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RE: What is the correct result? - KeithB - 04-15-2024 04:31 PM
(04-15-2024 02:27 PM)SlideRule Wrote: Normally, I don't chime in on these discussions, however, I have a simple observation - the expression axn seems to affirm arguments both favoring and disfavoring implied multiplication {neh?}? It seems we have lost sight of the original argument over ambiguity in math notation and the posturing of parenthesis free notation (RPN, etc.) as a working solution {praxis}.
my pennies worth
BEST!
SlideRule
Kind of like Duff's comments about Duff's device:
In his announcement of the technique to C's developers and the world, Duff noted that C's switch syntax, in particular its ``fall through'' behavior, had long been controversial, and that ``This code forms some sort of argument in that debate, but I'm not sure whether it's for or against.''
RE: What is the correct result? - dm319 - 04-15-2024 04:44 PM
1 ÷ 2a (or 1÷2a) will always mean 1÷(2 x a) for me.
BODMAS/PEMDAS is just a helpful acronym to help school children. I don't know when it attained the rank of an international standard.
There is no such thing as implicit multiplication or juxtaposition. There are just 'terms', 'products' or 'distributive law', of which contain coefficients or factors. New terminology is not needed.
There isn't ambiguity - once you've identifed that something is a term, it is clear how the expression is calculated. People claiming ambiguity are being deliberately obtuse because they don't want to accept they're doing it wrong.
There is plenty or historical mathematical context and modern school textbooks that support this concept, but not the reverse.
RE: What is the correct result? - SlideRule - 04-15-2024 05:03 PM
An edifying tome, Polysemy of symbols: Signs of ambiguity; give it a go & then hang on!
BEST!
SlideRule
ps: interesting term polysemy!
RE: What is the correct result? - Matt Agajanian - 04-15-2024 05:40 PM
(04-15-2024 04:44 PM)dm319 Wrote: 1 ÷ 2a (or 1÷2a) will always mean 1÷(2 x a) for me.
BODMAS/PEMDAS is just a helpful acronym to help school children. I don't know when it attained the rank of an international standard.
There is no such thing as implicit multiplication or juxtaposition. There are just 'terms', 'products' or 'distributive law', of which contain coefficients or factors. New terminology is not needed.
There isn't ambiguity - once you've identifed that something is a term, it is clear how the expression is calculated. People claiming ambiguity are being deliberately obtuse because they don't want to accept they're doing it wrong.
There is plenty or historical mathematical context and modern school textbooks that support this concept, but not the reverse.
Yes. I see. What about expressions where there is no telling the full extent of a term of a numerator, denominator, or the end of a summation, Σ since there is no ending notation while dx signifies the end of the expression for integrals & derivatives, for example?
RE: What is the correct result? - dm319 - 04-16-2024 02:05 PM
(04-15-2024 05:03 PM)SlideRule Wrote: ps: interesting term polysemy!
That's an interesting read and I do think it is good to revisit what we mean by our established conventions. However, we should probably not complicate school-level maths with asking what we mean by '1' unless we are going to confuse a whole generation!
RE: What is the correct result? - dm319 - 04-16-2024 02:26 PM
(04-15-2024 05:40 PM)Matt Agajanian Wrote:(04-15-2024 04:44 PM)dm319 Wrote: 1 ÷ 2a (or 1÷2a) will always mean 1÷(2 x a) for me.
BODMAS/PEMDAS is just a helpful acronym to help school children. I don't know when it attained the rank of an international standard.
There is no such thing as implicit multiplication or juxtaposition. There are just 'terms', 'products' or 'distributive law', of which contain coefficients or factors. New terminology is not needed.
There isn't ambiguity - once you've identifed that something is a term, it is clear how the expression is calculated. People claiming ambiguity are being deliberately obtuse because they don't want to accept they're doing it wrong.
There is plenty or historical mathematical context and modern school textbooks that support this concept, but not the reverse.
Yes. I see. What about expressions where there is no telling the full extent of a term of a numerator, denominator, or the end of a summation, Σ since there is no ending notation while dx signifies the end of the expression for integrals & derivatives, for example?
The discussion around the viral maths problem is that people are not getting the concept of terms, even though we know generally what they are when we see them, i.e. 2a, 3b2, 4(3+2), 5ab are all terms.
I think you are asking what defines the end of an expression, but that doesn't relate to the BODMAS issue from what I can see. I e.
8÷2(1+3) is the whole expression, 8 and 2(1+3) are terms separated by the ÷ operator. It is no different to:
8÷2a
where a = 1+3
RE: What is the correct result? - Matt Agajanian - 04-16-2024 05:45 PM
Makes perfect sense to me. Thanks.
RE: What is the correct result? - Matt Agajanian - 04-16-2024 10:59 PM
So, since terms include 4ac (from the quadratic equation), 1/3a (meaning 1 divided by 3a because 3a is a term), 32e, 29c, and so forth, why does TI of all companies put implied multiplication down the hierarchy list? You’d think TI would have a handle on the dynamics of mathematics. And then again, TI had the hierarchy of implied multiplication's higher in priority before. So why the switch?
