We’ve often pointed out that pattern or sequence problems, when nothing is given but a list of numbers, are not really math, in the sense that there is no one correct answer. They are psychology questions: What would a math teacher think is an interesting sequence to ask about? Mathematically, any number could come next, …
In the past (last May and November), we discussed ways to find patterns or sequences in numbers, sometimes leading to a formula. This included an example where the sequence turned out not to be just a provided list of numbers, but a process that generated the numbers. I want to focus on that type of …
(A new question of the week) Having just looked at L’Hôpital’s Rule, we can conclude with a look at a recent question about it, to illustrate the reality of struggling to apply it (and the process we go through to help a student find an error).
Last time we looked at the basics of L’Hôpital’s Rule, which applies to limits of the form or , and ways to understand or prove it. Here, we’ll consider a variety of questions we’ve received about less direct application of the rule. We’ll see ways to apply it to other indeterminate forms (, , ), and what …
The next few posts will look at a powerful technique for finding limits in calculus, called L’Hôpital’s Rule. Here, we’ll introduce what it is, and why it works. In the next post we’ll examine some harder cases.
We’ve looked at various aspects of turning English sentences into logical statements, and modifying them by negation, converse, and so on. Let’s finish by looking at some questions about standard rules of inference, such as Modus Ponens and the Law of Syllogism.
(An archive question of the week) Usually when we discuss converses (and inverses and contrapositives) we use clear, idealized examples. But statements in real life — even in real math — are not quite so straightforward. The difficulty is not merely in the language, but in the complexity of our statements. A question in the …
This is the third post in a series on logic, with a focus on how it is expressed in English. We’ve looked at basic ideas of translating between English and logical symbols, and in particular at negation (stating the opposite). Now we are ready to consider how to change a given statement into one of …
Last time, I started a series exploring aspects of the translation of English statements to or from formal logical terms and symbols, which will lead to discussions of converse and contrapositive, and eventually of logical arguments. We’ve looked at how to translate concepts of “or” (disjunction) and “if” (conditional); but our goals will also require …
The next few posts will examine aspects of logic, both symbolic logic, and how we talk about theorems in general. We’ll start here with issues in interpreting the wording of logic, and some of the semantic difficulties we face. English isn’t logical. (Well, I suppose humans in general aren’t logical.)