I will happily point out that Wolfram Alpha does this wrong. So do TI calculators, but not Casio or Sharp.
Go to any mathematics professor and give them a problem that includes 1/2x and ask them to solve it. Don’t make it clear that merely asking “how do you parse 1/2x?” is your intent, because in all likelihood they’ll just tell you it’s ambiguous and be done with it. But if it’s written as part of a problem and they don’t notice your true intent, you can guarantee they will take it as 1/(2x).
According to Casio, they do juxtaposition first because that’s what most teachers around the world want. There was a period where their calculators didn’t do juxtaposition first, something they changed to because North American teachers were telling them they should, but the outcry front the rest of the world was enough for them to change it back. And regardless of what teachers are doing, even in America, professors of mathematics are doing juxtaposition first.
I think this problem may ultimately stem from the very strict rote learning approach used by the American education system, where developing a deeper understanding of what’s going on seems to be discouraged in favour of memorising facts like “BIDMAS”.
To be clear, I’m not saying 1/2x being 1/(2x) rather than 0.5x is wrong. But it’s not right either. I’m just pretty firmly in the “inline formulae are ambiguous” camp. Whichever rule you pick, try to apply it consistently, but use some other notation or parenthesis when you want to be clearly understood.
The very fact that this conversation even happens is proof enough that the ambiguity exists. You can be prescriptive about which rules are the correct ones all you like, but that’s not going to stop people from misunderstanding. If your goal is to communicate clearly, then you use a more explicit notation.
Even Wolfram Alpha makes a point of restating your input to show how it’s being interpreted, and renders “1/2x” as something more like
Even Wolfram Alpha makes a point of restating your input to show how it’s being interpreted
This is definitely the best thing to do. It’s what Casio calculators do, according to those videos I linked.
My main point is that even though there is theoretically an ambiguity there, the way it would be interpreted in the real world, by mathematicians working by hand (when presented in a way that people aren’t specifically on the lookout for a “trick”) would be overwhelmingly in favour of juxtaposition being evaluated before division. Maybe I’m wrong, but the examples given in those videos certainly seem to point towards the idea that people performing maths at a high level don’t even think twice about it.
And while there is a theoretical ambiguity, I think any tool which is operating counter to how actual mathematicians would interpret a problem is doing the wrong thing. Sort of like a dictionary which decides to take an opinionated stance and say “people are using the word wrong, so we won’t include that definition”. Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.
Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.
You’re literally arguing that what you consider the ideal should be rigidly adhered to, though.
“How mathematicians do it is correct” is a fine enough sentiment, but conveniently ignores that mathematicians do, in fact, work at WolframAlpha, and many other places that likely do it “wrong”.
The examples in the video showing inline formulae that use implicit priority have two things in common that make their usage unambiguous.
First, they all are either restating, or are derived from, formulae earlier in the page that are notated unambiguously, meaning that in context there is a single correct interpretation of any ambiguity.
Second, being a published paper it has to adhere to the style guide of whatever body its published under, and as pointed out in that video, the American Mathematical Society’s style guide specifies implicit priority, making it unambiguous in any of their published works. The author’s preference is irrelevant.
Also, if it’s universally correct and there was no ambiguity in its use among mathematicians, why specify it in the style guide at all?
Mathematicians know wolfram is wrong and it was warned in my maths degree that you should “over bracket” in WA to make yourself understood. They tried hard to make it look like handwritten notation because reading maths from a word processor is typically tough and that creates the odd edge case like this.
1/2x does not equal 0.5x or it’d be written x/2 and I challenge you to find a mathematician who would argue differently. There’s no ambiguity and claiming there is because anyone anywhere is having this debate is like claiming the world isn’t definitely round because some people argue its flat.
Woo hoo! I hadn’t heard of anyone else pointing this out (rather, I’m always on the receiving end of “But Wolfram says…”), so thanks for this comment! :-) Now I know I’m not alone in knowing that Wolfram is wrong.
like claiming the world isn’t definitely round because some people argue its flat
OMG, I’ve run into so many people like that. They seem to believe (via saying “look, this blog says it’s ambiguous too”) that 2 wrongs make a right. No, you’re both just wrong! Wolfram, Google, ChatGPT(!), the guy who should mind his own business, are all wrong.
Sometimes people are wrong
Yes, they are… and unfortunately a whole bunch of the time they’re unwilling to face it and/or admit it, even when faced with Maths textbooks which clearly show what they said is wrong.
I will happily point out that Wolfram Alpha does this wrong. So do TI calculators, but not Casio or Sharp.
