PEMDAS and the Ambiguity of Mathematical Notation in Physics

[John3509]

In case you have not seen the equation that has gone viral recently https://www.popularmechanics.com/science/math/a28569610/viral-math-problem-2019-solved/

This lead me to this

Similarly, there can be ambiguity in the use of the slash symbol / in expressions such as 1/2x.[5] If one rewrites this expression as 1 ÷ 2x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:
1 ÷ 2 × x = 1 × 1/2 × x = 1/2 × x.
With this interpretation 1 ÷ 2x is equal to (1 ÷ 2)x.[1][6] However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. This higher precedence itself implies the need for an updated mnemonic PEIMDAS, with I = Implied multiplication.
For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[7] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[a]

Why are physicists allowed to break the rules? What reasoning does the Physics Review and Feynman have for making multiplication of higher precedence than division with a slash? And doesn't this cause problems for consistency within Physics?

 

[Ibix]

In Britain an 18 year old may drink alcohol in a bar. In America they'd risk arrest. Why do Brits get to break the rules?

Answer: we don't. Not everybody agrees on the same set of rules. As long as everyone is clear what the rules are in their circumstances there's no problem. The "puzzle" you linked (what is 8/2(2+2)?) boils down to whether you believe you should apply strict left-to-right evaluation or not. It's just an excuse for a fight. The correct answer, of course, is that it depends what convention you use.

Also many physics equations use quantities with units, which frequently resolve any ambiguity. For example, deliberately being sloppy, kinetic energy is $1/2mv^2$. Knowing that's an energy, interpreting it as $1/(2mv^2)$ is clearly wrong since it has the wrong units so I must, in this case, mean $(1/2)mv^2$. That means that an editorial instruction to never write $1/2mv^2$ is largely a style choice.

Edit: just to add, multiplication having a higher precedence than addition is also a convention. In this case it's a natural one, since everyone interprets "I've got three twenty pence pieces and two ten pence pieces" as me having 80p. It makes sense for 3×20+2×10 to be interpreted the same way as the verbal version. But systems that don't respect that rule are easy to write and (as long as I make clear that I'm writing in such a convention) they aren't wrong, just different.

 

[fresh_42]

This entire discussion about the order is ridiculous. It only shows that division signs lack context in a linear word. I liked @Ibix's comparison. If I write "either", is it eether or ither? Is it is-sue or ish-you? There is no way to figure it out.

My personally preferred answer to such questions is: Divisions and subtractions are multiplications and additions. There is no such thing as a division or a subtraction, ergo no problem. We only have two operations, not four, and what we really do is $a \div b = a \cdot b^{-1}$ and $a-b = a + b^{-1}$. As the latter leads to confusion if we used both operations and the convention to write inverse elements as ${*}^{-1}$, it is acceptable to write $b^{-1}=-b$ in the additive case, whence $a-b=a+ -b$.

To debate a convention which is wrong by nature is ridiculous.

 

[ZapperZ]

In case you have not seen the equation that has gone viral recently https://www.popularmechanics.com/science/math/a28569610/viral-math-problem-2019-solved/

This lead me to this

Why are physicists allowed to break the rules? What reasoning does the Physics Review and Feynman have for making multiplication of higher precedence than division with a slash? And doesn't this cause problems for consistency within Physics?

This is nothing to do with mathematical rules. It has everything to do with typesetting style!

Many publications have their own set of styles, and more often, it is based on what the common readers of that publication are familiar with. The Physical Review editor isn't trying to rewrite the math rules, but rather to clarify that if you type it that way, it will be interpreted by their editor and copywriters as such, and will be typeset as such!

Sometime people make a mountain out of a molehill.

Zz.

 

[Mark44]

This is nothing to do with mathematical rules. It has everything to do with typesetting style!

I agree completely.
This expression --
$$\frac 1 {2x}$$
-- is unambiguous, but this one --
$1/2x$
-- is ambiguous.
In the first expression, the vinculum clearly indicates that $2x$ is the denominator, thereby playing the same role as parentheses in the equivalent expression $1/(2x)$.

