To close out this series that started with postulates and theorems in geometry, let’s look at different kinds of facts elsewhere in math. What is commonly called a postulate in geometry is typically an axiom in other fields (or in more modern geometry); but what about those things we call properties (in, say, algebra)?
Last time we looked at the question of why we have to have postulates, which are not proved, rather than being able to prove everything. Often, this question is mixed together with a different question: Why do different texts give different lists of postulates, so that what one calls a postulate, another calls a theorem? …
As I slow down the site for the summer, I plan to run a couple series of connected posts, one per week, on subjects that have seemed too large to cover in one post. I’m starting with questions about the structure of mathematics, particularly the postulates and theorems that are common in geometry classes. This …
Why Does Geometry Start With Unproved Assumptions? Read More »
(A new question of the week) In Monday’s post about fallacies in calculus, one of them used the definition of the derivative (or rather, misused it). Today we’ll look at a short question about applying that same definition, that came in last month.
(An archive question of the week) In searching for answers to include in Monday’s post on calculus fallacies, I ran across a long discussion that illustrates some important aspects of methods of integration. In particular, there are often multiple ways to find an integral (the best not necessarily being the one taught in your textbook); …
Integration: More Than One Way, More Than One Answer Read More »
“False Proofs”, where seemingly good logic leads to nonsensical conclusions, can be a good way to learn the boundaries of reality — what to look out for when you are doing real math. We have a FAQ on the subject; there we discuss several well-known fallacies based in algebra, and have links to others. Today, …
(A new problem of the week) We usually look here at problems or concepts that are relatively basic and generally applicable; that could give a wrong impression of the kinds of questions we get. Here I want to show a recent example of a discussion about a problem, related to a geometric figure called the …
(An archive question of the week) An interesting question came to us in 2016, where rather than using a well-known formula, it was necessary to work out both what data to use, and how to calculate the desired radius.
Students often wonder where the formula for the area of a circle comes from; and knowing something about that can help make it more memorable, as I discussed previously about other basic area formulas.
Sometime soon I will do a series of posts on word problems, which are a common point of difficulty with students. But here is one recent example from a high school student, where language was the main difficulty, but the algebra is worth discussing as well. We’ll look a little more deeply into the problem …