Question about PEMDAS/BODMAS ambiguity

[FactChecker]

Regarding $2^{3^2}$:

Yes. PEMDAS does not address that specific condition.

We should not confuse the order of operations with ambiguity caused by bad typesetting. With proper typesetting, $2^{3^2}$ would not be confused with $(2^3)^2$ and PEMDAS would give the correct result.

 

[Mark44]

We should not confuse the order of operations with ambiguity caused by bad typesetting. With proper typesetting, $2^{3^2}$ would not be confused with $(2^3)^2$ and PEMDAS would give the correct result.

This has nothing to do with order of operations (AKA precedence), but rather with associativity (grouping) of operators. PEMDAS (or BODMAS) doesn't have anything to say about how exponents should be grouped when there are more than one of them. Modern programming languages specify exactly how to deal with these kinds of situations. IOW, whether operators group left-to-right or right-to-left. I've said before that it seems to me that mathematics should take a leaf from programming languages in this regard.

 

[FactChecker]

This has nothing to do with order of operations (AKA precedence), but rather with associativity (grouping) of operators. PEMDAS (or BODMAS) doesn't have anything to say about how exponents should be grouped when there are more than one of them.

The proper typesetting position of an exponent is clearly defined. There is no confusion between $xy$ versus $x^y$. Likewise, there is no confusion between $2^{3\cdot 2}$ versus $2^{3^2}$.

Modern programming languages specify exactly how to deal with these kinds of situations. IOW, whether operators group left-to-right or right-to-left. I've said before that it seems to me that mathematics should take a leaf from programming languages in this regard.

PEMDAS does not define the proper typesetting of exponents, subscripts, or superscripts, and it does not try. Also, programming languages do not allow for advanced typesetting to identify exponents, subscripts, or superscripts. [UPDATE] In those cases it is up to the individual language to specify how an exponent will be identified, like '^' or '**'.

 

[DaveC426913]

...there is no confusion between $2^{3\cdot 2}$ versus $2^{3^2}$.

OK, but that's not what's in question.

Does
$2^{3^2}$
equal 64? Or 512?

 

[Mark44]

The proper typesetting position of an exponent is clearly defined. There is no confusion between xy versus xy. Likewise, there is no confusion between $2^{3\cdot 2}$ versus $2^{3^2}$.

PEMDAS does not define the proper typesetting of exponents ...

What I'm saying is that PEMDAS or some modification of it should explicitly define how $2^{3^2}$ is to be evaluated. Even with the typesetting as shown here, it's not clear to many people that the correct grouping is $2^9$ rather than $8^2$.

 

[DaveC426913]

While

What I'm saying is that PEMDAS or some modification of it should explicitly define how $2^{3^2}$ is to be evaluated.

While I am fine with how this thread is playing out, my initial query was about the ambiguity that's already within the scope of PEMDAS - which is not all-encompassing. To-wit: PEMDAS is a high school mnemonic that is intended to help with basic arithmetic operations

It seems it implicitly considers nested powers to be too rare in regular use to warrant being included in its compact rules. I'm OK with that. After all, it can't cover everything and still be a mnemonic.

But since it already has rules for brackets and for M/D, it should be internally consistent to clarify how the two should be reconciled, since the causal user of PEMDAS will surely encounter values next to brackets but lacking an operator.

 

[PeroK]

If I understand it correctly, the correct resolution is:
6/2(1+2)
= 6/2(3)
= 6/2x3
= 3x3
= 9

To go back to your original question. I can't see the rationale behind splitting up the term after the division symbol. Naively, I would interpret $?/?$ in the same way as $\frac ? ?$.

and not
6/2(1+2)
= 6/2(3)
= 6/6
= 1

That seems more natural to me.

 

[FactChecker]

What I'm saying is that PEMDAS or some modification of it should explicitly define how $2^{3^2}$ is to be evaluated. Even with the typesetting as shown here, it's not clear to many people that the correct grouping is $2^9$ rather than $8^2$.

