Biggest science or math pet peeve

[late347]

Sure you can - you just have to use braces.
$2^{3^{4^3}}$

Here's the LaTeX I used:
##2^{3^{4^3}}##

Where should you start to calculate according to pemdas rule?

 

[micromass]

Where should you start to calculate according to pemdas rule?

From top to bottom. What's on the top must be computed first.

 

[micromass]

It seems to me that expressions can be grouped together in several ways (in addition to parenthesis) and essentially the simple rule suggested by PEMDAS is not so simple.

Sure, if you're going to misrepresent PEMDAS, then it's very easy to show that it doesn't work.

 

[Mark44]

It seems to me that expressions can be grouped together in several ways (in addition to parenthesis)

Offhand, I can't think of any other grouping mechanism, other than the bar that I mentioned, which is also used in typesetting to separate the numerator from the denominator, as in
$$\frac {3 + 4} {8 + 6}$$
The P in PEMDAS and the B in BODMAS refer to any enclosing symbols, including parentheses, brackets, braces, single or double vertical bars (as in |x + y| and $||\vec{x} + \vec{y}||$).

and essentially the simple rule suggested by PEMDAS is not so simple.

Can you provide an example where it doesn't?

 

[Mark44]

the simple rule suggested by PEMDAS is not so simple.

PEMDAS/BODMAS is a piece of cake in comparison to the precedence rules of programming languages.

In the Microsoft documentation for C++ operator precedence and associativity (https://msdn.microsoft.com/en-us/library/126fe14k.aspx), there are 18 groups, several of which list 10 or more operators.

 

[gmax137]

I thought the original peeve was the way people fixate on this PEMDAS "rule" and spend all their energy on it, rather than learning something about numbers or mathematics. If so, it seems the thread proves the point...

Tying in the associative property is worthwhile, but it seems there would be more direct ways to bring that up in class.

 

[clope023]

Calling Entropy 'disorder'.

That and the religious dislike of rote learning and memorization.

 

[micromass]

I thought the original peeve was the way people fixate on this PEMDAS "rule" and spend all their energy on it, rather than learning something about numbers or mathematics. If so, it seems the thread proves the point...

Tying in the associative property is worthwhile, but it seems there would be more direct ways to bring that up in class.

Order of operations is important to teach in the classroom. The students need to evaluate the expressions in the right way. They need to do $6x^2 +5x + 7$ correctly. This means evaluating the square first, then the multiplications and then the additions. The students need to be given clear rules on how to do this.

Sure you can say that people will need to use their own judgement and that the typesetting dictates how it will be read. But that is for us experienced mathematicians. We have no problem with this. But for novice students, they need to be taught how to evaluate expressions correctly. If you're not spending time on this, then they will end up very confused. Take it from somebody who taught mathematics to young students: this is important.

 

[Mark44]

That and the religious dislike of rote learning and memorization.

+1

 

[jack action]

What's wrong with simply?

(6+3) + (-1/3) + (1*0) + (-4^3) + (1x2)

All those extra brackets are not needed not because of PEMDAS but because addition is associative. If you give up PEMDAS you do not lose the associativity of addition or muliplication as many of you seem to assume!

Absolutely nothing is wrong with that equation. The fact is that somehow, somewhere, somebody created a set of rules to further simplify the parenthesis use. @Mark44 gave you a lot of examples of these rules and in your equation, according to PEMDAS, all your parenthesis can be implied.

For some reason that I don't understand, you seem to refuse recognizing the validity of those rules.

This is like if I was saying $a^3$ can too easily be mixed up with $a3$ which means $a \times 3$, hence, exponents shouldn't be thought in classroom and we should all write $aaa$ to make it clear to everyone.

You know what? $aaa$ is not really clear either. What is the sign implied in between variables? Is it addition, subtraction, multiplication, division or even a mix of any of those? Who knows? How can we know for sure?

What's wrong with simply writing:

$a \times a \times a$

There, now it's clear! No ambiguity, since multiplication is associative!

