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Rigor is something that is emphasized frequently in higher levels of mathematics and physics, and it has always been something that I appreciated. Unfortunately, with increased rigor often comes a decreased number of people who can understand an argument. One pedagogical ploy that has been used to great effect has been to offer "proofs" of rather difficult concepts on the basis of certain tricks that are not themselves rigorous. I call these things "lazy proofs", and they suffer from the problem of leading to outright contradictions and nonsense if taken to far. This kind of problem, usually, is swept under the rug by the person (usually a teacher) offering the proof in hopes that the misconceptions that could arise never rear their ugly head. Sometimes they never do. Other times, they cause problems down the road. One example of such a lazy proof is the following argument that the centripetal acceleration is   a = v2/r.   (1)   Imagine... read more

Proof of the Assertion that Any Three Non-Collinear Points Determine Exactly One Circle This is an interesting problem in geometry, for a couple of reasons. First, you can apply some earlier, basic geometry principles; and secondly, you can choose two different strategies for solving the problem. The basic geometry underlying: any three non-collinear points determine a plane, somewhere in 3D space. Once that has been done, imagine that the plane has been rotated into the x-y plane, which will make the problem much easier to solve! The two strategies for solution are: (Proof A) actually solve to find the circle. This is equivalent to finding the center of the circle (finding the equation of the circle is simple from there). But, you actually have to do some math to get this! If, while doing this, there is no possibility to obtain other values for the coordinates of the center of the circle, you have proved the assertion as well as obtained a method (and perhaps... read more

In mathematics, we start with the natural numbers (or more simply the 'counting' numbers) and learn how to count, starting with 1 and moving up the positive number line. But something special about counting numbers is usually overlooked- primes.    Looking at the naturals, we have {1, 2, 3, 4, 5, 6, 7, ... }  all the way to infinity.    Now if you pick a number, any number, and analyze it, you can see its basic properties such as its factors or its multiples. Let's take the number 4 for example.    4 is a multiple of 2, which means it can be divided by a number other than itself and 1. We write "2|4" meaning "two divides four."    Obviously, most other numbers have these factors and are built on them. But there is a type of number that you may be familiar with but not realize its significance in mathematics. One of the most interesting things in mathematics, though a basic concept,... read more

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