You don’t, because the second 2 is associated with a Division that has to be done before the addition. Maybe go back to school and learn how to do Maths 🙄
Right, so you cannot derive precedence order from the definition of the operations. Your argument based on the definition of multiplication as repeated addition is wrong.
or you get wrong answers
This is begging the question. We are discussing whether the answers are flat wrong or whether there is a layer of interpretation. Repeating that they are wrong does nothing for this discussion, so there’s no need to bother.
You have nothing to say that I can see about why the different interpretations are impossible, or contradictory, or why they ought to qualify as “wrong” even though maths works regardless; you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority. I’m sorry, but lots of things we get taught in high school are wrong, or only partially right. I see from your profile that you are a maths teacher, so it’s actually your job to understand maths at a higher level than the level at which you teach it. It may be easier to to teach high school maths this way, but it’s not a good enough level of understanding for an educator (or for a mathematician).
Left to right is a convention (as is not writing the brackets in RPN). Brackets before Multiplication before Addition are rules.
OK, let’s try a different tack. When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system. But there is no such rule or axiom in, for example, first order Peano arithmetic. So what are you talking about? Can you find somewhere an enumeration of all the rules you’re talking about? Because maybe we’re just talking at cross-purposes: if you deviate from the axioms of Peano arithmetic then we’re fundamentally not doing arithmetic any more. But I contend that you will not find included in any axiomatisation anything which specifies order of operations. This is because from the point of view of the “rules” (i.e. the axioms) the addition and multiplication operations are just function symbols with certain properties. Even the symbols themselves are not really part of the axiomatisation; you could just as well get rid of the + symbol and write A(x, y, z) instead of “x + y = z”; you’d have the exact same arithmetic, the exact same rules.
If you’re able to answer this, we can get away from these vague terms which you keep introducing like “notation definition”, and we can instead think about what it means to be a convention versus whatever it is you mean by “rule”. (For example, Peano arithmetic has a privileged position amongst candidates for arithmetic because it encompasses our intuition about how numbers work: you can’t just take an alternative arithmetic, like say arithmetic modulo 17, and say that’s an “alternative convention” because when you add an apple to a bowl of 16 apples, they don’t all disappear. But there’s no such intuition about how to write mathematics to express a certain thing. I contend that is all convention.)
It works because it treats every operation as bracketed without writing the brackets. Also that’s only a Maths notation, not the Maths itself.
So, you understand that a notation can evaluate things in a certain order with what you call “treating every operation as bracketed without writing brackets.” What does it mean to be “bracketed without writing brackets”? There are exactly two aspects to brackets:
the symbols themselves - but we’re not writing them! So this isn’t relevant.
the effect they have - the effect on the order of evaluation of operations
So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.
So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5 and hey look, this notation now has left-to-right order of evaluation, not the usual multiplication first. If you prefer to think of there being invisible brackets there, go right ahead, but the effect is the same.
So, how do we decide whether our usual notation “has bogus brackets” or not? Convention. We could choose one way or the other. Nothing breaks if we choose one or the other. Symmetrically, we could say that left-to-right evaluation is the notation “without bogus brackets” and that BODMAS evaluation is the notation “with bogus brackets”. Which choice we make is entirely arbitrary. That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again.
They don’t actually. Welcome to most e-calcs give wrong answers because the programmers failed to deal with it correctly.
What problems does it cause? Are the problems purely that they don’t have the order of operations you expect, and so get different answers if you don’t clarify with brackets? Because that, again, is begging the question.
To re-iterate, you are in a discussion where you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition. The fact that you’ve written a post in which you document how some calculators don’t follow this convention and said that they’re wrong is not evidence of that. It’s just your opinion. Indeed, it’s really (weak) evidence that your opinion is wrong, because you’re less of an authority than the manufacturers of calculators.
On calculators, there’s something important you need to realise: basic, non-scientific, non-graphing calculators all have left-to-right order of operations. You can test this with e.g. windows calculator in “standard” mode by typing 2, +, 3, x, 5 (it will give you 25, not 17). Switch it to “scientific” mode and it will give you 17.
Why is it different? Because “standard” mode is emulating a basic calculator which has a single accumulator and performs operations on that accumulated value. When you type “x 2” you are multiplying the accumulator by 2; the calculator has already forgotten everything that you typed to get the accumulator. This was done in the early days of calculators because it was more practical when memory looked like this:
Now, you can go on about your bogus brackets until you’re blue in the face, but the fact is that this isn’t “wrong”. It has a different convention for a sensible reason and if you expect something different then it is you who are using the device wrong.
