What is a Ratio, Really?
A recent question reminded me I hadn’t yet written about the complexity surrounding the definition of ratio (and related terms, like rate and fraction). Here are four questions about the words.
Definitions
What is the Meaning of “Locus”?
A recent question asked about an interesting locus, which led me to realize we haven’t talked about that topic in general. Here we’ll look at what a locus is, using three simple examples, and then dig into a question about the wording.
Why Can’t a Logarithm Have a Negative Base?
We’ve looked at the basics of logs and how they work; now we have some questions testing the limits of the definition. We’ll focus on the inverse idea of exponential functions with a negative base, looking at this from several perspectives.
Why Are There Different Definitions of Range?
A recent question about two interpretations of the range of a data set in statistics leads us into some older questions and some mysteries. Is “range” defined as the interval containing the data, or the difference between largest and smallest values, or 1 more than that? Yes! All three are used, and are useful.
Types of Data: Discrete, Continuous, Nominal, Ordinal, …
Last time, we looked at some ideas about appropriate graph types, and the references I found put this in the context of identifying types of data. Here we’ll look at questions about two such classifications: nominal/ordinal/cardinal (with variants), and continuous/discrete. We’ll see that classifications can become distorted as they filter down from higher levels to …
Types of Data: Discrete, Continuous, Nominal, Ordinal, … Read More »
Trig Terminology: What Do Those Words Mean?
Students sometimes wonder why the trigonometric functions (sine, cosine, tangent, secant, and so on) have the names they do, and how they relate to the corresponding terms in geometry. How are the tangent and secant functions related to tangent and secant lines in trigonometry? And what in the world is a sine? Here we’ll look …
Zero Factorial: Why Does 0! = 1 ?
We’ve been talking about the oddities of zero, and I want to close with another issue similar to last week’s \(0^0\). All our questions will be essentially identical apart from details of context: “We know zero factorial equals 1; but why?” This isn’t nearly as controversial as the others, but will bring closure to the …
Anything to the Zero Power: Why 1?
We’ve been looking at oddities of zero. Because “nothing” behaves differently than “something”, operations with it can be surprising. Although students learn that \(x^0=1\) for any non-zero number x, they often wonder, why?? I’ve selected a few out of at least a dozen such questions in our archive.
Is Zero Positive or Negative? Even or Odd?
Last week we looked at some basics about zero; now let’s look at whether zero is positive or negative, and then at the topic of the recent comment that triggered this series: whether zero is even or odd.
Polynomials: A Matter of Degrees
Last time we examined why polynomials are defined as they are. This time, let’s look at some tricky aspects of the concept of “degree”, mostly involving something being zero.