We’ll work through a textbook exercise that encourages students to discover what it’s like to invent a new number system, as well as why some ideas work but others do not. The topic: What would happen if we changed the definition of the imaginary unit i so that its square is 1 rather than -1?
What do you do when you are given a problem that starts with a “lie” and ends with a wrong answer? We’ll go in several directions with this problem, a system of two exponential equations in two variables.
Looking back at interesting questions I had to skip over when there were too many to choose, I found this interesting discussion of a functional equation.
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.