From your other comment, since having two threads seems pointless:
So if you have one “notation definition” as you call it which says that 2+2*3 means ”first add two to two, then multiply by three” and another which says “first multiply two by three, then add it to two”, why on earth do the “rules” have anything further to say about order of operations?
No we don’t. We have another notation which says to do paired operations (equivalent to being in brackets) first.
What do you mean “we don’t”? I just made the definition. It exists. This is why terms like “notation definition” are not actually helpful IMO, so let’s be precise and use terms that are either plain english (like “convention”) or mathematical (like “axiom”, “definition”, etc).
Right, so you cannot derive precedence order from the definition of the operations.
Yes you can. I’m not sure what you’re not understanding about Division before Addition 😂
Your argument based on the definition of multiplication as repeated addition is wrong
No it isn’t! 😂
We are discussing whether the answers are flat wrong or whether there is a layer of interpretation
Flat wrong, as per the rules Of Maths 🙄
Repeating that they are wrong does nothing for this discussion, so there’s no need to bother
So stop doing wrong things and I can stop saying you’re doing it wrong 😂
why they ought to qualify as “wrong” even though maths works regardless
If you have 1 2 litre bottle of milk, and 4 3 litre bottles of milk, even a 3rd grader can count up and tell you how many litres there are, and that any other answer is wrong. 🙄 2+3x4=2+3+3+3+3=14 correct 2+3x4=5x4=5+5+5+5=20 wrong See how the Maths doesn’t work regardless? 😂
you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority
Nope, I’ve proven it myself - that’s the beauty of Maths, that anyone at all can try it for themselves and find out. I’m guessing that you didn’t try it yourself 😂
lots of things we get taught in high school are wrong
says person failing to give a single such example 🙄
it’s actually your job to understand maths at a higher level than the level at which you teach it
No it isn’t. I’m required to to get the Masters degree which is required to be a teacher here, and that’s the end of it.
It may be easier to to teach high school maths this way
The correct way, yes 😂
When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system
Nope, neither.
So what are you talking about?
What don’t you understand about 20 being a wrong answer for 2+3x4??
whatever it is you mean by “rule”
Thing which results in wrong answers if disobeyed - like 2+3x4=20 - not complicated. This is what we teach to students - if you always obey all the rules then you will always get the correct answer.
arithmetic modulo 17, and say that’s an “alternative convention”
Of course not, just a different function of Maths, that doesn’t involve Arithmetic at all (other than the steps along the way in doing the long division), unlike 2+3x4 🙄
I contend that is all convention
Nope! Just a different rule to Arithmetic 🙄
What does it mean to be “bracketed without writing brackets”?
Same thing as we’re adding the 2 in 2+3 without writing a plus (or a zero) in front of the 2 - all Arithmetic starts from zero on the number-line. Maths textbooks explicitly teach this, that we can leave the sign omitted at the start if it’s a plus.
the symbols themselves - but we’re not writing them! So this isn’t relevant
Just like we aren’t writing the plus sign in 2+3 🙄
So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.
Nope. Same order as though we did write it in a notation using Brackets, same as we always start with adding the 2 even though we didn’t write a plus sign in 2+3.
So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5
No you can’t, because you get a wrong answer 🙄
this notation now has left-to-right order of evaluation
No it doesn’t, Multiplication before Addition 🙄
If you prefer to think of there being invisible brackets there
You know we were writing this without brackets for several centuries before we started using brackets in Maths, right?? 😂
So, how do we decide whether our usual notation “has bogus brackets” or not? Convention
Nope. proven rules 🙄
We could choose one way or the other.
No we can’t. Even a 3rd grader who is counting up can tell you that 🙄
Nothing breaks if we choose one or the other.
