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.. Why we use 9: Modular arithmetic We’ll start with a question from 1997: Casting Out Nines and Elevens At a parent-teacher meeting this evening, the teacher asked the parents why nine is used in proving a math answer. She did not know the answer, and the parents didn't either. Can you help? Our fourth-grade students ..read more

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. 1: Averages; failing to rethink the plan Here’s the first question (all are from Ivka): Could you please help me solve this problem? I believe the solution might be incorrect. I solved it a few times but kept getting the same answer. The givens are: There are eight students in the class. Seven of the eight grades are: 71, 76, 82, 87, 89, 92, 95. If the class average of all eight student ..read more

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 ger really weird! Powers and logs of a negative  base We’ll start with a basic question from 1997 about real powers, which will open the door to complex numbers: Exponentiation How do I calculate x^y using only exp, ln, log, and the trigonometric functions? I know exp(y*ln(x)) works, but only for x > 0. We could calculate, say, \(2^3\), as $$\left(e^{\ln(2)}\right)^3=e^{3\ln(2)}=e^{3\cdot0.693…}=e^{2.07 ..read more

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. Does a root cancel a power, or not? First, a question from Cindy in 1997: Square Root of a Negative Number Squared I'm reading popular math books in addition to my textbook and am really trying to understand the material. Here's my question: I understand that the nth root of x^n = x^(n/n) or x^1. My textbook gives two problems: Sqrt(-6)^2 and the 4th root of (-3)^4. Thi ..read more

(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 of time? And what about other powers? It gets a little subtle! Square root of a square: simplifying vs solving The first question is from Debkanta, who had just learned that \(\sqrt{x}=|x|\), not \(x\): Dr. today I have encountered a question on finding range of f(x) = x + √(x2). I have solved ..read more

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. Finding a minimum without calculus – but with three monotonic functions This question came from Amia in March: Hi Dr math, I have a question about minimizing the function below. Can you help me to understand it? Minimize the function √(ex^2 – 1). Doctor Fenton answered ..read more

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. Here, we’ll look at attempts to prove one view or the other. Most of them are variations on the misuse of the distributive property that was discussed in the original post; but there are some new things to learn from them. For Strict PEMDAS: “You just need to add brackets ..read more

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? Online calculators are misleading Last October, Julia from Wisconsin had a simple version of the question: I’m seeking assistance because I’m genuinely curious … as to who is right and who isn’t. Okay so basically I was on Reddit today and a viral math problem appeared in r/YoungPeopleYouTube. It was qu ..read more

We keep getting new questions related to Order of Operations: Implicit Multiplication?, where we discussed expressions like 6/2(1+2) that keep showing up in social media arguments. Since I closed comments on that page some time ago, because of the toxicity of some of them, further questions have come through our Ask a Question page (as they should, when your goal is to learn something rather than just express your opinion in public). In the next few posts, I’ll excerpt some of the interesting paragraphs from the 17,000+ words of discussion we’ve had in the last two years. Here we’ll focus on w ..read more

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 topic; and I’m always interested in seeing multiple perspectives on the same fact. Combinations: Two people shaking hands Our first question is from 1998: Why does 0 factorial equal 1? Why does 0! = 1? Is there a reason or is this like anything to the power of 0 = 1 - there i ..read more