Having looked at words for large numbers, let’s turn to the less-asked, but still interesting, question of where our words (in English) for ordinary little numbers come from. We’ll look at the origins of one, two, three, then eleven, twelve, thirteen, and then first, second, third.
Last time we looked at names of anniversaries (like “semiquincentennial” for the 250th), and other words based on Latin numbers (like “septuplets” for 7 babies). Let’s back up and look at the names in English for large numbers themselves, like million, trillion, and vigintillion. We’ll work up to the big ones!
This is the 250th anniversary of the United States, and as a result we’ve heard the term “semiquincentennial” a lot. (Well, a lot more than usual.) That reminded me of several past questions about such terms, so we’ll take a look at them now, starting (after a quick introduction) with one asked 25 years ago …
Recently, a teacher in Bangladesh asked us how to answer students’ common questions about his version of the proof that the square root of 2 is an irrational number. Some are questions about apparent gaps in the proof itself, others about how one would decide what to do in writing the proof. We’ll then compare …
Last time we looked at what hyperbolic functions are, as a parallel world to trigonometric (circular) functions. But there’s another parallel: Each is related to the other through complex numbers. We’ll look at several questions that open the door to this relationship, finding how to calculate trig functions of complex numbers, and then delving into …
Hyperbolic and Circular Functions and Complex Numbers Read More »
Hyperbolic functions are functions that form a sort of parallel universe to the trigonometric functions. They are typically introduced as an aside while teaching calculus – because they happen to be useful there, and also because they can’t really be understood without calculus. As a result, we can easily misunderstand them, forget about them, and …
A recent question asked about quadratic inequalities; we’ve touched on that in Domain, Range, and Quadratic Inequalities, but here we’ll introduce three basic approaches. We’ll follow that with old questions about polynomial inequalities (mostly quadratic) and rational inequalities (fractions), expanding on the basic methods.
We’ve discussed the Rational Root Theorem in the past, but not a theorem that is often taught along with it, namely Descartes’ Rule of Signs, which predicts the numbers of positive and negative zeros (roots) of a polynomial. Both are ascribed to Rene Descartes; both are often taught without proof. Here we’ll introduce the theorem, …
A recent question reminded me that we haven’t yet covered completing the square, a technique important for solving quadratic equations, and also in several other applications. We’ll see the traditional method, and a modified method that avoids fractions, including a nice alternative to the quadratic formula.
A couple recent questions about integration turn out to be related to our last couple posts, in which we discussed substitutions that are not one-to-one. It’s better disguised here (and isn’t intertwined with symmetry issues!). You’ll also see a couple errors I made in correcting the student’s errors, which I am not hiding because you …