Casio anomaly or gremlin or bug...

[Matt Agajanian]

On Facebook, the 6/2(1+2) math quiz has been quite active. Between BODMAS, PEDMAS, PPMDAS, my collection from TI-58C, HP-32SII/33s/35s (to be fair, in ALG mode), Sharp EL-W516X, TI-36X Pro responding with 9 as the answer, my Casio fx-115ES Plus responds with a result of 1. Why are Casio's hierarchy rules the anomaly?

[Joe Horn]

Both Casio and Mathcad follow an "order of operations" rule which says that scaling (written as an implied multiplication) comes before divisions. But apparently it's more common for calculators to omit that rule, and to give all multiplications the same precedence, no matter how they are written. If you search for "implied multiplication" on this page, you'll see a good explanation of both interpretations.

Bottom line: this notation is ambiguous. Therefore it should simply be avoided, which is why HP's RPL and Prime's CAS don't allow it.

[John Cadick]

(09-05-2017 02:26 AM)Joe Horn Wrote:  Bottom line: this notation is ambiguous. Therefore it should simply be avoided, which is why HP's RPL and Prime's CAS don't allow it.

Joe: you have hit on one of my pet peeves. I have been seeing more and more flame wars on the internet (not in the museum forum) about the correct evaluation of a poorly written or ambiguous expression.

The so-called standard precedence cannot be used on very complicated expressions and sometimes on simpler expressions like this one. We have been blessed with two very powerful tools that should always be used in these types of problems. To wit: parentheses and RPN.

Rant off/

John

[ijabbott]

(09-05-2017 02:26 AM)Joe Horn Wrote:  Bottom line: this notation is ambiguous. Therefore it should simply be avoided, which is why HP's RPL and Prime's CAS don't allow it.

At least the Casio PRIZM series (or at least the fx-CG50 I have) disambiguates it for you. If you enter 6÷2(1+2), it rewrites it as 6÷(2(1+2)) and gives the answer 1.

[Matt Agajanian]

(09-05-2017 02:26 AM)Joe Horn Wrote:  Both Casio and Mathcad follow an "order of operations" rule which says that scaling (written as an implied multiplication) comes before divisions. But apparently it's more common for calculators to omit that rule, and to give all multiplications the same precedence, no matter how they are written. If you search for "implied multiplication" on this page, you'll see a good explanation of both interpretations.

Bottom line: this notation is ambiguous. Therefore it should simply be avoided, which is why HP's RPL and Prime's CAS don't allow it.

So, you're saying the TI and HP results are correct? What rules are present when +/- or x/: are on the same level in an expression?

[John Cadick]

(09-05-2017 06:20 PM)Matt Agajanian Wrote:  So, you're saying the TI and HP results are correct? What rules are present when +/- or x/: are on the same level in an expression?

All precedence of operations start with parentheses and then exponentiation. (PE) in some countries B (brackets) is used instead of P.

In some parts of the world (the US primarily) the whole sequence is PEMDAS. parentheses, exponentiation, multiplication, division, addition and subtraction. I believe that some European countries use PEDMSA instead.

The way that many people do it is to go through the entire expression and do anything in parentheses first. Then they go back to the left and go all the way through doing exponents

To be consistent they should then go back to the left of the expression do each of the operations individually from left to right.

Doing your example using PEMDAS the answer would be developed as:
6/2(1+2) => 6/2*3 => 6/6 = 1

If however the PEDMSA method is used:

6/2(1+2) => 6/2*3 => 3*3 = 9

Thus the "correct answer" depends on which precedence sequence you use and whether or not you go back to the left and work to the right for each of the six operations.

That is why I personally avoid ever entering an expression unless there are parentheses to to clarify.

John

[Matt Agajanian]

(09-05-2017 07:21 PM)John Cadick Wrote:  

(09-05-2017 06:20 PM)Matt Agajanian Wrote:  So, you're saying the TI and HP results are correct? What rules are present when +/- or x/: are on the same level in an expression?

All precedence of operations start with parentheses and then exponentiation. (PE) in some countries B (brackets) is used instead of P.

In some parts of the world (the US primarily) the whole sequence is PEMDAS. parentheses, exponentiation, multiplication, division, addition and subtraction. I believe that some European countries use PEDMSA instead.

The way that many people do it is to go through the entire expression and do anything in parentheses first. Then they go back to the left and go all the way through doing exponents

To be consistent they should then go back to the left of the expression do each of the operations individually from left to right.

Doing your example using PEMDAS the answer would be developed as:
6/2(1+2) => 6/2*3 => 6/6 = 1

If however the PEDMSA method is used:

6/2(1+2) => 6/2*3 => 3*3 = 9

Thus the "correct answer" depends on which precedence sequence you use and whether or not you go back to the left and work to the right for each of the six operations.

That is why I personally avoid ever entering an expression unless there are parentheses to to clarify.

John

Your point brings up another perspective. With PPMDAS/PEDMAS in mind, I usually work the statement from left-to-right, breaking down each sub-expression at its hierarchical level. Once those are evaluated, I then move through the statement with the PPMDAS hierarchy again and so forth.