Yes it does. Again ask the 3rd grader how many litres we have, and then try doing Addition first to get that answer 😂
we could say that left-to-right evaluation is the notation “without bogus brackets”
No we can’t. Ask the 3rd grader, or even try it yourself with Cuisenaire rods
Which choice we make is entirely arbitrary
Nope. proven rules 🙄
That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again
Count up how many litres we have 🙄
What problems does it cause?
wrong answers 😂
you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition
As per Maths textbooks 😂
you’ve written a post in which you document how some calculators don’t follow this
rule
said that they’re wrong is not evidence of that
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
It’s just your opinion
Nope! proven rules as found in Maths textbooks 🙄
it’s really (weak) evidence that your opinion is wrong,
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
you’re less of an authority than the manufacturers of calculators
Demonstrably not 😂
basic, non-scientific, non-graphing calculators all have left-to-right order of operations
No they don’t! 😂
e.g. windows calculator in “standard” mode
The Windows calculator is an e-calc which was written by a programmer who didn’t check that their Maths was correct. 🙄 Now try it with any actual calculator 🙄
Why is it different?
Written by a different programmer, but one who didn’t know The Distributive Law, so even in Scientific mode it gives wrong answers to 8/2(1+3) 🙄
Because “standard” mode is emulating a basic calculator
No it isn’t. All basic calculators obey Multiplication before Addition, 🙄 and if the programmer had tried it then they would’ve found that out
performs operations on that accumulated value
Instead of using the stack, to store the Multiplication first, like all actual calculators do 🙄
When you type “x 2” you are multiplying the accumulator by 2
No, the dumb programmer is. All actual calculators did the Multiplication first and put the result on the stack
the calculator has already forgotten everything that you typed to get the accumulator
But actual calculators have put that result on the stack 🙄
This was done in the early days of calculators
No it wasn’t. All calculators “in the early days” used the stack
It has a different convention for a sensible reason
Nope, it’s just disobeying the rules of Maths because dumb programmer didn’t check their Maths was correct 🙄
it was more practical when memory looked like this:
And even then the stack existed 🙄
the fact is that this isn’t “wrong”
Yes it is! 😂 Again, ask the 3rd grader to count up and tell you the correct answer
if you expect something different then it is you who
knows the rules of Maths 🙄
What do you mean “we don’t”?
What don’t you understand about “we don’t”?
I just made the definition
Of the notation, not the rules 🙄
We have another notation which says to do paired operations (equivalent to being in brackets) first
And this notation says to do paired operations first, same as if they were in Brackets. You so nearly had it 🙄
plain english (like “convention”)
says person who keeps calling the rules “convention” 🙄
mathematical (like “axiom”, “definition”, etc)
You know we have Mathematical definitions of the difference between conventions and rules, right??
Your habit of snipping replies into tiny segments and replying shortly to each makes the discussion much harder to follow. Try and collect your thoughts into something coherent, if you can.
If you have 1 2 litre bottle of milk, and 4 3 litre bottles of milk, even a 3rd grader can count up and tell you how many litres there are, and that any other answer is wrong. 🙄 2+3x4=2+3+3+3+3=14 correct 2+3x4=5x4=5+5+5+5=20 wrong See how the Maths doesn’t work regardless? 😂
So this is the most interesting thing you’ve said.
In mathematical notation with BODMAS order of operations, you can write your 14 litres of milk as 2 + 3 x 4, sure. But if you had right-to-left order of operations you could just write 2 + (3 x 4). So why is 2 + 3 x 4 the correct way to describe the situation? Writing out a real-life situation in mathematical notation is a question of correctly using the notational conventions to express reality.
Consider another scenario where you have two three litre bottles of milk and two three litre bottles of orange juice - how much liquid do you have in total? With BODMAS order, you could not write this as 2 + 2 x 3 = 8 litres; you’d have to insert brackets: (2 + 2) x 3 = 12 litres. But with left-to-right order you could write this as 2 + 2 x 3 = 12.
So what we have are two scenarios, where one translates readily to BODMAS order without brackets, and the other translates readily to L2R order without brackets. Neither tells you which is the superior or correct order. Neither leads to a contradiction, or problems, or incorrect results, as long as it is interpreted correctly. Yes, if you incorrectly translate my scenario as 2 + 2 x 3 with BODMAS order, you get the wrong answer. But the problem is that you translated the problem into mathematical notation using L2R order, then evaluated the expression using BODMAS order.
I’ll certainly agree that translating the problem with one convention then evaluating that with another is wrong! It leads to answers that don’t reflect reality! But of course, if you translate the problem into mathematical notation with L2R order, then evaluate it with L2R order, you get the right answer, and all is fine.
Nope, I’ve proven it myself - that’s the beauty of Maths, that anyone at all can try it for themselves and find out.
