# Higher math

## Frequently Questioned Answers: Uncountable Infinities

We could continue forever discussing questions whose answers are frequently questioned; but let’s finish by looking at infinity itself. The concept is impossible to fully grasp, because we are finite, and all of our experience is finite. Mathematicians have worked out ways to deal with infinity, though, and the results are often counter-intuitive. That means …

## A Very Different Kind of Sequence

(An archive problem of the week) While gathering sequence/pattern questions for my last post, I ran across a very different problem. Here we are told what the pattern is (a good example of one that you would probably never discover on your own), and asked some questions about later terms. It can be understood either …

## Greatest Common Divisor: Extending the Definition

Having just talked about definition issues in geometry, I thought a recent, short question related to a definition would be of interest. We know what the Greatest Common Divisor (GCD, also called the Greatest Common Factor, GCF, or the Highest Common Factor, HCF) of two numbers is; or do we?

## Edges and Faces: A Matter of Definition

(An archive question of the week) Having looked at the matter of faces, edges, and vertices from several different perspectives, I want to look at one more question and answer, to tie it all together.

## More on Faces, Edges, and Vertices: The Euler Polyhedral Formula

Last time we looked at how to count the parts of a polyhedron, and a mention was made of Euler’s Formula (also called the Descartes-Euler Polyhedral Formula), which says that for any polyhedron, with V vertices, E edges, and F faces, V – E + F = 2. We should take a close look at that simple, yet amazing, …

## What is Multiplication … Really?

I want to close out this series on multiplication with a very different kind of question. We have seen that multiplication of natural numbers can be modeled as a repeated sum of the multiplicand, taken the number of times indicated by the multiplier; and that the terms “multiplier” and “multiplicand” reflect only this model, not …

## A Quadratic Diophantine Equation

(New Question of the Week) One of the strengths of the Math Doctors is the breadth of knowledge represented by our volunteers; we are all different. I have tended here to show the topics that I myself deal with, because those are what I can say most about; but whereas I focus mostly on elementary …

## When Is a Definition Not a Definition?

(New Question of the Week) From time to time a student will ask for help understanding what he reads in his textbook. This often requires some back-and-forth as we try to understand both what the textbook is saying in its context (without having a copy of it to look at), and how the student is …