Go to any mathematics professor and give them a problem that includes 1/2x and ask them to solve it. Don’t make it clear that merely asking “how do you parse 1/2x?” is your intent, because in all likelihood they’ll just tell you it’s ambiguous and be done with it. But if it’s written as part of a problem and they don’t notice your true intent, you can guarantee they will take it as 1/(2x).
Famed physicist Richard Feynman uses this convention in his work.
In fact, even around the time that BIDMAS was being standardised, the writing being done doing that standardisation would frequently use juxtaposition at a higher priority than division, without ever actually telling the reader that’s what they were doing. It indicates that at the time, they perhaps thought it so obvious that juxtaposition should be performed first that it didn’t even need to be explained (or didn’t even occur to them that they could explain it).
According to Casio, they do juxtaposition first because that’s what most teachers around the world want. There was a period where their calculators didn’t do juxtaposition first, something they changed to because North American teachers were telling them they should, but the outcry front the rest of the world was enough for them to change it back. And regardless of what teachers are doing, even in America, professors of mathematics are doing juxtaposition first.
I think this problem may ultimately stem from the very strict rote learning approach used by the American education system, where developing a deeper understanding of what’s going on seems to be discouraged in favour of memorising facts like “BIDMAS”.
To be clear, I’m not saying 1/2x being 1/(2x) rather than 0.5x is wrong. But it’s not right either. I’m just pretty firmly in the “inline formulae are ambiguous” camp. Whichever rule you pick, try to apply it consistently, but use some other notation or parenthesis when you want to be clearly understood.
The very fact that this conversation even happens is proof enough that the ambiguity exists. You can be prescriptive about which rules are the correct ones all you like, but that’s not going to stop people from misunderstanding. If your goal is to communicate clearly, then you use a more explicit notation.
Even Wolfram Alpha makes a point of restating your input to show how it’s being interpreted, and renders “1/2x” as something more like
1 - x 2to make very clear what it’s doing.
This is definitely the best thing to do. It’s what Casio calculators do, according to those videos I linked.
My main point is that even though there is theoretically an ambiguity there, the way it would be interpreted in the real world, by mathematicians working by hand (when presented in a way that people aren’t specifically on the lookout for a “trick”) would be overwhelmingly in favour of juxtaposition being evaluated before division. Maybe I’m wrong, but the examples given in those videos certainly seem to point towards the idea that people performing maths at a high level don’t even think twice about it.
And while there is a theoretical ambiguity, I think any tool which is operating counter to how actual mathematicians would interpret a problem is doing the wrong thing. Sort of like a dictionary which decides to take an opinionated stance and say “people are using the word wrong, so we won’t include that definition”. Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.
You’re literally arguing that what you consider the ideal should be rigidly adhered to, though.
“How mathematicians do it is correct” is a fine enough sentiment, but conveniently ignores that mathematicians do, in fact, work at WolframAlpha, and many other places that likely do it “wrong”.
The examples in the video showing inline formulae that use implicit priority have two things in common that make their usage unambiguous.
First, they all are either restating, or are derived from, formulae earlier in the page that are notated unambiguously, meaning that in context there is a single correct interpretation of any ambiguity.
Second, being a published paper it has to adhere to the style guide of whatever body its published under, and as pointed out in that video, the American Mathematical Society’s style guide specifies implicit priority, making it unambiguous in any of their published works. The author’s preference is irrelevant.
Also, if it’s universally correct and there was no ambiguity in its use among mathematicians, why specify it in the style guide at all?
Mathematicians know wolfram is wrong and it was warned in my maths degree that you should “over bracket” in WA to make yourself understood. They tried hard to make it look like handwritten notation because reading maths from a word processor is typically tough and that creates the odd edge case like this.
1/2x does not equal 0.5x or it’d be written x/2 and I challenge you to find a mathematician who would argue differently. There’s no ambiguity and claiming there is because anyone anywhere is having this debate is like claiming the world isn’t definitely round because some people argue its flat.
Sometimes people are wrong.
Woo hoo! I hadn’t heard of anyone else pointing this out (rather, I’m always on the receiving end of “But Wolfram says…”), so thanks for this comment! :-) Now I know I’m not alone in knowing that Wolfram is wrong.
OMG, I’ve run into so many people like that. They seem to believe (via saying “look, this blog says it’s ambiguous too”) that 2 wrongs make a right. No, you’re both just wrong! Wolfram, Google, ChatGPT(!), the guy who should mind his own business, are all wrong.
Yes, they are… and unfortunately a whole bunch of the time they’re unwilling to face it and/or admit it, even when faced with Maths textbooks which clearly show what they said is wrong.