This should be easy for you to verify: pick your axiomatisation and write the proof! Or link it; that’s fine too. But you’ll have a struggle given that order of operations is about notation and that is not a (first-order) mathematical concept.
Unfortunately I suspect you think that your scenario above constitutes a proof. It does not. Here is the mathematical definition of a proof in a first order theory: It is a finite sequence of formulae in the theory, where each formula in the sequence is either an axiom of the theory or follows from one or more previous formulae by some rule of inference. The proof is then said to prove the last formula in the sequence.
There is no room for milk and bottles in a proof, unless they have first order definitions in the language of your theory. But the language of arithmetic only has the symbols for addition, multiplication, successor and zero, plus the logical symbols (quantifiers, and, or, brackets).
So I can specify these fake brackets to always wrap the left-most operation first: (2 + 3) x 5
No you can’t, because you get a wrong answer 🙄
You’re trying to establish that it’s wrong. You’re still begging the question. Maybe you’re referring back to the milk, in which case, see above. Either way though, this is an example of a pointless comment; it’s adding nothing beyond restating what you’re already saying.
No we can’t. Even a 3rd grader who is counting up can tell you that 🙄
Count up how many litres we have
Yes it does. Again ask the 3rd grader how many litres we have, and then try doing Addition first to get that answer 😂
No we can’t. Ask the 3rd grader, or even try it yourself with Cuisenaire rods
Yes it is! 😂 Again, ask the 3rd grader to count up and tell you the correct answer
Your imaginary third-grader would be quite capable of looking at the milk and orange juice and writing down 2 + 2 x 3 = 12 and get the correct answer, if you taught him or her the right-to-left convention.
The Windows calculator is an e-calc which was written by a programmer who didn’t check that their Maths was correct. 🙄 Now try it with any actual calculator 🙄
Demonstrably not 😂
No they don’t! 😂
Instead of using the stack*, to store the Multiplication first, like *all actual calculators do
No, the dumb programmer is. All actual calculators did the Multiplication first and put the result on the stack
But actual calculators have put that result on the stack
No it wasn’t. All calculators “in the early days” used the stack
And even then the stack existed 🙄
Wow, 8 separate replies from you all expressing the exact same thing, and all confidently incorrect.
I have no idea how you have forgotten these old, basic calculators.
So, now we’ve established that you’re confidently incorrect about “all actual calculators” having a stack, and about Windows calculator being “wrong” in its emulation of stackless calculators, let’s bring this back to the point: calculators are perfectly usable even though their order of operations is left-to-right. As I said before: it had a different convention for a sensible reason, and if you expect something different it is you who are using the device wrong. How to use the device is written in the manual, so every user of it can use it correctly.
By the way, if you want to continue this discussion, please acknowledge that you were wrong about this. This is a simple, verifiable matter of fact that you’ve been shown to be wrong about, and if you can’t cough to that then you certainly won’t cough to something more nebulous.
wrong answers
Nope! proven rules as found in Maths textbooks 🙄
As per Maths textbook
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer
So, as above, the different calculators have different conventions. But let’s stick with textbooks. Because you are saying all through this that order of operations is not merely a convention, but a rule. So, it’s not actually about textbooks, is it? Yet they are, in fact, the best resource you have: your spilled milk establishes the opposite of what you want it to, so textbooks are all you have.
So consider, if all the textbooks were edited overnight to teach L2R order of operations, what would happen? Children would learn that to add 2 litres of milk to 3 bottles of 4 litres, they ought to write 2 + (3 x 4), which they would calculate and get 14. They would learn that to add the volume of two three litre bottles to two three litre bottles you would write 2 + 2 x 3 and get 12.
The textbooks are, in fact, how you can see that this is just a convention. If the textbooks changed, only what people write would change. The answers would stay the same.
But the textbooks you’ve been linking haven’t been about order of operations, but about the “primitive meaning of multiplication”. Yet, here are the axioms of arithmetic:
For all x (0 = S(x))
For all x, y (S(x) = S(y) -> x = y)
For all x (x + 0 = x)
For all x (x * 0 = 0)
For all x, y (x + S(y) = S(x + y))
For all x, y (x * S(y) = (x * y) + x)
The axiom schema of induction
There is no “definition of multiplication” here because when you get down to it, definitions are things for human beings, not mathematics. Axiom 6 no more (partly) “defines multiplication” than it (partly) “defines addition.”