The expression $2^{3^2}$ is well defined as $2^{(3^2)}=2^9$. The expression $(2^3)^{{}^2}$ is not valid; the final 2 is raised too high to be an valid exponent. EDIT: $(2^3)^{{}^2}$ does not equal $(2^3)^2$ or $2^{3^2}$. It is not a valid expression.

 

[Mark44]

The expression $2^{3^2}$ is well defined as $2^{(3^2)}=2^9$.

Exactly where is this well-defined? I don't recall reading it in algebra textbooks and it's certainly not stated as part of PEMDAS that exponents should be grouped right-to-left.

The expression $(2^3)^{{}^2}$ is not valid; the final 2 is raised too high to be an valid exponent. EDIT: $(2^3)^{{}^2}$ does not equal $(2^3)^2$ or $2^{3^2}$. It is not a valid expression.

I would bet that at least 99 out of any 100 people would not see $(2^3)^{{}^2}$ as being appreciably different from $(2^3)^2$.

 

[DaveC426913]

To go back to your original question. I can't see the rationale behind splitting up the term after the division symbol. Naively, I would interpret $?/?$ in the same way as $\frac ? ?$.

That seems more natural to me.

It seems natural, but it flies in the face of the PEMDAS rules.

X(Y) means X times Y; it's just shorthand.
So Z/X(Y), means start from left - i.e. the division first

 

[DaveC426913]

F-18 Hornet - apex predator of the sky

No aircraft buffs here, eh?

 

[Ibix]

No aircraft buffs here, eh?

It's an F-15, either the A or C variant. It's off-topic, though...

 

[PeroK]

It seems natural, but it flies in the face of the PEMDAS rules.

What about $\frac 6 {2(1+2)}$?

Division first, remember!

 

[FactChecker]

Exactly where is this well-defined? I don't recall reading it in algebra textbooks and it's certainly not stated as part of PEMDAS that exponents should be grouped right-to-left.

An exponent is on the right and one level up, not two levels up. Just because it is hard to see visually that it is wrong does not make it right. I don't know where this is defined, but I know that the exponent, $c$, of $a^{b^c}$ is not on the same level as $b$ is.

I would bet that at least 99 out of any 100 people would not see $(2^3)^{{}^2}$ as being appreciably different from $(2^3)^2$.

It's not a question of how hard it is to visually see. This is a mathematics question, not an eye chart. They certainly are very different mathematically.

 

[DaveC426913]

An exponent is on the right and one level up, not two levels up. Just because it is hard to see visually that it is wrong does not make it right. I don't know where this is defined, but I know that the exponent, $c$, of $a^{b^c}$ is not on the same level as $b$ is.

It's not a question of how hard it is to visually see. This is a mathematics question, not an eye chart. They certainly are very different mathematically.

Are we arguing whether an given exponent looks too small, and that somehow that changes the meaning? Surely we are not.

Because, unless I am mistaken, you can't arbitrarily pick levels, can't you can't skip any. Which means text size is not an indicator - only location (i.e. up-and-right) is.

To-wit:

The expression $(2^3)^{{}^2}$ is not valid; the final 2 is raised too high to be an valid exponent.

I say they are the same thing.

The fact that you were able to construct them visually using LaTex does not mean the distinction has mathematically rigorous meaning. How would that rule apply if this were written in pencil? How could you even write it? (In other words, you are calling out a LaTeX artifact, not a math artifact.)

They are both, simply, 2^3^2. One just uses some janky LaTeX.


Still, that seems be a tangent.


This is what I'm arguing:

The expression $2^{3^2}$ is well defined as $2^{(3^2)}=2^9$.

I do not grant this without convincement.

That could be interpreted as either 2^(3^2) or (2^3)^2 until and unless we have some conventions to follow.

 

[FactChecker]

Are we arguing whether an given exponent looks too small, and that somehow that changes the meaning? Surely we are not.

An exponent is shown in typeset documents as being just to the right and above the base number. In $a^{b^c}$, $c$ is the exponent of the base $b$ and the result is the exponent of $a$. $c$ is not the exponent of $a^b$. That would be denoted by $(a^b)^c$.
You can do it your way but, IMO, it will cause you trouble if you try to publish.

 

[DaveC426913]

In reviewing my responses, I see they could be misinterpreted as sarcastic. I am not being sarcastic.

