Minimizing an Angle
(New Question of the Week) An interesting trigonometry problem came through about a month ago, answered by Doctor Rick. It gives a nice example of how our process works at its best. It is also an interesting problem!
Kids Ask About Even and Odd Numbers
(Archive Question of the Week) Although high school and up probably constitute the majority of our questions, I always enjoy answering younger children. For today’s look at the archives, I thought I’d look at two such questions, both from 1999, and very memorable. The first is almost certainly the youngest “patient” we’ve ever had, and …
Why, in Logic, Does “False” Imply Anything?
In a class on symbolic logic, students are taught the truth tables that define the “logical connectives” ∧ (and), ∨ (or), ¬ (not), and → (if … then). Everything makes sense until they are told that if p is false, then is true whether or not q is true. How can we say that “If pigs fly, then 2 is …
When Can a Function Equal Its Inverse?
(New Question of the Week) This week I answered a seemingly simple question that can be solved in several different ways when presented as multiple choice, but is rather difficult as a straightforward algebra problem. Trying to guess what the “patient” had done yielded an invalid method that gave the right answer — or was it really invalid? …
How Do You Know That Events Are Equally Likely?
(Archive Question of the Week) Having recently discussed a couple different issues that touched on the relationship of math to reality, I was reminded of this old favorite – a question that is not asked often enough, and reveals some of the “dirty little secrets” behind math. Math is not reality; it is often used …
Extraneous Solutions: Causes and Cures
The other day a student I was helping face to face asked how she can know when to check for extraneous solutions of an equation. I gave her a quick version of my standard answer, and the light went on! Today I want to share these thoughts here, because they are very important in several …
Arrhenius Equation: Which Graph is Right?
(New Question of the Week) Occasionally we get questions challenging the correctness of a textbook problem, or of a test grade. And sometimes we get questions about mathematics used in sciences like physics or chemistry, which lead us to explore unfamiliar fields. The most interesting question this week is one of these. It is one …
Averages, Probability, and Reality
(Archive Question of the Week) Recently I discussed the definition of the median of a data set, pointing out how it needs refinements that are not often discussed. In searching for questions in our archive on that topic, I ran across a discussion of an opposite issue: the breadth of the general term “average”, which …
Rounding to the Nearest: The Basics
We frequently get questions about how to round; so many different issues arise that I won’t try to fit them all into one post. Children have trouble learning how to do it, and sometimes their parents are surprised to find that they are being taught a different way than they learned. There are several common …
Unfamiliar Geometry Notation
(New Question of the Week) One of the benefits of being a Math Doctor is interacting with the math of many cultures around the world, as we attract an international following. We have observed variations in terminology and notation from country to country, as well as variations in the content of math education (some better …