[pier4r]

(09-05-2017 06:20 PM)Matt Agajanian Wrote:  So, you're saying the TI and HP results are correct? What rules are present when +/- or x/: are on the same level in an expression?

Those questions (that are a bit embarrassing sometimes) rely on the fact that people hold different knowledge about conventions (and then they fight for it).

A convention is defined by one or more people, then people sticks to it and a result is given. So the "correct" part is valid only if one follows the mainstream convention.

If I decide that addition come first and subtraction comes after, of course the result "1" (or 9) is wrong, but because I am using a convention that is valid for myself only.

So defining a convention and sticking to it makes the result unambiguous.


Also in Europe, as far as I know, the mainstream definition is:

parentheses (from innermost to outmost)
exponentiation and roots (leftmost first. That is, the operation that happens first on the left of the expression gets evaluated)
multiplication or division (which one first? Left first)
addition and subtraction (leftmost first)

translating 6/2(1+2) in RPN it would be (RPN is really neat to show what I do first)
1
2
+
6
2
/
*

thus 9.

For instance
(1+2)2/6

would be
1
2
+
2
*
6
/

[Matt Agajanian]

I'm sorry. I meant no disrespect. At this point, I am trying to ascertain some clarity. Both sides can defend which is right. But isn't mathematics predicated on accuracy and disambiguous results?

In a science as disipline and structured as mathematics, I find that it becomes confusing when this kind of ambiguity surfaces. I thought that PEDMAS/BODMAS rules were identical, infallible, and rigid. Now, with US versus European interpretation differences, it's frustrating.

Please clarify because this variation of interpreting PEDMAS/BODMAS seems to say that mathematics rules are flexible instead of having a standard rule base.

[pier4r]

I think that the idea of mathematics as precise (and boring) comes from poor teachers. mathematics (as a lot of other things) is a set of consequences given a certain starting point agreed as convention.

[Joe Horn]

Lesson: Math is not a Single Eternal Truth. It's a tool, and like any tool, it comes in many designs, all of which are useful for various tasks. Pick a tool, learn how to use it, and then use it, but PLEASE don't waste time trying to convince the world that YOUR tool is the best one, or the only one that should exist, or some kind of Eternal Truth. That kind of debate belongs in a religion forum, not here.

Quiz: Which algebraic order of operations is Eternally True? or is the best one? or is the only one that should exist?

Answer: Those are stupid questions. Re-read the Lesson above.

EDIT: Whoa Nellie, sorry for being so easily annoyed. I didn't have my coffee this morning. Yeah, I'll blame that. <blush>

[SlideRule]

(09-05-2017 09:08 PM)Matt Agajanian Wrote:  ... isn't mathematics predicated on accuracy and disambiguous results?
In a science as disipline and structured as mathematics, I find that it becomes confusing when this kind of ambiguity surfaces... Please clarify because this variation ... mathematics rules are flexible ...

From the PREFACE of Philosophy of Mathematics
[attachment=5157]
my emphasis
"One of the most striking features of mathematics is the fact that we are much more certain about what mathematical knowledge we have than about what mathematical knowledge is knowledge of. Mathematical knowledge is generally accepted to be more certain than any other branch of knowledge; but unlike other scientific disciplines, the subject matter of mathematics remains controversial.

In the sciences we may not be sure our theories are correct, but at least we know what it is we are studying. Physics is the study of matter and its motion within space and time. Biology is the study of living organisms and how they react and interact with their environment. Chemistry is the study of the structure of, and interactions between, the elements. When man first began speculating about the nature of the Sun and the Moon, he may not have been sure his theories were correct, but at least he could point with confidence to the objects about which he was theorizing. In all of these cases and others we know that the objects under investigation - physical matter, living organisms, the known elements, the Sun and the Moon - exist and that they are objects within the (physical) world.

In mathematics we face a different situation. Although we are all quite certain that the Pythagorean Theorem, the Prime Number Theorem, Cantor's Theorem and innumerable other theorems are true, we are much less confident about what it is to which these theorems refer. Are triangles, numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that force philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics".

I find this tome is counted among my more recurring reads.

BEST!
SlideRule

[Matt Agajanian]

(09-05-2017 10:11 PM)Joe Horn Wrote:  Lesson: Math is not a Single Eternal Truth. It's a tool, and like any tool, it comes in many designs, all of which are useful for various tasks. Pick a tool, learn how to use it, and then use it, but PLEASE don't waste time trying to convince the world that YOUR tool is the best one, or the only one that should exist, or some kind of Eternal Truth. That kind of debate belongs in a religion forum, not here.

Quiz: Which algebraic order of operations is Eternally True? or is the best one? or is the only one that should exist?

Answer: Those are stupid questions. Re-read the Lesson above.