An exponent is shown in typeset documents as being just to the right and above the base number.

I'm sorry. I do not follow.
Are typeset documents the authorities? Is that where the rule originates?

In $a^{b^c}$, $c$ is the exponent of the base $b$ and the result is the exponent of $a$.

Are you asserting this?

$c$ is not the exponent of $a^b$.

Why not?

That would be denoted by $(a^b)^c$.

You're saying we must use brackets to denote the difference?

In other words, the absence of brackets makes the tower of exponents ambiguous, yes? Which means 2³²
might be interpreted either way.

 

[FactChecker]

In other words, the absence of brackets makes the tower of exponents ambiguous, yes? Which means 2³²
might be interpreted either way.

No. The brackets are necessary for it to be a valid expression equivalent to $(2^3)^2$.
Notice that the final $2$ is at different levels in $2^{3^2}$ versus $(2^3)^2$. It can not be at both levels. They are mutually exclusive.
If you agree that the $c$ in $a^{b^c}$ is at the right level to be the exponent of $b$ then it can not be the exponent of $a^b$. The two are mutually exclusive because the $c$ can not be at both levels.
$2^{3^2}$ should not be interpreted as $(2^3)^2$.

 

[pbuk]

Exactly where is this well-defined? I don't recall reading it in algebra textbooks and it's certainly not stated as part of PEMDAS that exponents should be grouped right-to-left.

No it is not in PEMDAS because at the level PEMDAS is taught repeated exponentiation is not encountered, however exponentiation is a right-associative operation. If it were left-associative you would have $x^{a^b} = (x^a)^b = x ^ {ab} \implies a^b = ab$.

This is not taught explicitly for the same reason that we do not teach explicitly that in e.g. $e^{i\theta}$ the multiplication comes before the exponentiation, breaking PEMDAS.

As said somewhere above, PEMDAS is not mathematics, it is simply a mnemonic that helps us do arithmetic and algebra over first the integers, then the rationals and finally the polynomials without starting with the field axioms, which would be a bit tough in 5th grade.

 

[Mark44]

I would bet that at least 99 out of any 100 people would not see $(2^3)^{{}^2}$ as being appreciably different from $(2^3)^2$.

It's not a question of how hard it is to visually see. This is a mathematics question, not an eye chart. They certainly are very different mathematically.

I don't buy your argument at all. Since both expressions have $2^3$ in parentheses, which are at the highest level of precedence, I'm going to rewrite the two expressions as $8^{{}^2}$ and $8^2$. To claim that these are "very different mathematically" seems pedantic to me.

 

[FactChecker]

I don't buy your argument at all. Since both expressions have $2^3$ in parentheses, which are at the highest level of precedence, I'm going to rewrite the two expressions as $8^{{}^2}$ and $8^2$. To claim that these are "very different mathematically" seems pedantic to me.

You have to know that $8^{{}^2}$ has the '2' in the wrong position to be an exponent of the 8 whereas $8^2$ is proper. It is hard to see unless there are other properly placed exponents to compare it with. The difference is significant. The first example is simply invalid -- it has an exponent with a missing base. And the original post example, $2^{3^2}$, has an "exponent tower" with both a properly positioned exponent and an exponent for the exponent. The mathematical interpretation is unambiguous. Mathematics has many "exponent tower" examples.

IMO, the fundamental misconception of the OP is that PEMDAS completely defines mathematical syntax. It does not. If it was only up to PEMDAS, we would have $\sin(x)=s\cdot i\cdot n\cdot x$.

 

[Mark44]

No it is not in PEMDAS because at the level PEMDAS is taught repeated exponentiation is not encountered, however exponentiation is a right-associative operation.

With regard to "however exponentiation is a right-associative operation," you know this and I know this, but very many people don't know it. My concern is that it should be taught when PEMDAS is presented so that mathematics at this level is at least on par with how programming languages operate and are taught. There seem to be a number of implicit assumptions about how PEMDAS should be used to parse an algebraic or arithmetic expression, such as that multiplication and division have the same precedence, with neither being at a higher level than the other. And the same for addition and subtraction.

