Copyright© 1996-2015 by Gary and Kenny Felder
This is a collection of papers that we have written to explain various concepts in math and physics. But first, the obligatory list of links...
Math
What you will find below are essays explaining various topics, more or less arranged from the most basic to the most advanced.
In addition to these essays, Kenny has also written a complete textbook for Advanced Algebra II. You can download the entire thing, or individual pieces and topics, for free, from the wonderful folks at Connexions at Rice University: go to http://cnx.org/content/m19435/latest.
Gary and Kenny have also written a college-level textbook called Math Methods for Engineering and Physics. That book contains (among other things) "exercises" designed to facilitate active learning in that class. Click here to read more about the textbook, or click here to download the exercises for free.
Math and the Obvious (Kenny)
- For consistency, I probably should have called this "Think Like a Mathematician." I also considered "The Math Teacher's Manifesto." Written for both students and teachers, this brief essay describes the right way—and the wrong way!—to think about math. As such, it is arguably more important than any of the specific papers on specific topics.
Negative Times Negative is What? (Kenny)
- Mike, a ninth grader in Westbrook, Maine, emailed me the question: "Why is it when you multiply two negative numbers you get a positive number?" This is the answer that I emailed back.
Base Eight And other math for people who are missing fingers (Kenny)
- Before you read this, you should know how to count.
- After you read this, you should be able to count in base eight and other non-standard bases.
Triangles, Circles, and Waves (oh my!): An Overview of Trigonometry (Kenny)
- Before you read this, you should have taken—or be taking—trig, and be frustrated. This will not really teach you trig, because I don't give you any problems to work, and you can't learn trig without doing a lot of problems. But I do offer a general description of the triangles, circles, and curves that all somehow mean the same thing, so this may help clarify a few things that have been confusing to you.
- After you read this, trig should make a little more sense than it did before.
Quick-and-Dirty Guide to the TI-83, TI-83+, TI-84, and TI-84+ (Kenny)
- Before you read this, you should probably be at least marginally familiar with how calculators work. Not much experience is necessary, but I do skip over some basic basics like "how to turn the Calculator on."
- As you read this, you should have a TI-83 or TI-84 in your hand.
- After you read this, you will know most of what I know about the calculator. I should also mention that, unlike the other papers on this site, this one can be used as a reference—instead of reading the whole thing through, you may just want to say "where is that darned absolute value?"
Kenny's Overview of Hofstadter's Explanation of Gödel's Theorem (Kenny)
- Before you read this, you should grab a big cup of coffee. No prior knowledge is required, but this is a real brain-strainer. It's also really cool.
- After you read this, you should be able to dazzle your friends with your explanations of Gödel's theorem, and why truth is always one step beyond logic.
How to Draw a Five-Dimensional Cube (Gary)
- Before you read this, you should have some general sense of what we mean by "2 dimensions" (flat things such as squares or circles) vs. "3 dimensions" (cubes and spheres), and have wondered what comes next.
- After you read this, you will have a technique that can be used to answer questions such as "how many corners does a 4-dimensional hypercube have?" not by doing any math, but by visualization.
What dx Actually Means (Kenny)
- Before you read this, you should have taken a few semesters of calculus. This requires more mathematical background than almost anything else I've written: you should know how to take a derivative and an integral, and perhaps you should be a bit curious about that dx thing that you keep seeing.
- After you read this, you should have a renewed understanding of what you've been doing all this time, and be able to solve some problems with integrals that you couldn't solve before.
Partial Fractions by Don Methven
- This paper (not written by Kenny or Gary) describes a technique for rewriting a complicated fraction as a sum of simpler ones. Although the technique can be performed by anyone who understands high school algebra, it is usually taught as a part of college-level Calculus because it lets you turn something hard-to-integrate into something easy-to-integrate.
Physics/Science
Think Like A Scientist: An Induction Fable (Kenny)
- Before you read this, you don't need to know anything in particular. You could be anywhere from junior high to senior citizen, with some interest in science.
- After you read this, you should have an improved understanding of the scientific method, and in particular of "induction" as a way of learning about nature.
