Degenerate cases are instances of a concept that are just on the edge of fitting its definition. They occur when we stretch a definition to its limits, at which point some of the original properties remain, but others break. We’ll start here with common instances of the phenomenon, in conic sections, pursuing the elusive case …
Degenerate Conics I: Mystery of the Missing Case Read More »
Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such proofs.
One of the more common questions we’ve been asked is, How can the product of two negative numbers be positive? Between this post and the next, I’ll put together many of the answers we have given, starting here with examples from the “real world” (gradually getting more abstract), and next time we’ll look at proofs. …
Negative x Negative = Positive? Concrete Illustrations Read More »
A question just after we recently discussed quartic equations, has special features that lead to a unique solution method. We’ll be showing how to use synthetic division, and seeing some interesting graphs.
Last time we looked into terminology related to negative numbers; one subtopic was too big to fit, so I’ve broken it out into a separate post. How are the concepts of “negative” and “minus” (subtraction) related? How much do we need to distinguish them? We’ll look at two questions, the first from a child focused …
Last week we looked at what negative numbers mean; here we’ll consider a number of questions we’ve been asked about the terminology of signed numbers: what “negative” means, and other words for negative numbers. Up, down, and opposite This question from 1998 asks about translating words to signed numbers: Converting Words to Numbers Can you …
This week we’ll look at some Ask Dr. Math questions like, “How can a number be less than zero?” and “Why do we need negative numbers?” We’ll see a number of examples of their use, and how negative numbers make life easier.
(A new question of the week) It is not uncommon for students to ask about why they get different answers using different methods. Usually the answer is that the answers are really equivalent. This time, the answers really are different! This was partly the result of being taught an incomplete technique, omitting important cautions. And …
(A new question of the week) Last week I discussed several Ask Dr. Math questions about factoring quartic polynomials, which had been on my list of potential topics. That list also included a question on that topic from three years ago, that didn’t make it into the blog at the time. That will lead us …
Factoring a quadratic polynomial (degree 2) is a standard topic in algebra; but for higher degrees, things get a lot harder. Here we’ll look at some old questions from the Ask Dr. Math site about factoring quartic (degree 4) polynomials. There is no standard method, but several interesting tricks you might want to know about.