# Algebra

## 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?

## Implied Multiplication 1: Not as Bad as You Think

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 …

## 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 …

## Zero to the Zero Power: Indeterminate, or Defined?

Last week we looked at numbers raised to the zero power, as part of our series on oddities of zero. We’ve looked at zero divided by zero in the past, and just recently observed how 0 to the 0 power relates to degree in polynomials, which is part of the motivation for this series. But …

## 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.