Think Like a Physicist (Kenny, and my personal favorite)
- Before you read this, you should have taken—or be taking—an introductory Physics course. You may be wondering why there is so much math involved, when all you really want to do is understand nature. Or you may have the frustrated feeling that you understand the Physics concepts, and you can do the math, but you can't quite connect the two.
- After you read this, you should have begun to develop some intuition for how math fits into Physics (and other sciences). You should be able to spot unreasonable answers to problems based on common sense.
- And by the way—my original paper uses British units (measures length in feet). Even though scientists dislike these units, I think it makes it more intuitive—and the whole point of this paper is to be intuitive!—for American or British students. However, it makes it pretty nonsensical for students in metric countries! So Ray Forma of the Methodist Ladies' College in Australia has kindly donated a version of this same paper that uses SI units. Click here for that paper!
One of These Things is Not Like the Other: A Discussion of Units (Kenny)
- Before you read this, you should have taken—or be taking—an introductory science course (Physics or Chemistry).
- After you read this, you should have a good feeling for units (also called "dimensional analysis"): why they are important, and how to work with them.
When Units Go Square (Kenny)
- Before you read this, you should have read and understood the above paper, "One of These Things is Not Like the Other," in which I explain the basic concept of units.
- After you read this, you should be able to apply dimensional analysis to square units (which measure area) as well as regular ones.
The Day the Universe Went All Funny: An Introduction to Special Relativity (Kenny)
- Before you read this, you should probably have made your way through at least one Physics course, in high school or college, that covered words like "velocity," "acceleration," and "momentum."
- After you read this, you should have a beginning understanding of some key concepts in "classical relativity" (a part of Newtonian mechanics that they often gloss over), and "special relativity" (the start of 20th century Physics).
Bumps and Wiggles: An Introduction to General Relativity (Gary)
- After Special Relativity (see Kenny's paper above), Einstein developed General Relativity, which is even more far out. In this paper, Gary gives an explanation that you can follow without any math of the basic concepts, including curved space and black holes.
General Relativity II: A Deeper Look (Gary)
- More on General Relativity, including some math. Make sure to read the first paper first! You can do the first paper without this one, but you probably won't understand this paper without the first one.
Quantum Mechanics: The Young Double-Slit Experiment (Gary and Kenny)
- Before you read this, you should be ready to think very hard. No background is technically required, but some basic familiarity with "waves" in classical mechanics (such as what you learn in a high school Physics class) would help.
- After you read this, you should have a pretty good layman's (non-mathematical) understanding of some of the fundamental concepts of quantum mechanics, and some of the experimental evidence that underlies these concepts.
Quantum Mechanics: What Do You Do with a Wavefunction? (Kenny and Gary)
- Before you read this, you should really be taking—or have taken—a quantum mechanics class. This is not the conceptual overview: instead, this gets into the math, including partial differential equations, Fourier transforms, and things too fierce to mention. But if you are struggling through your first Quantum class, this may help.
- After you read this, you should have a good understanding of what Y represents, how to use Y to predict the results of measurements, and how to use Schrödinger's Equation to predict the evolution of Y.
Things Fall Apart: An Introduction to Entropy (Gary)
- Entropy is one of those concept that gets mystified more often than explained. What does it mean to quantify "disorder?" Why is entropy refered to as "the arrow of time"? The answers make a surprising amount of sense...at least, in Gary's hands!
The Expanding Universe (Gary)
- This paper is a lively introduction to the field of "cosmology" which basically means the study of the universe (it's a large subject). With no math, it answers questions such as: what is the big bang, and how do we know it happened? Is the universe finite or infinite? Will the universe come to an end?
Beyond the Big Bang: Inflation and the Very Early Universe (Gary)
- Before you read this paper, you should read Gary's cosmology paper (above), which introduces the big bang and the expanding universe. This paper, which assumes the previous one as background, goes on to explain the more modern cutting edge of cosmology—not a replacement for the big bang theory, but an enhancement of it—called "inflation."
Spooky Action at a Distance: An Explanation of Bell's Theorem (Gary)
- This paper describes "Bell's Theorem," a quantum mechanical theory—and experiment—which shows conclusively that what happens here can have an instant effect on something else there no matter how far apart "here" and "there" are. This is a recent and disturbing aspect of quantum mechanics, which is actually surprisingly easy to understand (at least when Gary explains it).