EDIT: Whoa Nellie, sorry for being so easily annoyed. I didn't have my coffee this morning. Yeah, I'll blame that. <blush>

Apology accepted. But yes, like I said before, I mean and meant no offence. To clarify, I was taught, for example, my earlier illustration 6/2(1+2) because it is written as such and I thought PEDMAS and BODMAS lead to the same answer. But, I guess, PEDMAS and BODMAS lead to two different solutions.

[Joe Horn]

(09-05-2017 11:31 PM)Matt Agajanian Wrote:  But, I guess, PEDMAS and BODMAS lead to two different solutions.

Those two should NOT lead to different solutions, since in both of them multiplication and division have the same precedence and are both performed left to right. The fact that M is before or after D in them is NOT an indication of which one has precedence; they have equal precedence in both Order Of Operations systems.

Also, please note that in the USA it's usually spelled PEMDAS, not PEDMAS, but both would be equivalent, since the order of M and D is irrelevant. Therefore, if I understand the Wikipedia page about Order of Operations correctly, PEMDAS and PEDMAS and BODMAS are all exactly equivalent systems, just given different names.

The only Order of Operations system that I'm aware of which yields a different answer is, as I said above, the one which includes a hierarchy which other systems don't even have, namely, SCALING, which is symbolized by implied multiplication. In those systems, THAT comes before all other multiplications (and divisions). There is a paragraph in the above-linked article which talks about the ambiguity of "implied multiplication" in expressions such as 1÷2x. Apparently not only Casio and Mathcad consider implied multiplication to come before other multiplications; so do "the manuscript submission instructions for the Physical Review journals" and "prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics."

[AlexFekken]

(09-05-2017 10:11 PM)Joe Horn Wrote:  Lesson: Math is not a Single Eternal Truth.

I agree completely (but probably referring to something else, e.g. the axiom of choice, the inclusion of 0 in the natural numbers; not whatever tools are used to do math). But this discussion isn't even about math.... Would you call a discussion about grammar, spelling or punctuation a discussion about literature?

[toml_12953]

(09-05-2017 07:21 PM)John Cadick Wrote:  In some parts of the world (the US primarily) the whole sequence is PEMDAS. parentheses, exponentiation, multiplication, division, addition and subtraction.

Doing your example using PEMDAS the answer would be developed as:
6/2(1+2) => 6/2*3 => 6/6 = 1

IJohn

No, In the US, Multiplication and division have the same precedence so they're done left to right (as are addition and subtraction). Thus

6/2(1+2) Do the operation in parentheses first.

6/2(3) Of the remaining operators, division is the first, left to right so it is done in precedence over the multiplication.

3(3) There's only a multiplication left to do.

9

Implied multiplication has the same precedence as a multiply sign in most implementations.

[Matt Agajanian]

(09-06-2017 12:17 AM)toml_12953 Wrote:  

(09-05-2017 07:21 PM)John Cadick Wrote:  In some parts of the world (the US primarily) the whole sequence is PEMDAS. parentheses, exponentiation, multiplication, division, addition and subtraction.

Doing your example using PEMDAS the answer would be developed as:
6/2(1+2) => 6/2*3 => 6/6 = 1

IJohn

No, In the US, Multiplication and division have the same precedence so they're done left to right (as are addition and subtraction). Thus

6/2(1+2) Do the operation in parentheses first.

6/2(3) Of the remaining operators, division is the first, left to right so it is done in precedence over the multiplication.

3(3) There's only a multiplication left to do.

9

Implied multiplication has the same precedence as a multiply sign in most implementations.

A-HA (not the 'Take on Me' group)!!!! That's what I thought! Growing up in California, this US variation of PPMDAS, PEDMAS, left-to-right, Multiplication & Division have equal precedence to each other as do Add & Subtract makes more valid sense.

But, toml_12953, if your statement about implied multiplication is to hold, why does the Casio fx-115ES Plus calculate 1 for the problem?

[Joe Horn]

(09-06-2017 12:45 AM)Matt Agajanian Wrote:  But, toml_12953, if your statement about implied multiplication is to hold, why does the Casio fx-115ES Plus calculate 1 for the problem?

I already answered that above in reply #2. Notice that Tom said, "Implied multiplication has the same precedence as a multiply sign in most implementations." But other implementations do exist, and have good reasons to exist (as explained above).

[mfleming]

Operator precedence, along with cut 'n paste, always trips up programmers. This brings to mind Iverson's solution to the many weird functions and operators in APL - there is no defined precedence. The order of evaluation is "more or less" right to left with parentheses serving to provide some ordering of evaluation. Hence

2 * 3 + 4 = 2 * 7 = 14 or

6/2(1+2) = 6/2*3 = 6/6 = 1

No need to memorize which squiggly symbol outranks another. Hard enough as it is to remember what each symbol does...

Good for Ken!
~Mark

[Dan]

(09-05-2017 09:39 PM)pier4r Wrote:  I think that the idea of mathematics as precise (and boring) comes from poor teachers.

Teachers are always blamed for all of society's ills! Mathematics is not popular with many people because it requires one to think. Teachers do the best they can with the curriculum they are required to teach and the limited time they have to prepare students for their examinations.