Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …
Having recently tutored a number of students through their finals, and given a lot of the same advice repeatedly, this seems like a good time to share a couple recent questions on how to approach learning and doing math. We’ll see one student who wants to stop making mistakes, and another who needs to learn …
In looking into combinatorics for last week, I ran across several questions about the topic of “derangements” (permutations of objects in which none of them are in their original positions). Let’s look at those, first at probability, and then at the closely related matter of counting. This will also bring us to the Inclusion-Exclusion Principle. …
Certain kinds of word problems tend to be easy to misinterpret or to misstate. That is particularly true in combinatorics. Let’s look at two of those, one recent and one a few years old, where we are assigning people to groups, and the wording is not quite clear.
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 …