Why Do Logarithms Work That Way?
Last time, we introduced logarithms by way of their history. Here, we’ll look at their properties.
Algebra
Where Do Logarithms Come From?
Having answered many questions recently about logarithms, I realized we haven’t yet covered the basics of that topic. Here we’ll introduce the concept by way of its history, and subsequently we’ll explore how they work.
Casting Out Nines: Why It Works
Last week we looked at how to “cast out nines” to check arithmetic, and touched only briefly on its relationship with modular arithmetic and remainders. Here we’ll look at several explanations of why it works, aimed at different levels of students, with varying levels of success..
Algebra Word Problems: Learning from Mistakes
A recent series of questions from an insightful high school student about word problems, provided a number of opportunities to discuss how to find and correct your mistakes – or the book’s! We’ll look at five.
Complex Powers of Complex Numbers
Having looked at issues surrounding powers and roots of complex numbers, including fractional powers, let’s go even further and consider complex powers of complex bases. Things will get a little weird as we work toward \((2+3i)^{3+2i}\)!
Powers of Roots and Roots of Powers
Last time, we looked at two recent questions about combining squares and roots, and implications for the properties of exponents. We didn’t have space for some older questions that we referred to. Here, we will go there.
Squares, Roots, and Negative Numbers
(New questions of the week) Two recent questions (five days apart, from high school students in different countries) were about nearly the same thing, and fit nicely together: What do you get when you square a square root, or take the square root of a square, but don’t know the sign of the number ahead …
Monotonic Functions, Inequalities, and Optimization
Looking for a cluster of questions on similar topics, I found several from this year in which monotonic functions (functions that either always increase, or always decrease) provide shortcuts for various types of problems (optimization with or without calculus, and also algebraic inequalities). We’ll look at a few of these.
Implied Multiplication 3: You Can’t Prove It
This is the last of a series on our discussions, since I closed comments at the end of 2021, of Implied Multiplication First (IMF), the idea that multiplications written by juxtaposition, rather than with a symbol, are to be done before other multiplications or divisions. Last time, we saw that there is no “official” answer. …
Implied Multiplication 2: Is There a Standard?
This is part 2 of a series of extracts from discussions we have had on whether multiplication implied by juxtaposition is to be done before division (which I call IMF, for Implied Multiplication First). Some people write to us claiming that there is one official correct answer. Are they right?