It can be tricky deciding how to approach a proof; this problem, whose answer requires going in a very different direction than you might expect, provides some interesting insights into the nature of proof. The proof itself, in fact, is far less interesting than the process of getting there!
One of the more frequent questions we had on Ask Dr. Math was about how to find the length of material (carpet, paper, wire, etc.) on a roll, knowing only the inner and outer diameters and something else: the thickness of the material, or the number of turns, or the original size of the roll. …
Last time we looked at the meaning of the concept of locus. This time, we’ll explore seven examples, from two students. We’ll look at both algebraic (equation) and geometric (description) perspectives.
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..
A recent series of questions from an insightful high school student about word problems, provided a number of opportunities to discuss how to find and correct your mistakes – or the book’s! We’ll look at five.
Having looked at issues surrounding powers and roots of complex numbers, including fractional powers, let’s go even further and consider complex powers of complex bases. Things will get a little weird as we work toward \((2+3i)^{3+2i}\)!