You know we have Mathematical definitions of the difference between conventions and rules, right??
There’s a mathematical definition of an axiom in a first order theory, but there’s certainly no mathematical definition of a convention, because a convention is a social construct.
What don’t you understand about “we don’t”?
The definition exists. Saying “we don’t have it” doesn’t make sense. I’ve told it to you, so now you have it; you can choose to ignore it, but that’s just making the choice of convention I’m saying you’re making.
Nope, neither.
So… what is it then?
Same thing as we’re adding the 2 in 2+3 without writing a plus (or a zero) in front of the 2 - all Arithmetic starts from zero on the number-line. Maths textbooks explicitly teach this, that we can leave the sign omitted at the start if it’s a plus.
In first-order arithmetic, the + symbol is a binary operation. We’re not “leaving it out” in front of the 2, because it would make no sense to put it there.
Your repeated talk of “wrong answers” makes it sound like you’re a slave to the test. A test has right and wrong answers, after all, and if you read 2 + 2 x 3 on a test and answer 12 you’d be marked wrong. But your job is to establish not that the answer is wrong in this situation, but that if you changed the test then it would be wrong. How are you going to do that? So far you have not even tried to write down what it would mean for the test to be wrong. But I can lay out my definition of “it’s a matter of convention” easily: It’s a matter of convention because humans have agreed to do it one way even though all of maths, all totalling of milk and orange juice, everything could be done another way, and be consistent with itself and with physical reality. Maybe you can say what you find defective with that.
snipping replies into tiny segments and replying shortly to each makes the discussion much harder to follow
Says person who did it in a random order, and included stuff that wasn’t even in this thread to begin with, thus making it impossible to follow 🙄
this is the most interesting thing you’ve said
You on the other hand haven’t said anything interesting, so do us all a favour and give it a rest
you can write your 14 litres of milk as 2 + 3 x 4
You “can” write it the way it’s always been written, yes 😂
But if you had right-to-left order of operations
Which we don’t 🙄
you could not write this as 2 + 2 x 3 = 8 litres
Right, you would write 3x2+3x2 😂
you’d have to insert brackets: (2 + 2) x 3 = 12 litres
Or you just write it correctly to begin with, then Factorise
But with left-to-right order you could write this as 2 + 2 x 3 = 12
No you can’t. As you already pointed out 2+2x3=8. 😂 Have you forgotten that we already do evaluate left to right??
where one translates readily to BODMAS order without brackets
Dating back many centuries before we even started using brackets in Maths 😂
the other translates readily to L2R order without brackets
Umm, it’s the same one 😂
interpreted correctly
Welcome to the order of operations rules - so glad you could finally join us
Yes, if you incorrectly translate my scenario as 2 + 2 x 3 with BODMAS order, you get the wrong answer
What you mean is you get the wrong answer, having written it out wrongly to begin with 🙄
the problem into mathematical notation using L2R order, then evaluated the expression using BODMAS order
They’re the same order 😂
the problem with one convention then evaluating that with another is wrong!
No it isn’t! 😂 All conventions give the same answer. Disobeying the rules on the other hand…
axiomatisation and write the proof
Umm, there’s no axioms involved, and I already showed you the proof 🙄
order of operations is about notation
Nope. It’s about rules. That’s why everyone the world over gets the same answers regardless of the notation they use in the different countries
constitutes a proof. It does not
says someone revealing they only know about the two types of proof, not all the others ones as well 🙄
Here is the mathematical definition of a proof in a first order theory
Which is one type of proof 🙄
no room for milk and bottles in a proof
There’s room for Cuisenaire rods though. Welcome to even a 3rd grader can prove it 😂
trying to establish that it’s wrong
I already proved it’s wrong 🙄
it’s adding nothing beyond restating what you’re already saying
And yet, you keep ignoring that it’s been proven correct Mr. Ostrich, hence I need to keep repeating it 🙄
imaginary third-grader
I can assure you that they aren’t imaginary! 😂
writing down 2 + 2 x 3 = 12
Ah, nope! They would write 3x2+3x2
if you taught him or her the right-to-left convention
We taught them first how to use Cuisenaire rods, then the order of operations rules, which follows on logically from there 🙄
all confidently incorrect.