 

[fresh_42]

What's wrong with simply writing:

$a \times a \times a$

There, now it's clear! No ambiguity, since multiplication is associative!

... well, it depends ...

 

[micromass]

... well, it depends ...

Well, you just need power-associativity.

 

[PeroK]

For some reason that I don't understand, you seem to refuse recognizing the validity of those rules.
!

For the reason that the rules cause ambiguity. This is evidenced by the fact that the two Microsoft calculators gave different answers for the same expression. And that 76% of the population gave the "wrong" answer.

If 76% of the population can't do basic arithmetic - and unfortunately that includes me - then maybe the rules are a bit obscure?

I also question whether these rules help or hinder maths education - a question apparently I'm not at liberty to ask.

I did Dr Peter Price a disservice in a previous post. He was responsible for the 76% survey and, although he is a supporter of PEMDAS, this caused him at least to question the status of these rules, if so few people know them.

Are you really denying my inalienable human right to have a pet peeve? Must I kowtow to the sacred cow that is PEMDAS?

 

[micromass]

For the reason that the rules cause ambiguity. This is evidenced by the fact that the two Microsoft calculators gave different answers for the same expression.

And how exactly would calculators be programmed without PEMDAS? How would the removal of PEMDAS over some other convention like white spaces by beneficial here?

And that 76% of the population gave the "wrong" answer.

Also 60% of the US population says the evolution is false or is not sure about it.
Also, this http://blog.sciencegeekgirl.com/2009/11/09/myth-because-the-astronauts-had-heavy-boots/
Referring to the ignorance of the total population isn't really helpful.

 

[S.G. Janssens]

Referring to the ignorance of the total population isn't really helpful.

I thought this is the principle behind national elections?

 

[micromass]

I thought this is the principle behind national elections?

Sadly...

 

[PeroK]

And how exactly would calculators be programmed without PEMDAS? How would the removal of PEMDAS over some other convention like white spaces by beneficial here?
Also 60% of the US population says the evolution is false or is not sure about it.
Also, this http://blog.sciencegeekgirl.com/2009/11/09/myth-because-the-astronauts-had-heavy-boots/
Referring to the ignorance of the total population isn't really helpful.

I'm not the only one:

http://www.math.harvard.edu/~knill/pedagogy/ambiguity/

This issue is not as clear cut as many of you would like to pretend.

 

[PeroK]

There may be more on this forum but the brow beating I've taken would deter most from uttering a word in my defence!

 

[micromass]

I'm not the only one:

http://www.math.harvard.edu/~knill/pedagogy/ambiguity/

This issue is not as clear cut as many of you would like to pretend.

Again, how would the removal of this rule be beneficial in programming?

 

[S.G. Janssens]

There may be more on this forum but the brow beating I've taken would deter most from uttering a word in my defence!

No, I agree with you on this, but I have other peeves to pet

 

[PeroK]

Again, how would the removal of this rule be beneficial in programming?

Programmers would have to tighten their syntax rules. That may be no bad thing. An over reliance on obscure and complicated rules of precedence might be deemed poor programming practice.

I wonder how uniformly implemented the current rules are, in any case.

 

[micromass]

I also question whether these rules help or hinder maths education - a question apparently I'm not at liberty to ask.

No, please do address this. I'm interested how you would teach this to children. How would you teach children to evaluate $2p+3q$?

 

[ShayanJ]

There may be more on this forum but the brow beating I've taken would deter most from uttering a word in my defence!

Its not "the brow beating" that you've taken, but that this doesn't seem to me as black and white as its being implied by this discussion. It seems to me that those rules are like Bohr's model for arithmetic. They seem important for educating children but you should get rid of them as soon as possible, i.e. when you're sure the children have the intuition about algebra(cases mentioned by micromass that imply we seem to use some rules there) that we grown ups have now.

 

[micromass]

Its not "the brow beating" that you've taken, but that this doesn't seem to me as black and white as its being implied by this discussion. It seems to me that those rules are like Bohr's model for arithmetic. They seem important for educating children but you should get rid of them as soon as possible, i.e. when you're sure the children have the intuition about algebra(cases mentioned by micromass that imply we seem to use some rules there) that we grown ups have now.

