Prime Numbers: What About 0 and 1?
Last week we looked at the definitions of prime and composite numbers, and saw that 1 is neither. The same is true of 0. What, then, are they? That raises some deep questions that we’ll look at here.
Prime Numbers: What and Why
I’ll begin a short series of posts on prime numbers with several questions on the basics: What are prime (and composite) numbers, and why do they matter?
Three Trigonometric Inequalities
(A new question of the week) We often solve basic trigonometric equations; but a recent set of questions dealt with challenging trigonometric inequalities, which bring with them a new set of issues. We’ll look at several of those here, which combine trig with polynomials, rational functions, and more. Each will illustrate something new to watch …
One-Variable Equations in a Two-Variable World
(A new question of the week) This week we have a short discussion of a question that takes a basic concept one step further: How do you graph an equation on the plane, that contains only one variable? It’s a simple question when applied to linear equations, but takes on new dimensions when we generalize …
Writing a Riemann Sum as an Integral
(A new question of the week) Riemann sums are used in defining the definite integral. But they can also be used in reverse: Sometimes you can be given the limit of a summation and asked to read it as a Riemann sum, and then turn it into an integral. Usually this is fairly straightforward; but …
Fundamental Theorem of Calculus: a Tale of Two Parts
(A new question of the week) A recent question about the application of the Fundamental Theorem of Calculus provided an opportunity to clarify what the theorem means in practice, and specifically how the two parts are and are not related. Misunderstandings like these are probably more common than many instructors realize! We’ll also glance at …
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Parallel Vectors: Missing a Solution
(A new question of the week) We were recently asked to check work on an interesting little question about parallel vectors, and I was almost convinced that there was no solution … until I realized there was one! How was it missed? How can we avoid doing that? That’s our goal today.
A Surprising Route to a Differential Equation
(A new question of the week) We are often asked to help a student understand a solution to a problem, obtained from a book or a website, that is not fully explained there. Here, we’ll look at a rather odd demonstration that a function satisfies a differential equation, both figuring out what the author did, …
Why Properties Matter: Beyond Addition and Multiplication
We’ve been looking at the commutative, associative, and distributive properties of operations, starting at an introductory level. But why are these properties important? Why do they have names in the first place? And what other operations have them?
Commutative, Associative, and Distributive Properties
In working on last week’s post, I realized that I haven’t yet covered the general idea of properties, such as commutative, associative, and distributive. Here I’ll collect some introductory answers on that topic. Next week, we’ll dig a little deeper.