We’ll first look at several old questions about proving a relationship between permutations or combinations, where we’ll see some algebraic proofs using formulas, and others that center on the meaning of the symbols as ways of counting. The latter are called “combinatorial proofs”. We’ll end with a recent question of the same type, which suggested …
Combinatorial Proofs: Counting the Same Thing in Two Different Ways Read More »
Several recent questions involve things that go wrong with signs in integrating, and reveal some subtleties that are easily overlooked. We’ll also see some creative thinking!
Having looked at improper integrals last time, let’s look at some questions we’ve had involving integrals that either look improper but aren’t, or are improper but were missed, or that have other issues with their interval of integration.
We have a question about an improper integral, where one is strongly tempted to take a shortcut that makes it convergent, though the proper definition does not. Why can’t we do this? We’ll see something of the freedom mathematicians have in the matter of definitions, as well as why the standard definition has to be …
Two-sided Improper Integrals: Can I Take Both Limits at Once? Read More »
Over the years, we have received several questions about problems that give an “assertion” and a “reason”, and ask you to decide whether each is true, and also whether the latter is “the correct explanation” (that is, a valid reason) that the former is true. These can involve subtle reasoning, and subtle errors. Since some …
Let’s look at three similar questions we’ve received about Least Common Multiples, Greatest Common Factors, and so on, starting with a recent question and going back in time. We’ll see a bad question, a good question, and an interesting challenge.
Last time, we looked a little more deeply than usual at an epsilon-delta proof of a simple function of one variable. Here, we have two questions about such proofs for a function of two variables, to illustrate how you can question aspects of such a proof and satisfy yourself that it is valid – or …
Some time ago we looked at the meaning of the definition of limits, and I included several links to additional discussions on the subject. Now I want to took at three of those, which fit together rather nicely. We’ll look deeply into the proof that the limit of \(x^2\), as x approaches 2, is 4, …
Having looked into our explanations of transformations and symmetry, over the last weeks, let’s turn to the recent questions that triggered this series. Here we have an adult studying the topic from a good book, but tripping over some issues in the book. We’ll be touching on some topics we haven’t yet looked at, such …
Having looked at geometrical transformations, we can now apply them to the idea of symmetry. We’ll focus on symmetry of figures in a plane.