I would most definitely agree with this assessment.

 

[PeroK]

No, please do address this. I'm interested how you would teach this to children. How would you teach children to evaluate $2p+3q$?

You don't have to hit that with a sledgehammer like PEMDAS.

I've already answered that above: the precedence of multiplication is a universal rule.

What I wouldn't do is insist on PEMDAS and then have to explain:

$\frac{a+b}{c+d}$

And why you do the additions before the division. That's what I as a 15 year old would have taken exception to! I fail to see it ad a logical consequence of PEMDAS.

I think I would just treat fractions on their own merit. This is how we evaluation a fraction. There are all sorts of other things to deal with. Common factors, addition of fractions, partial fractions. Order of operations is the least of it.

 

[micromass]

I wonder how uniformly implemented the current rules are, in any case.

They're not, and that's a big problem. I would much prefer there to be one standard that everybody follows, no matter what that standard is. Note however that most professional scientific software does seem to follow the standard convention.

 

[ShayanJ]

But I should say that I have no memory of learning the order of operations in my elementary school years. But I don't have a good memory so I can't remember how I did it!

 

[S.G. Janssens]

I am tempted to say one thing specifically about programming.

In the programming languages that I know, there are many more unary and binary (and ternary) operators than in basic school arithmetic. (Probably C++ tops it all in this regard.) Textbooks and references usually come with a table of precedence. I found that in practice it did not at all contribute to the clarity of code when these rules of precedence were fully exploited by the programmer. Usually I found it much, much clearer when parentheses were used to remove ambiguity from complicated expressions, such as those involving both ordinary and pointer arithmetic.

Ok, now I go back to my own peeves, although I do enjoy reading along with this discussion.

 

[ShayanJ]

I am tempted to say one thing specifically about programming.

In the programming languages that I know, there are many more unary and binary operators that in basic school arithmetic. (Probably C++ tops it all in this regard.) Textbooks and references usually come with a table of precedence. I found that in practice it did not at all contribute to the clarity of code when these rules of precedence were fully exploited by the programmer. Usually I found it much, much clearer when parentheses were used to remove ambiguity from complicated expressions, such as those involving both ordinary and pointer arithmetic.

Ok, now I go back to my own peeves, although I do enjoy reading along with this discussion.

I think even programmers forget most of that and just use intuition and parentheses. At least that's the case about me!

 

[gmax137]

I have no recollection of learning "order of operations" until I was doing programming in Basic on a PDP something. I thought it was just a convention to save space (no need for the parentheses). As far as the equations on the blackboard, I never had any question that when teacher wrote "3 X-squared" she meant the 2 goes with the X alone, not with the 3. I certainly never saw the word PEMDAS before this morning. I have no basis for this notion, but I bet some kids get turned off by these memory devices, and the associated testing ("oh look, here's a pathological example, can you evaluate it correctly?"). If that's how they teach math in fourth grade now, I'd hate it too.

Spelling bees aren't literature.

 

[Mark44]

For the reason that the rules cause ambiguity. This is evidenced by the fact that the two Microsoft calculators gave different answers for the same expression.

I used to work in the Windows team at Microsoft, but not with the bunch that does the calculator. If I had to guess, the intent of the designers of the "four-banger" calculator, was to do simple (i.e., with two operands) add/subtract/multiply/divide calculations. I would further guess that it's stack-based, meaning that it takes the two operands and an operator (+, - *, /) and carries out the operation.

And that 76% of the population gave the "wrong" answer.

It wouldn't be the first time in history that 76% of the population gave the wrong answer, so I'm not impressed by that statistic.

I'm not the only one:

http://www.math.harvard.edu/~knill/pedagogy/ambiguity/

This issue is not as clear cut as many of you would like to pretend.