RE: What is the correct result? - toml_12953 - 04-17-2024 12:03 AM
(04-16-2024 10:59 PM)Matt Agajanian Wrote: So, since terms include 4ac (from the quadratic equation), 1/3a (meaning 1 divided by 3a because 3a is a term), 32e, 29c, and so forth, why does TI of all companies put implied multiplication down the hierarchy list? You’d think TI would have a handle on the dynamics of mathematics. And then again, TI had the hierarchy of implied multiplication's higher in priority before. So why the switch?
In mathematics, you don’t write everything on one line. There’s no ambiguity since you can clearly see what terms go where. In the physical sciences, it’s tougher to determine. TI uses the mathematical conventions.
RE: What is the correct result? - Steve Simpkin - 04-17-2024 12:05 AM
For reference, here is a link to a brief discussion of how various calculator manufacturers have interpreted implied multiplication over the years (with examples) and links to further discussion on this subject.
Order of operations - what is 6÷2(1+2)?
RE: What is the correct result? - vaklaff - 04-17-2024 04:23 PM
(04-15-2024 04:44 PM)dm319 Wrote: here are just 'terms'A term in algebraic expressions (a not only there) is a syntactic construct.
(04-15-2024 04:44 PM)dm319 Wrote: There isn't ambiguity - once you've identifed that something is a term, it is clear how the expression is calculated. People claiming ambiguity are being deliberately obtuseThere is ambiguity because the semantics haven’t been clearly established for expressions where terms and the ÷ symbols are used together. I can live with whatever consensus will prevail. I can think of three possible outcomes but frankly, I don’t really care. What I don’t like is when fans of one side call the other sides being obtuse. That’s not the way how to support one’s preference, in my humble opinion.
RE: What is the correct result? - dm319 - 04-18-2024 12:33 PM
(04-17-2024 04:23 PM)vaklaff Wrote: the semantics haven’t been clearly established for expressions where terms and the ÷ symbols are used together.
They are clear though. ÷ is an operator, which can work on terms. I.e:
2b ÷ 2a
Very clearly defines the terms (2b and 2a), with an operator in between. I've haven't seen an example where the terms aren't clear. The viral expression:
8÷2(1+3) or 8/2(1+3)
Is straightforward, when you apply the concept that 2(1+3) is a term.
(04-17-2024 04:23 PM)vaklaff Wrote: What I don’t like is when fans of one side call the other sides being obtuse. That’s not the way how to support one’s preference, in my humble opinion.
Sorry I mean no personal disrespect. I'm seeing a lot of 'it's ambiguous' as a refusal to address the existence of terms. Instead, confusion is generated by calling it 'implied' multiplication, or suggesting the real rule is that multiply comes before division (it doesn't, but this was proposed by Lennes, and was shot down). I call it being 'deliberately obtuse' when there is a refusal to acknowledge that we're talking about terms, here is a nice example of an explanation using terms, with a reply that doesn't address that.
RE: What is the correct result? - Thomas Klemm - 04-19-2024 11:25 AM
(04-18-2024 12:33 PM)dm319 Wrote: here is a nice example of an explanation
That was a interesting read. Thanks for posting it here.
Quote:(…) it highlighted something I long ago realized: that many people — among them some school mathematics teachers — have a perception of algebra that mathematicians abandoned in the seventeenth century.
(…)
Yet I fear that not all school teachers convey just what a powerful tool it is — in part I suspect because they are constrained by a curriculum centuries out of date. Indeed much of current K-12 “algebra instruction” seems to go no further than the pre-modern algebra used prior to the 17th Century.
No surprise then, that the population at large views “algebra” as something different from the discipline that is taught universally in colleges and universities. Those Internet debates result from that division.
I couldn't agree more.
RE: What is the correct result? - Maximilian Hohmann - 04-19-2024 12:31 PM
Hello!
(04-19-2024 11:25 AM)Thomas Klemm Wrote:Quote:No surprise then, that the population at large views “algebra” as something different from the discipline that is taught universally in colleges and universities. Those Internet debates result from that division.
I couldn't agree more.
I don't think so. This debate was completely nonexistent before calculators appeared on the market that attempt to take the thinking away from their users. When I - and I consider myself to be a totally average part of the "poulation at large" - went to school and later university, there were no such calculators yet. There was no difference either about what was taught at school and university with respect to mathematical notation. The words "algebra", "calculus", "mathematics" all meant the same then (and still do for me now). At least outside the faculties of mathematics and computer science about which I know nothing.
You had to look at your task and formulate equations for it's solution, or alternatively copy down fitting equations from your textbook. Then you would enter values in your equations and calulate a numerical result, either using pen and paper, a slide rule or a pocket calculator. When you were graded for your work, you would get 90% of the marks for the analytical part and 10% for the correct numerical result. It didn't matter much if you or your calculator got the operator precedence wrong, at least not to an extent that would be the source of a world wide discussion on the internet.
