I was on one of my favorite subreddits today, /r/learnmath (of course I sort by New), and a question came up that ultimately came down to understanding order of operations. These always frustrate me because they stem from a poor understanding of how the order of operations work; furthermore, any problem involving order of operations can be fixed and cleared up with an additional set of parentheses.
Consider this math problem below — the type I see making its way around Facebook.
The answer is 288. Your calculator can do it, because your calculator knows the order of operations. First you perform the addition in parentheses. Then you do division and multiplication (no preference for one over the other) from left to right. 48 divided by 2 is 24, then multiply 24 by 12. You get 288. And obviously you’ll only get this correct if you’re smart.
But here’s the thing: the order of operations are accepted to remove ambiguity and because mathematicians agreed we need some set of rules for this. However, they fall apart in their usefulness if people don’t remember them. In reality, no person should ever write the statement above. I would write it in plaintext as (48/2)*(9+3). There. No ambiguity. We recognize we should do parentheses first, then multiply. If I wanted everything divided on the right, I could write 48/(2*(9+3)). Again, no ambiguity.
I teach mathematics online with my company. One important aspect of these classes is the lack of any audio or video component. It is entirely text-based, and that is one of its strengths. However, it is incumbent on myself as the instructor to teach good habits and proper mathematical writing to the students, as they must type in all of their answers. If a student writes “1/2x”, regardless of context and whether it is technically correct, I will comment to them to fix it. Strictly speaking, 1/2x means precisely (1/2)*x, or more easily x/2. But almost any student I’ve come across would write that to mean 1/(2x). So which is it? The correct answer is “It doesn’t matter. Write it in a way that allows the reader to understand you immediately.”
When I was trying to explain this to a the user on /r/learnmath, I came up with the phrase that entitles this post: “That’s a bad pirson.” Without any context, it’s entirely unclear whether that’s a typo of person, or a typo of prison. Depending on what precedes the phrase, it could suddenly become clear.
I remember my visit to Alcatraz last summer. I thought to myself, “That’s a bad pirson.”
It’s pretty clear someone did a small goof and meant to type “prison”. No big deal. It still feels a little clunky, but I know what you mean. In the same way, context can make math clear when typed out:
We know the volume V is equal to half the reciprocal of x, and thus we can write 1/2x = V.
The reciprocal of x is 1/x, and half of that is 1/(2x). While the order of operations strictly says the usage in the quote above is incorrect, the context before it makes it clear enough to the reader. Of course, sometimes context is insufficient. This is where care needs to be taken.
I think back on that place, and particularly how the guard treated us and all I can tell people is “That’s a bad pirson.”
Well, who or what is a bad pirson? The whole place, or just the guard? Who’s to say in this moment? Math can be dealt with similarly. It is incredibly important that you take care to communicate clearly. Don’t rely on context clues to cover for your laziness or inattention to detail. Be explicit and clear; if you choose not to be explicit or clear, then have a reason (this applies to casual writing, certainly not mathematical writing.)
Good writing, both technical and creative, is developed from the details. Focusing on consistency and clarity for the reader will do ninety percent of the leg work, and style can come in after. But if you are unclear, nobody will care about the panache with which you make your cluttered argument. Doing a degree in mathematics made me a significantly better writer. I must focus on the details, understand what is essential to the argument, make decisions on who my audience is, then add in the necessary scaffolding and style to communicate the essential elements to the target audience. This process works for any piece of writing. Whether you are writing an essay in history, a short story for the campus magazine, or a research paper on chemistry, you need to understand what you want to say; then, focus on saying it clearly with no ambiguity. If you don’t, your words can be misconstrued and people can view you as a bad pirson. We wouldn’t want that.