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. The “rules” are only descriptive First, here are a couple paragraphs from the 2017 answer …
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 …
(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 »
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. PEMDAS vs properties, I: Meaning vs. …
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. Why not just left to right? First, an overview of reasons for the …
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 …
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. …
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. Multiplication before division? Here is a question from 2005 from a teacher, “WRW”: Confusion over Interpretation of PEMDAS In telling students to “do multiplication and division IN …
The order of operations in algebra (also called operator precedence) is a very common source of questions; I count at least 50 archived discussions explicitly about the topic (not just mentioning it in passing), in addition to the Ask Dr. Math FAQ on the subject. I’ll devote the next few posts to looking at various aspects …
Here is another puzzle we have received and answered many times. (I count 7 that have been archived.) It has several variations, which make it even more interesting. The story varies, too; sometimes the monkeys are the stars, other times they just get the leftovers. Someone could to an interesting folklore study on this one. …