The way you learned it in the 70s/80s is still the same way it’s being taught now. If it makes no sense to you now, that’s an indicator that you didn’t remember it very well.
My HP 48 emulator is the only calculator I ever use. It’s always good to see someone else who appreciates the old classics. Honestly, RPN is closer to how computers, and humans, actually think through problems: It’s a lot easier than “algebraic”, once you get used to it. But maybe I need to check out this “RealCalc” (which I presume is also RPN?).
A big one that I see coming up often in the real world is things like f = \nu/2\pi . What that (almost always) actually means is that \nu should be in the numerator, and 2\pi should be in the denominator. But a TI calculator will interpret it as (\nu/2)\pi (some other calculators take other interpretations, which just makes things even worse). Fortunately, I’ve learned that when 2pi shows up anywhere in a calculation, when a student is off by a factor of about 10, that’s almost always the reason: I ask them to show me typing it into their calculator, and stop and tell them to put in parentheses.
The order of operations is not ambiguous. Like most of mathematics, it’s clear and precise. Just because many people don’t know the rule doesn’t mean the rule doesn’t exist.
If I had a friend who was a Spanish speaker and they talked about el perro they had as a pet when they were a child, I might not know whether they had a cat or a dog. But that is an indication of my ignorance of the Spanish language; not a sign that the Spanish language is ambiguous on the subject. It would be foolish of me to insist that perro can mean cat or dog because I don’t know which one it is. It would be foolish of me to insist that using the word perro is just an arbitrary convention because people speaking other languages use other words. And it would be foolish of me to argue that the majority of people in the world don’t speak Spanish and their opinions on the meaning of the word perro is more valid than the minority of the world’s people who actually speak Spanish.
A lot of problems in our society are caused by people who have strong opinions on subjects they have very little knowledge of. They appear to assume that if they don’t understand something then it must mean that nobody understands it and therefore their guess is as good as anyone else’s - even if the other person actually knows the subject. We see this attitude in everything from how words are spelled to how vaccinations work.
The idea there is a “rule” in the first place is exactly the problem and what we have not gotten through to some posters.
It really is a convention. And convention can and does change. There are literal examples in math texts from 150 years ago that need to be read carefully because the . vs x convention are different from how we’d use them today.
It goes the other way around. Insisting there is a single, absolutely correct answer here is the ignorant one that we’re trying to fight. In years of answering questions about math, I’ve rarely seen such strong emotions raised that I’ve seen around the idea that the order of operations is not an end-all, be-all set of rules.
In historic hindsight, we should have used and/or invented a different algebraic notation system for which there is no ambiguity by design.
Instead, we used one that makes more sense when natural languages are taken into account, i.e. we need an order of operations because human language isn’t great for mathematics but we created our algebraic notation based on those very languages. For example, we say “two plus two”, so writing “2 + 2” makes sense. But notationally, this leads to the ambiguities that require an order of operations.
And the solution we developed, even from the early days is to add parentheses or brackets everywhere to avoid ambiguity. But that’s sometimes cumbersome or unwieldy, so it gets skipped a little more than it should.
years?? More like centuries, more or less, for the basic notation. Officially according to today’s official standard one should simply avoid expressions like a/bc, if there is reason to worry about potential ambiguity.
NB, ÷ is disqualified — I agree with the ISO here — as it is not even an unambiguous division sign.
Which is why no less than Knuth writes explicitly, in the preface to one of his books, “An expression of the form ‘a/bc’ means the same as ‘a/(bc)’.”, just to make sure nobody gets confused [This is someone who cares greatly about communicating mathematical ideas clearly and effectively, and has given several courses on that theme.]
Sure, but the people posting these types of order of operations problems aren’t trying to spark a debate about how math has changed since the Victorian era. Ancient Sumerians aren’t on Facebook.
It’s a pretty straightforward gotcha: “You were given this exact sort of problem during an order of operations lesson in 4th or 5th grade. Can you still get it right?” And the reality is that a lot of people would do poorly if they were thrust into a 5th grade math class on this particular day. And trolls find that funny.
It can work either way, but I only use it in RPN. Since it’s not trying to duplicate everything the 48G did, there are fewer, larger buttons, which makes it easier to use.
Right, but depending on how it’s written out, it can be confusing. Take a look at Chronos’s example, with 2pi in the denominator. Strictly speaking, that equation would really have pi in the numerator, since first you divide v by 2, then multiply by pi. But, if I saw that, I would think 2pi should be together in the denominator.
I don’t understand what the ambiguity is. The order of operations is what, four hundred years old? And has been universally taught for a century or more in public education? It’s way past being an opinion.
There are 2 ways to think of the order of operations, one is more correct than the other.
The order of operations is a firm set of rules universally agreed upon and to be followed without thinking
The order of operations is a set of best practices, most of which are universally applied without exception, to facilitate natural language communication of arithmetic/algebraic expressions
The second is the actual case. The only real rule is disambiguation. We only have an order of operations because natural human languages don’t lend themselves well to arithmetic expressions.
The purpose of having an order of operations is disambiguation. If a significant number of people get different results from the same expression, sure, we can just say “they’re doing it wrong” but that defeats the purpose. We want them to be able to understand the intent of what we are writing. In that case, it is on the writer to take reasonable care to form arithmetic/algebraic expressions in a way that almost everybody will get the same answer.
