Combinatorics and Coefficients
(A new question of the week) A question from last August gave us some nice problems reminiscent of the Binomial Theorem, which were very deserving of discussion.
Algebra
Does a Linear Function Have to be Continuous?
(A new question of the week) As we approach a new year, I want to start cleaning up a backlog of recent questions, and start posting more typical interactions, rather than waiting for the most momentous. Many of these will therefore be relatively short! This one goes back to last July, but it connects to …
Order of Operations: Historical Caveats
To finish up this long series on the order of operations, I want to look at where the “rules” came from, which will also demonstrate why some aspects are not fully agreed upon, finishing up the discussion from last time.
Order of Operations: Implicit Multiplication?
I want to close this series with a topic that arises constantly, both in classrooms and on social media: How do you evaluate an expression like or , where the multiplication is indicated without a specific symbol? There are several reasons one might want to interpret this differently than the rule we’ve discussed, that multiplication …
Order of Operations: Fractions, Evaluating, and Simplifying
(An archive problem of the week) Last time we looked at the subtle distinction between the order of operations, which defines the meaning of an expression, and properties that allow us to do something other than what an expression literally says. Here I want to look at one longer discussion that brings out these issues …
Order of Operations: Fractions, Evaluating, and Simplifying Read More »
Order of Operations: Subtle Distinctions
Some questions we have been asked about the order of operations go beyond the what and why, pondering the relationship of the conventions both to theoretical matters (properties of operations) and to practical matters (evaluating and simplifying expressions). We will see here an important distinction between meaning and processes.
Order of Operations: Why These Rules?
Last time we looked at some questions about why we need rules for Order of Operations at all, with some hints in the answers as to why the rules we use make sense. This time I want to survey some deeper explanations.
Order of Operations: Why?
Having looked at what the order of operations convention means, another common question is, why is it what it is? We’ll look at some basic ideas here, focusing on why we need a convention at all, and why the one we have makes sense; then next time we’ll dig in a little deeper, examining some …
Order of Operations: Neglected Details
The basic statement of the order of operations covers the five main operations (exponents, multiplication, division, addition, and subtraction). But what about other operations like square roots? How about trigonometric functions? And are operations at the same level always carried out left to right? Here are some questions about the details we don’t often mention.
Order of Operations: Common Misunderstandings
Last time I started a series looking at the Order of Operations from various perspectives. This time I want to consider several kinds of misunderstandings we often see.