Regards
Max
RE: What is the correct result? - Matt Agajanian - 04-21-2024 07:43 PM
Hi all.
Here’s another question that popped up. Since TI postures itself as the pinnacle of calculators in education, how is it that at what is their current order of hierarchy, they can claim that their way is the right hierarchy when Casio’s, Sharp’s, and other hierarchies are improper evaluation methods?
(04-17-2024 12:03 AM)toml_12953 Wrote:(04-16-2024 10:59 PM)Matt Agajanian Wrote: So, since terms include 4ac (from the quadratic equation), 1/3a (meaning 1 divided by 3a because 3a is a term), 32e, 29c, and so forth, why does TI of all companies put implied multiplication down the hierarchy list? You’d think TI would have a handle on the dynamics of mathematics. And then again, TI had the hierarchy of implied multiplication's higher in priority before. So why the switch?
In mathematics, you don’t write everything on one line. There’s no ambiguity since you can clearly see what terms go where. In the physical sciences, it’s tougher to determine. TI uses the mathematical conventions.
Since that is so, why does Casio provide Line IO and TI provide Classic?
RE: What is the correct result? - Maximilian Hohmann - 04-21-2024 09:44 PM
Hello!
(04-21-2024 07:43 PM)Matt Agajanian Wrote: Since TI postures itself as the pinnacle of calculators in education...
Where does Ti do that? On their "education" website (https://education.ti.com/en/) they only state: „That’s why TI is the most recommended brand by math teachers and used by millions of students each year.“
I would understand "pinnacle" in the sense of "most technically advanced", which is not something that Ti claims for themselves.
And anyway, since when do PR depatments tell the truth? ;-)
Regards
Max
RE: What is the correct result? - Steve Simpkin - 04-21-2024 09:49 PM
(04-21-2024 07:43 PM)Matt Agajanian Wrote: Hi all.
Here’s another question that popped up. Since TI postures itself as the pinnacle of calculators in education, how is it that at what is their current order of hierarchy, they can claim that their way is the right hierarchy when Casio’s, Sharp’s, and other hierarchies are improper evaluation methods?
...
That is a hard argument for TI (or Casio) to make considering "their way" has changed over the years. I am sure in both cases the changing method used to interpret implied multiplication and order of hierarchy was driven by educational customer demand based on regional customs.
Solution 11773: Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators.
TI example
Casio example
More discussions on this subject
RE: What is the correct result? - Matt Agajanian - 04-21-2024 11:18 PM
(04-21-2024 09:49 PM)Steve Simpkin Wrote:(04-21-2024 07:43 PM)Matt Agajanian Wrote: Hi all.
Here’s another question that popped up. Since TI postures itself as the pinnacle of calculators in education, how is it that at what is their current order of hierarchy, they can claim that their way is the right hierarchy when Casio’s, Sharp’s, and other hierarchies are improper evaluation methods?
...
That is a hard argument for TI (or Casio) to make considering "their way" has changed over the years. I am sure in both cases the changing method used to interpret implied multiplication and order of hierarchy was driven by educational customer demand based on regional customs.
Solution 11773: Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators.
TI example
Casio example
More discussions on this subject
So, I guess it all boils down to the ‘When in Rome, do as the Romans do.’
Although, what would you say, for example, Russia’s space & astronomy divisions & headquarters use PEJMDAS and our NASA, CalTech, JPL use PEMDAS or is there a standard between engineering, medical, scientific discipline regardless of location or organization? What then?
RE: What is the correct result? - carey - 04-22-2024 12:50 AM
(04-21-2024 11:18 PM)Matt Agajanian Wrote: Although, what would you say, for example, Russia’s space & astronomy divisions & headquarters use PEJMDAS and our NASA, CalTech, JPL use PEMDAS or is there a standard between engineering, medical, scientific discipline regardless of location or organization? What then?
NASA, JPL, and other scientific agencies conduct their business using computer software, not calculators, where operator precedence is determined by the software used.
Note that there are many more operators than + - * / (e.g., logical, relational operators and more), so PEMDAS vs PEJMDAS is a false dichotomy. For example, both MATLAB and Python’s NumPy largely align with PEMDAS, yet their documentation both list extensive and somewhat different precedences for their many operators.
RE: What is the correct result? - Matt Agajanian - 04-22-2024 01:44 AM
(04-22-2024 12:50 AM)carey Wrote: NASA, JPL, and other scientific agencies conduct their business using computer software, not calculators, where operator precedence is determined by the software used.
Note that there are many more operators than + - * / (e.g., logical, relational operators and more), so PEMDAS vs PEJMDAS is a false dichotomy. For example, both MATLAB and Python’s NumPy largely align with PEMDAS, yet their documentation both list extensive and somewhat different precedences for their many operators.
That clears things up quite a bit!
So, let me ask, would TIs adherence to
PEMDAS comply with both MATLAB and Python’s NumPy? Or is it more extensive than that?
Thanks!