If you use an RPN calculator you will have to decide on the precedence yourself… but anyone using an RPN calculator is likely to do it correctly! 
(I have a really good RPN calculator on my phone that I use all the time.)
Totally disagree with this.
The point of math problems is not to try to put out gotchas to trick students but to educate them. And using deliberately ambiguous gotchas not only fails to do that but shames them for making a perfectly understandable “error”.
No working mathematician (or scientist or engineer) would write an expression like this. Ever. Or even one remotely close. If they did, it would be inadvertent and when pointed out, would likely not take offense if asked to make the meaning clearer. They wouldn’t, by contrast, get huffy and say “Order of Operations!”.
This expression really IS ambiguous. Appealing to the authority of an order of operations defeats the purpose, which is a shared convention, not an absolute, unambiguous rule. If a lot of people interpret the expression in the “wrong” way, then perhaps that ‘convention’ isn’t as conventional as we hope or need.
Back in my Ask Dr Math days, that’s more or less how I would have written a response to this kind of answer. Not “it’s 104 - use the Order of Operations” but “They’re probably expecting 104 for such and such a reason but we can see where 4 is a valid response because of such and such a reason”.
And I’m not even blaming math teachers. Online, this is more the realm of people who have an almost pathological need to be ‘correct’ with cute gimmicks rather than the more realistic and useful concept that we all be able to communicate effectively with one another. People think math (arithmetic, really) is some rigid robotic process for walking calculators when it’s not really anything of the sort.
So is all this “old math” my dad learned in the 50s, “new math” I learned in the 70s/80s, or the “new new math” my kids learned in the 2010s that makes absolutely no sense to me?
This is just math. It applies at all times.
It’s all the same math, just explained differently. Some would argue more or less successfully but that’s a separate debate.
The current Order of Operations (yes, it’s changed a bit over the years) was more or less settled late 19th century/early 20th century in the US. A little earlier or later in other countries, i.e. the convention is regionally and temporally specific.
Oh, that’s interesating. I would have thought the current order would have been settled on a bit earlier than that. What other common orders were there in different time periods? (And feel free to talk about different areas, as well.)
I only have one calculator, the one that came with my iPhone. It provides the answer 104.
I alway did well in math, especially in high school. I honestly don’t remember being taught Order of Operations. But then I graduated in ‘73, maybe my old mind just can’t remember.
What I’ve seen on Facebook and other sites of ill repute where “gotcha” expressions like this are shared, is there is an underlying concept that math doesn’t exist for any higher, practical purpose. Rather math is just some arcane wizardry with lots of gotcha rules that only exist so people can show you that you are dumb for forgetting or misapplying those rules.
I agree fundamentally with @Great_Antibob here, the correct response to things like this is to explain that order of operations having a standard in some contexts, is to create an easily shared convention for people trying to solve common problems that involve math. That in some situations a different convention makes more sense, and that is fine too. As long as it is spelled out. If someone is solving a particular problem with math, and has any need to vary the expectations as to how an expression is written, the norm would be to communicate that in whatever they were writing–not put it out there as a gotcha game.
I’ve always felt math gotcha puzzles like this reinforce the unfortunate concept that a lot of K-12 students get, that math isn’t tied directly into the practical/real world, but is just an arbitrary and esoteric thing. Math is a tool, order of operations are tools, different tools work in different situations. Math expressions in real application are not usually designed to confuse, and it’d be considered a defect in style and writing if you did so.
Is it RealCalc? I have that one and an HP-48G emulator, but the RealCalc one is easier to use, given the buttons are all virtual.
I guess the answer is 1 or 9? 9/(3*(2+1)) is 1 and 9/3*(2+1) is 9. It’s weird because they use a division sign but not a multiplication sign.
I was staring at that myself, and that’s what I came up with. To me, the answer is instictively 1, as “3(2+1)” evaluates to 9 and 9/9 is 1, but, yeah, you could read it as 9 / 3 * (2+1), hence 9/3*3 is 9, as you have it.
Ugh, that’s ugly.
This and @Martin_Hyde posts both deserve “Likes” if this board had it. I had students in Computer Science who would have gotten 104, but if it was a class in Excel their answer would be 4 until they tried it as an Excel formula (which uses algebraic precedence). I bring up the Windows desktop calculator, which by default shows 4, and the scientific version is 104. No wonder they get confused.
Exactly. As I noted above, it tricks the mind due to the quick visual parsing that we do. We see the 3(2+1) and group it as a one symbol, just as we would always do, not twigging that the \div is of a higher precedence, mostly because it is so out of context to the use of the familiar concatenation \equiv product we are so used to. So the notation is ill formed, mixing conventions. It is expressed as either:
9 \div 3 * (2 + 1), or
{9\over3}(2+1)
but not the hybrid.
It is RealCalc, actually!
It’s not often that I come across an app that’s so well designed that I enjoy using it. It’s by far the best calculator app I’ve ever found.
Definitely. I think I even paid the couple of bucks to support them and get rid of ads.
PEMDAS for me. (e=exponents, rather than orders/indices)
~Max
(2021-2021)!
~Max
The free version doesn’t have ads, but if you want the extra features you have to pay a couple of bucks for the Plus version. It’s worth it.
Yeah, maybe that’s what I paid for. Anyway, I don’t think I’ve used the built-in calculator on any phone since I’ve got this one.
Algebraic notation (and our system of numerals) as we know it didn’t really even exist until the last 500 years or so anyway, so of course it was going to go through some changes while things settled down. It didn’t spring forth whole and complete from the beginning.
Exponents first was a rule from pretty much the beginning and makes a lot of sense. The notation of adding brackets or parentheses was an invention that was added later, because it’s purely a notational thing and not an operation by themselves, but also very practical (notice how practicality and functionality have always been key considerations!). Multiplication before addition/subtraction was also early for obvious reasons.
The really big one was settling that multiplication and division were the same level of precedence and should be done left to right. This was early in the 20th century but both ways - all multiplications before division OR multiplications/divisions at the same time as you come across them left to right - were both taught into the 1910s or even later depending on the textbook used. And even when that was settled in the US (mostly), other countries still had that difference of opinion. For all I know, multiplication before division might still be taught in some countries.
This is sort of true for addition/subtraction as well but for those it was accepted fairly early that they should be done left to right at the same time.
And in a few textbooks from early in the 20th century, it is indeed recommended to add brackets or parentheses in cases to avoid ambiguity, i.e. the “rule” according to some was to avoid the situation in the OP in the first place.
And, as mentioned above, RPN neatly avoids the issue to begin with.
And yeah, this is really a convention and not absolute rules. It’s like how English grammar came about after people starting speaking English. The “rules” are post-hoc justifications and we sometimes see violations of those “rules” that are perfectly acceptable in common practice.
And another is the use of a dot (.) or cross (x) for multiplication. Some people have historically treated them differently. Others not.
As I stated, it’s a convention and disambiguation is the real rule. It is on the writer to make intention clear instead of relying on conventions that may change with time, place, or author.