The expression in this article of the link seems clear-cut to me.
As written in the post, it is
$2x/3y - 1$, which we're supposed to evaluate for x = 9 and y = 2.
If it had been written like this:
$$\frac{2x} {3y} - 1$$
it would have been clear that 2x is to be divided by 3y, as the fraction bar serves to separate the numerator as a group from the denominator as a group.

As it was written, the expression on the left should be interpreted to mean 2 * x / 3 * y. The M and D operations of PEMDAS are at the same level in the hierarchy of operations (as are the A and S). If a multiplication appears before a division (going left to right), you do the multiplication first. If a division occurs first, you do the divison and then the multiplication.

Here's a simple example that should elucidate my reasoning. It uses addition and subtraction instead of multiplication and division, but it should throw some light on the expression of the blog by Knill.
a) 3 + 2 - 1
b) 3 - 1 + 2
Both BEDMAS and PEMDAS have A before S. If you interpret this to mean that additions should be done before subtractions, then the value of expression a) is 4, while the value of expression b) is 0.
OTOH, if you treat addition and subtraction as having the same precedence, then in any expression involving only these operators, you do whichever one comes first (i.e., in left to right order). With this in mind, expression a) yields 5 - 1 = 4, and expression b) yields 2 + 2 = 4, as well.

For Knill's expression, I maintain that the same idea holds with multiplication and division, as well.

 

[Mark44]

In the programming languages that I know, there are many more unary and binary (and ternary) operators than in basic school arithmetic. (Probably C++ tops it all in this regard.) Textbooks and references usually come with a table of precedence. I found that in practice it did not at all contribute to the clarity of code when these rules of precedence were fully exploited by the programmer. Usually I found it much, much clearer when parentheses were used to remove ambiguity from complicated expressions, such as those involving both ordinary and pointer arithmetic.

I agree, but then it wasn't a problem of ambiguity, but rather, a problem of comprehension by humans.

 

[Mark44]

You don't have to hit that with a sledgehammer like PEMDAS.

I've already answered that above: the precedence of multiplication is a universal rule.

What I wouldn't do is insist on PEMDAS and then have to explain:

$\frac{a+b}{c+d}$

And why you do the additions before the division.

As I have already explained, the above means exactly the same as (a + b)/(c + d). That is the purpose of the bar between the top and the bottom and the bar above the radicand in a radical. Why is this so hard?

That's what I as a 15 year old would have taken exception to! I fail to see it ad a logical consequence of PEMDAS.

I think I would just treat fractions on their own merit. This is how we evaluation a fraction. There are all sorts of other things to deal with. Common factors, addition of fractions, partial fractions. Order of operations is the least of it.

 

[jack action]

What I wouldn't do is insist on PEMDAS and then have to explain:

$\frac{a+b}{c+d}$

And why you do the additions before the division. That's what I as a 15 year old would have taken exception to! I fail to see it ad a logical consequence of PEMDAS.

But that is a fraction, not a division. The horizontal bar adds meaning to the division implied (i.e. the parenthesis, as told by @Mark44 earlier). See this example in Latex:
$$\frac{\frac{a}{b}}{\frac{c}{d}}$$
Latex (not me) makes one of the 3 horizontal bars longer than the other two. It adds meaning to how this equation must be evaluated. And you can't assume that it means $a \div b \div c \div d$. Of course, there are other ways that would make this particular equation a lot more clearer, no doubt, such as:
$$\frac{^a/_b}{^c/_d}$$
$$\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}$$
$$\frac{a \div b}{c \div d}$$
$$\left(a \div b\right) \div \left(c \div d\right)$$
Of course, you are allowed to your opinions and your pet peeves, but I'm not ready to say that those rules are useless and complicate everything.

 

[PeroK]

As I have already explained, the above means exactly the same as (a + b)/(c + d). That is the purpose of the bar between the top and the bottom and the bar above the radicand in a radical. Why is this so hard?

That's because you're determined to stick with PEMDAS, so you need your implied parenthesis. Whereas, I never learned PEMDAS so I'm free to say in this case we do the division last. As I have no a priori rule that operations must be done in a set order it doesn't upset my mathematic apple cart