 

[John3509]

In Britain an 18 year old may drink alcohol in a bar. In America they'd risk arrest. Why do Brits get to break the rules?

Answer: we don't. Not everybody agrees on the same set of rules. As long as everyone is clear what the rules are in their circumstances there's no problem. The "puzzle" you linked (what is 8/2(2+2)?) boils down to whether you believe you should apply strict left-to-right evaluation or not. It's just an excuse for a fight. The correct answer, of course, is that it depends what convention you use.

Also many physics equations use quantities with units, which frequently resolve any ambiguity. For example, deliberately being sloppy, kinetic energy is $1/2mv^2$. Knowing that's an energy, interpreting it as $1/(2mv^2)$ is clearly wrong since it has the wrong units so I must, in this case, mean $(1/2)mv^2$. That means that an editorial instruction to never write $1/2mv^2$ is largely a style choice.

Edit: just to add, multiplication having a higher precedence than addition is also a convention. In this case it's a natural one, since everyone interprets "I've got three twenty pence pieces and two ten pence pieces" as me having 80p. It makes sense for 3×20+2×10 to be interpreted the same way as the verbal version. But systems that don't respect that rule are easy to write and (as long as I make clear that I'm writing in such a convention) they aren't wrong, just different.

Right, but PEMDAS/BEDMAS is a standardized set of rules for math. There are no different "countries" in math, there is just...math. I do agree that multiplication before addition is more intuitive, that's probably why PEMDAS is the way it is, but what is the reason for Physicists choosing multiplication to take priority over division with a slash when it goes against PEMDAS. I know you said you can chose what ever convention you want but even when all technology and calculators they use are programed to strictly follow PEMDAS, this decision seems weird and pointless to me.

 

[fresh_42]

Right, but PEMDAS/BEDMAS is a standardized set of rules for math. There are no different "countries" in math, there is just...math. I do agree that multiplication before addition is more intuitive, that's probably why PEMDAS is the way it is, but what is the reason for Physicists choosing multiplication to take priority over division with a slash when it goes against PEMDAS. I know you said you can chose what ever convention you want but even when all technology and calculators they use are programed to strictly follow PEMDAS, this decision seems weird and pointless to me.

Read this thread. The notation is not uniquely defined, hence worthless. No mathematician or physicist would ever write down such a literal nonsense. It is a character string which needs further interpretation rules. Same as you cannot write $(123)(321)$ without telling us whether this has to be read from left to right or right to left. Langauge isn't unique, that's why scientists prefer mathematics. The character string in your example isn't mathematics, it's crap.

Edit: "A road crosses a chicken." is a syntactically correct sentence, but without semantics it is useless. The example in post #1 is also syntactically correct, but readers apply their own semantic when they read it. However, nobody has defined a semantic for $\div$.

 

[John3509]

Read this thread. The notation is not uniquely defined, hence worthless. No mathematician or physicist would ever write down such a literal nonsense. It is a character string which needs further interpretation rules. Same as you cannot write $(123)(321)$ without telling us whether this has to be read from left to right or right to left. Langauge isn't unique, that's why scientists prefer mathematics. The character string in your example isn't mathematics, it's crap.

But that is what i am trying to understand. Why do physicists think it is not uniquely defined or ambiguous? If you plug it into a ti84 it will 16 as the solution, programing has it clearly defined, / = (division symbol with 2 dots). When written as 1/2x why would anyone be unsure if that both 2 and x are in the denominator if there are no paranthesis around them?

 

[PeroK]

But that is what i am trying to understand. Why do physicists think it is not uniquely defined or ambiguous? If you plug it into a ti84 it will 16 as the solution, programing has it clearly defined, / = (division symbol with 2 dots). When written as 1/2x why would anyone be unsure if that both 2 and x are in the denominator if there are no paranthesis around them?

A computer is a computer. Mathematicians and physicist's are not mere computers, parsing strings of symbols mechanically according to a fixed set of rules.

For example, I would write $e^{i\pi}$, literally as you see it on the screen. But, to communicate via a computer I have to render that as a Latex string. Which is another arbitrary convention.

