A recent question asked about an interesting locus, which led me to realize we haven’t talked about that topic in general. Here we’ll look at what a locus is, using three simple examples, and then dig into a question about the wording.
(A new question of the week) Definite integrals can sometimes be solved by finding an antiderivative; but when that is either difficult or impossible, there may be special tricks available. Here we’ll lead a student gradually to a solution using symmetry; and then we’ll look at an earlier problem that used essentially the same trick …
(A new question of the week) Suppose we have a question that can be answered with Yes, No, or Maybe, and that whenever two people with different opinions meet, their discussion convinces each of them that neither can be right, so they both change to the other opinion. Given initial numbers of people with each …
Last time we considered negative bases for logarithms; in that discussion it was mentioned that complex numbers can change everything. This will allow us to do things like finding logs of negative numbers; but it will also make things, well, more complex! Let’s take a look.
We’ve looked at the basics of logs and how they work; now we have some questions testing the limits of the definition. We’ll focus on the inverse idea of exponential functions with a negative base, looking at this from several perspectives.
Last time, we introduced logarithms by way of their history. Here, we’ll look at their properties.
Having answered many questions recently about logarithms, I realized we haven’t yet covered the basics of that topic. Here we’ll introduce the concept by way of its history, and subsequently we’ll explore how they work.
Last week we looked at how to “cast out nines” to check arithmetic, and touched only briefly on its relationship with modular arithmetic and remainders. Here we’ll look at several explanations of why it works, aimed at different levels of students, with varying levels of success..
This old technique for checking arithmetic is both easy and hard to describe: easy to explain in advanced terms, but hard to explain in elementary terms. We’ll try to do it all here, but a fuller explanation of the “why” will come next week.
Last month, four students from the same class wrote to us with the same question: Which is more accurate, the Law of Sines or the Law of Cosines? Those led to a couple deeper discussions, as we explored the context.