Most people who have studied some physics or chemistry know that it is important to keep the units of our numbers straight when we do calculations. Failure to attend to units usually leads to wrong answers. What is not well known is that the analysis of units can often help scientists to derive formulas, even when the underlying physics is not well understood. How it works seems a bit mysterious, and the technique was not understood or appreciated until about 1870, when the great physicist James Clerk Maxwell laid out the principles of the technique, which is formally known as Dimensional Analysis.

Bayes Formula

curiousCharacter — Wed, 29 Dec 2010 07:11:55 +0000

Most people who have been exposed to probability and statistics have come across Bayes’ Formula, but I suspect that many have not fully understood and internalized what the formula tells us. This is unfortunate because, as we will see, the formula applies to situations where our intuition about probability can lead to wildly incorrect judgments. What is more, those situations often involve critical issues, such as interpreting the results of medical tests. The importance of the formula is such that a whole branch of thought in science and statistics, Bayesian inference, or Bayesianism, springs directly from the formula and its implications.

Measuring the Speed of Light in 1676

curiousCharacter — Mon, 25 Oct 2010 23:34:36 +0000

In 1676, the Danish astronomer Ole Romer did something quite remarkable for his time – he measured the speed of light. Although his value was not very accurate, it was the first demonstration that light does not travel instantaneously, a belief that been held by almost everyone from Aristotle on down. In this post I will describe how Romer did it, and then describe my little experiment to reproduce his measurements and calculations.

The Basel Problem

curiousCharacter — Wed, 29 Sep 2010 17:12:11 +0000

The history of mathematics has many instances where someone has posed a problem for the mathematical world at large to solve, and the problem was not resolved for decades, or even centuries. Often, new mathematics has been discovered in the process of working out a solution. This post is the story of one such case, the so-called Basel Problem, first posed as a challenge to European mathematicians in 1644. It withstood all attempts to solve it until, in 1734, young Leonard Euler found the answer. As the reader will see, Euler's solution is a work of astonishing ingenuity, even though the level of the mathematics does not go beyond Algebra I.

How Henry Cavendish Weighed the Earth

curiousCharacter — Tue, 21 Sep 2010 05:19:12 +0000

Most readers will be familiar with Newton's Law of Gravitation, which states that the attractive force between two masses is proportional to the product of the masses, and inversely proportional to the square of the distance. We can only determine the proportionality constant G if we can measure the force between two known masses separated by a known distance. Even Newton had no ideas for doing this, and thought the measurement might be {}``beyond the skill of man''.

Pythagorean Triples

curiousCharacter — Wed, 21 Jul 2010 17:30:54 +0000

When math textbooks need an example of a right triangle, they frequently use a triangle with sides of length 3, 4, and 5, since the numbers work out so nicely: \(3^{2}+4^{2}=5^{2}\) by the Pythagorean theorem. If that gets tiresome, 12, 5, 13 might be used: \(5^{2}+12^{2}=13^{2}\). Clearly, multiplies of these numbers work also, e.g. \(6^{2}+8^{2}=10^{2}\). Such [...]