The lesson for students for this particular case is not “this is a set of rules to memorize and follow without question” but “here are general guidelines because human languages suck and we should take care that our meaning is clear”.
And over the centuries we have largely gotten those guidelines down pat. But not entirely, as the case of 1/2pi shows. The implicit multiplication is still an outlier we haven’t dealt with very well.
And, yes, the multiplication probably should be done first and people should learn that better. But clearly they haven’t and communication is the important part rather than an unthinking adherence to something your teacher once taught you.
Are these two definitions mutually exclusive? I would think that a rule on written notation is in fact a set of practices defined by people; there is nothing inherently wrong with an idiosyncratic order of operations, only that there is an established ‘correct’ way.
ETA: It’s not that human languages “suck” when it comes to math. Mathematical notation (including order of operation) is a language for mathematics.
But when the purpose is to facilitate communication, the established ‘correct’ way should be the way most people parse the expression. If there is significant disagreement, it calls into question whether that ‘established’ way is actually all that well established.
The purpose may not be so broad as “to facilitate communication”. Modern mathematical notation is after all much more stringent and proper than normal language. You must recognize that as far as languages go, mathematical notation is not general-purpose. It is a specialty language that is almost always taught in a classroom setting according to standard rules. The language of mathematical notation has the purpose of facilitating unambiguous communication… between those who are learned in its rules.
You can forgive those who are unfamiliar with the rules just as we forgive foreigners who have trouble speaking English - it does not follow that the English language should change to reflect deficient understanding of English worldwide. Therefore mathematical notation does not necessarily need to adopt different operator precedence to account for people who are not familiar with the established rules.
Except that’s exactly what we do. That’s part of how languages evolve. Because they are based entirely on convention and not on any ultimate truth. If enough people find it useful to say things in a certain way, then it becomes acceptable. Descriptivism always beats prescriptivism.
In math, however, we do have actual truths that are not merely conventions. And it is quite useful to distinguish between those and things that are mere conventions. The conventions can change if they cause more trouble than they’re worth, while the truths won’t.
If you write a sentence in English that is technically correct but confusing, that is considered bad English. The same is true when using conventions in math. If your mathematical statement is confusing, it is still bad math communication, even if it is technically correct.
The way these things are written are just specifically designed to get people disagreeing, so that they go viral from all the replies and everyone discussing them. As math communication, they are quite poor.
(I wish I could call up a confusing but technically correct English sentence on command, but I have difficulty doing so. I know I’ve made them before, but it was entirely on accident.)
I ask what I asked before—citations/examples that people “breaking the rules” of notation constitute a big problem currently in mathematics, engineering, computer science, whatever.
@Great_Antibob is right that is it both true that there is an ISO 80000-2 “firm set of rules universally agreed upon”, and at the same time mathematical notation is the way it is as a set of best practices to facilitate communication; it’s not like the ISO committee made stuff up in contradiction to what people generally understand.
How about the famous:
Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
It’s technically a grammatically sound English sentence, most people need to read a multi-paragraph explanation of what’s going on and still struggle to wrap their heads around it.
For those unfamiliar, it uses three different meanings of ‘buffalo’ and a tendency of English to elide various helper words here and there.
Buffalo
n. Buffalo, NY
n. American Bison
v. to fool someone
Buffalo1 buffalo2 Buffalo1 buffalo2 buffalo3 buffalo3 Buffalo1 buffalo2.
Or
New York bison [that] New York bison fool, fool New York bison.
Surely you do not argue that if enough people think the order of operations should be left-to-right, the following equation will not hold?
50+50-25*0+2+2=104
If such a fundamental change to standard mathematical notation took place, I would say that the former equation was written in a different language and thus requires translation:
The citations will not be forthcoming, I jumped into a conversation where it was assumed a significant number of people have a problem with order of operations.
If a significant number of people get different results from the same expression
And over the centuries we have largely gotten those guidelines down pat. But not entirely, as the case of 1/2pi shows. The implicit multiplication is still an outlier we haven’t dealt with very well. And, yes, the multiplication probably should be done first and people should learn that better. But clearly they haven’t
To those saying that “the order of operations hasn’t changed for centuries”, how long has the / (forward slash) symbol been used to mean division? Any rules for the use of that symbol surely cannot be older than the symbol itself. There are other symbols that were used for division before the forward slash, of course, but you can’t just say that forward slash works like them, because they didn’t work the same way as each other.
For that matter, they’ve changed even since then. If a Casio and a TI give different answers for the same problem (which they sometimes do), then they’re using different orders of operations. Even if there was a fixed One True Order before those calculators, at least one of them changed it.
The diagonal fraction bar (also called a solidus or virgule) was introduced because the horizontal fraction bar was difficult typographically, requiring three terraces of type.
An early handwritten document with forward slashes in lieu of fraction bars is Thomas Twining’s Ledger of 1718, where quantities of tea and coffee transactions are listed, e.g. 1/4 pound green tea. This usage of the horizontal fraction bar was found by Hans Lausch, who believes there are likely even earlier occurrences.
Many other examples of early uses on the same page.
[This is someone who cares greatly about communicating mathematical ideas clearly and effectively, and has given several courses on that theme.]