Also, regarding your assertion that mathematics has no national boundaries, the Chinese might have something to say about that.

 

[Ibix]

Right, but PEMDAS/BEDMAS is a standardized set of rules for math. There are no different "countries" in math

You yourself cited a specific example of a journal using different rules from the ones you want to use, but you are here trying to tell me no-one uses different rules. Your position seems somewhat self contradictory.

what is the reason for Physicists choosing multiplication to take priority over division with a slash

Saves on typing, I imagine. With your rules, there's no way to write $1/(2x)$ without using brackets. With the journal's rules you can write $1/2x$ and with either set of rules you can write $x/2$ if you meant $(1/2)x$.

when all technology and calculators they use are programed to strictly follow PEMDAS, this decision seems weird and pointless to me.

So your value judgement differs from that of the editors of the journal. But note that your only argument is "follow the majority", not anything technical.

 

[Mark44]

...but even when all technology and calculators they use are programed to strictly follow PEMDAS...

This is not true of calculators in general. In the same wiki article that you cited (https://en.wikipedia.org/wiki/Order_of_operations#Calculators), it mentions several calculators, including Texas Instrument models and Casio models, that don't follow this convention. It also mentions that the Windows calculator evaluates $1 + 2 \times 3$ in two different ways, depending on whether the calculator is in Standard mode or Scientific mode.

Also, the same wiki article talks about an ambiguous representation of $1/2\sqrt N$ on a page in the Feynman lectures. On the page cited, this expression is written as
$$\frac 1 {2\sqrt N}$$
In a later paragraph it is written as $1/2\sqrt N$, so there is some context for interpreting the latter expression to mean what is written the first time.

As already mentioned, a major reason for writing this as $1/2\sqrt N$ is as a space-saving feature in typesetting.

 

[russ_watters]

Windows calculator evaluates $1 + 2 \times 3$ in two different ways, depending on whether the calculator is in Standard mode or Scientific mode.

This threw me just the other day!

And can I please get a 1/x in scientific mode?

 

[Ibix]

And can I please get a 1/x in scientific mode?

You just type 1/1x.

This is not true of calculators in general.

Nor programming languages. I've written a program in postscript, which is stack based and effectively uses reverse Polish notation. Like Yoda maths doing it is. Perfectly understandable once you spot how it works, but a completely distinct notation.

 

[fresh_42]

There is a simple rule which always leads to success, whether in programming or in writing formulas:
If in doubt, use a bracket!

 

[Mark44]

If in doubt, use a bracket!

And even better, use a pair of them...

 

[John3509]

You yourself cited a specific example of a journal using different rules from the ones you want to use, but you are here trying to tell me no-one uses different rules. Your position seems somewhat self contradictory.

Well then if many people break the rules in a consistent way I suppose you can call that "using different rules"

I think that is how dielects work too.

 

[John3509]

This is not true of calculators in general. In the same wiki article that you cited (https://en.wikipedia.org/wiki/Order_of_operations#Calculators), it mentions several calculators, including Texas Instrument models and Casio models, that don't follow this convention. It also mentions that the Windows calculator evaluates $1 + 2 \times 3$ in two different ways, depending on whether the calculator is in Standard mode or Scientific mode.

that's interesting, I did not notice that. I can understand why / would be interpreted as everything to the right is denominator, it saves the need to type parentheses, but why this?

 

[PeroK]

Well then if many people break the rules in a consistent way I suppose you can call that "using different rules"

I think that is how dielects work too.

Well, you are breaking the rules by spelling "dialect" your own way.

 

[Klystron]

Well then if many people break the rules in a consistent way I suppose you can call that "using different rules"

I think that is how dielects work too.

Well, you are breaking the rules by spelling "dialect" your own way.

Perhaps, but I read it as dialectic, more consistent with content but still a reach.

[Edit: It was dialect. See below.]

 

[John3509]

Well, you are breaking the rules by spelling "dialect" your own way.

Not sure it that's supposed to mock me or not but I did not know I was spelling it wrong,

But if some English professor or language expert published everything with, and required all papers submitted in class as well to have dialect misspelled it would be expected to raise questions among the students.