Abstract algebra can be a huge leap for many students, who may know algebra well, but are not used to abstraction – generalizing the concept of numbers so we can invent new kinds of “numbers” and “operations” and comparing their properties. Here we will look at a question from a student beginning the study of …
(A new question of the week) A recent question dealt with an apparent conflict between the right-triangle definition of sines and cosines, and the unit-circle definition, pertaining to multiples of 90° (angles on the axes). This provides an opportunity to look closely at the relationship between those two definitions. Two definitions Recall that the right-triangle …
(A new question of the week) A set is closed under an operation if, whenever that operation is applied to two elements of the set, the result is still an element of the set. It’s straightforward … until you look closely at some details! In the course of the discussion, we’ll dig into different definitions …
Is {0} Closed Under Division? Thoughts, and Second Thoughts Read More »
(A new question of the week) Average rate of change is a topic taught in pre-calculus and calculus courses, primarily as preparation for the derivative, though it has more immediate applications. A recent question asked about when the concept is valid, which I found interesting.
Last week, we looked at exactly what the mean is, referring specifically to the arithmetic mean, the one we first learn as the “average”. But just as we previously saw that there are several things called “average” (mean, median, mode), there are in fact several different kinds of “mean”. We’ll look here at the arithmetic, …
There are three different statistics that are commonly taught as “averages”, or “measures of central tendency”, of a set of numbers: mean, median, and mode. (There are others as well, which we will get to later.) What are they? How do they differ? How do you use them? We’ll look into questions like these as …
To close out this series on the definition of the derivative, I want to look at a few questions about alternative versions of the definition, primarily the “symmetric difference quotient”. We’ll see that this leads to a slightly different result, not always equivalent to the original, and we’ll observe some associated ways that calculators can …
To start a series of posts on differentiation (one of the basic concepts studied in calculus), I’d like to look at a number of answers we’ve given to the basic question, What is a derivative? This includes questions about the meaning of the concept and its definition, as well as examples. Something all these questions …
(An archive question of the week) We’ve looked at the scalar (dot) product and the vector (cross) product; but there is one answer in the Ask Dr. Math archives that was too long to fit in either post. Here we’ll see again where the two familiar products come from, while looking deeper into the math …
Last time, we looked at the scalar, or dot, product of vectors, focusing on proving the equivalence of two ways to define it. This time, we’ll look at the vector, or cross, product in the same way. The distinction between dot and cross product reflects the symbol used, u · v vs. u × v, …