This is not taught explicitly for the same reason that we do not teach explicitly that in e.g. $e^{i\theta}$ the multiplication comes before the exponentiation, breaking PEMDAS.

OTOH, this exponential alternatively can be written as $exp(i\theta)$, which makes it explicit that the multiplication takes precedence over the exponentiation. In textbooks, the placement of the exponent in $e^{i\theta}$ implies that the exponent is parenthesized. In a similar way, the vinculum in $\dfrac C {2\pi}$ implies that the denominator is parenthesized whereas writing $C/2\pi$ as a linear expression is ambiguous## unless we invoke the more complete parsing rules of programming languages.

As said somewhere above, PEMDAS is not mathematics, it is simply a mnemonic that helps us do arithmetic and algebra

And poorly, in its current state, IMO. As I said before, in my first algebra class, around the middle of the last century, only MDAS (with mnemonic My Dear Aunt Sally) was presented. Since then it has been expanded to include Parentheses and Exponents, to make the mnemonic more comprehensive and helpful. I don't see why it can't be improved on again to cover situations that seem to give people so much difficulty; That is, to be explicit on how all of the operators, and particularly exponents, group.

 

[Mark44]

You have to know that $8^{{}^2}$ has the '2' in the wrong position to be an exponent of the 8 whereas $8^2$ is proper.

Baloney...

 

[FactChecker]

Baloney...

Ok. Show me a reference with an exponent tower that is interpreted as $a^{b^c}=a^{bc}$.

 

[PeroK]

This is not taught explicitly for the same reason that we do not teach explicitly that in e.g. $e^{i\theta}$ the multiplication comes before the exponentiation, breaking PEMDAS.

This is the perfect example and goes to the heart of what I was trying to say. 99% of students would automatically and intuitively interpret that as $e^{i\theta}$ and not as $e^i\theta$. However, to render the Latex, you had to put the $i\theta$ in curly braces! It would be the same in any programming language: computer input is limited and cannot make use of the visual conventions that we use.

The hard-line PEMDAS/BODMAS advocates insist that we should give up our human visual skills and rely solely on computer-like parsing of strings of characters. This is bad enough. But, if you take PEMBAS literally, then we do need brackets in this case.

And then there really ought to be a debate on whether PEMDAS/BODMAS are fit for purpose.

 

[FactChecker]

Baloney...

My point is this.
Suppose you were reading a document and knew that the normal exponent was positioned like $8^2$. Then suppose you ran into $8^{{}^2}$ and know that there is a missing number, $b$, in the usual position of an exponent: $8^{b^2}$.
Wouldn't you say that the missing number was a serious omission?

 

[martinbn]

My point is this.
Suppose you were reading a document and knew that the normal exponent was positioned like $8^2$. Then suppose you ran into $8^{{}^2}$ and know that there is a missing number, $b$, in the usual position of an exponent: $8^{b^2}$.
Wouldn't you say that the missing number was a serious omission?

If you knew that there is a missing $b$, even if it had been written as $8^2$ you would still say that there is a serious omission. On the other hand if you just see $8^{{}^2}$, you may check if there is a footnote, but then you will read it as $8^2$. The way most people will read it anyway.

 

[FactChecker]

If you knew that there is a missing $b$, even if it had been written as $8^2$ you would still say that there is a serious omission.

But there is an empty position for the first exponent. That is not mathematically valid. It would be hard to notice, but this is not a thread about a vision test, it is a thread about proper mathematics.
I will leave it at that.

 

[martinbn]

But there is an empty position for the first exponent. That is not mathematically valid. It would be hard to notice, but this is not a thread about a vision test, it is a thread about proper mathematics.
I will leave it at that.

Well, then I envy you. It seems that you have been very lucky and have read books with perfect typesetting. I have books that have the formulas written by hand then printed.

 

[FactChecker]

Well, then I envy you. It seems that you have been very lucky and have read books with perfect typesetting. I have books that have the formulas written by hand then printed.

You have missed my point. There is an empty position in the exponent tower. That is mathematically invalid, whether it can be seen or not, whether the typesetting is good or bad. This is a mathematical issue.