says person about to prove that they are the one who is confidently incorrect… 😂
Note especially the phrase: “Many simple calculators without a stack”
Note the lack of a reference 🙄
chain calculation mode) is commonly employed on most general-purpose calculators
No it isn’t. It’s only employed by calculators designed to use chain calculations, which is another specialist, niche market, like RPN calculators. Note again the lack of a reference
an example of a calculator manual from the 70s showing (in Example 6) that the order of operations is left-to-right
BWAHAHAHAHAHAHA! No it doesn’t! 🤣🤣🤣 It shows you to press the +/= button after the bracketed part in order to evaluate that first, because, if you don’t, it will evaluate the Multiplication first, as per the order of operations rules, which it will use the stack for. 😂 When you press the x button, the parser know you meant the previous button press to be used as an equals and not as addition. You need to work on your reading/comprehension skills dude
the successor
A chain calculator, so this is just you rehashing your RPN argument with a different, niche notation
you have forgotten these old, basic calculators
says person who forgot to check that the manual agrees before posting it, leading to proof that they are the ones who have forgotten how they work! 🤣🤣🤣
now we’ve established that you’re confidently incorrect
No, we’ve established that you are the one who is confidently incorrect 😂
Windows calculator being “wrong” in its emulation of stackless calculators
We’ve established that isn’t what it’s doing, given it’s not called Chain mode, it’s called Standard mode, which it most definitely isn’t! 😂
let’s bring this back to the point
Yep, that point being that simple calculators, like the first one, will say 2+3x4=14. To get 20 you have to do 2+3=x4 😂
even though their order of operations is left-to-right
only chain calculators do it left to right. You’re making a false equivalence argument, just like RPN was a false equivalence argument
I said before: it had a different convention for a sensible reason
Which you just proved the first one doesn’t have a “different convention”. 😂 The second one does, but again that’s a false equivalence argument to all other calculators (same for RPN)
if you expect something different it is you who are using the device wrong
You proved they both do exactly what I expect 😂
How to use the device is written in the manual
Which you didn’t read carefully 🤣🤣🤣
so every user of it can use it correctly
As I have been, the whole time
if you want to continue this discussion, please acknowledge that you were wrong about this.
Except you just proved that you were the one who was wrong about this! 🤣🤣🤣 I expect you are now going to acknowledge that you were wrong about this, because otherwise you’re exposing yourself as a hypocrite
This is a simple, verifiable matter of fact that you’ve been shown to be wrong about
Nope, you were shown to be wrong 🤣🤣🤣
as above, the different calculators have different conventions
As above, only niche calculators like RPN and Chain have different conventions, and it’s right there in their manual, that you didn’t read carefully
all through this that order of operations is not merely a convention, but a rule. So, it’s not actually about textbooks
Which part didn’t you understand about the rules can be found in Maths textbooks?
your spilled milk establishes the opposite of what you want it to
Umm, no it doesn’t. It establishes that there is only one correct answer to 2+3x4, that being 14
textbooks are all you have
and calculators, and Cuisenaire rods, and counting up, and proofs 😂
if all the textbooks were edited overnight to teach L2R order of operations
They already do teach left to right! 😂
Children would learn that to add 2 litres of milk to 3 bottles of 4 litres, they ought to write 2 + (3 x 4)
No, they would learn the same thing they learn now 2+3x4. You know they haven’t been taught about brackets yet, right? They don’t learn about brackets until Year 5
The textbooks are, in fact, how you can see that this is just a convention
No, Cuisenaire rods show that this is a rule. 🙄 That’s why kids are shown how to use them before they first learn how to multiply
If the textbooks changed, only what people write would change
Because notations change but the rules don’t 🙄
you’ve been linking haven’t been about order of operations
In other words, not the right tool for the job. Glad you finally worked that out! 😂
a convention is a social construct
And the rules aren’t 🙄
The definition exists
In your mind maybe, not in Maths textbooks, as I would’ve told you at the time (wherever it was - you’re now referring to something that isn’t even in this thread originally, so I don’t even know what you’re talking about anymore)
Saying “we don’t have it” doesn’t make sense
And I still don’t know where you’re having trouble in understanding that
I’ve told it to you
And I told you that we don’t have that definition 🙄
so now you have it;
And I told you that you were wrong 🙄
the choice of convention I’m saying you’re making
I’ve been talking about rules the whole time Mr. Ostrich
what is it then?
Proof by disproof 🙄
first-order arithmetic, the + symbol is a binary operation
So now you’re resorting to the minority of the population that has studied that at University. Way to admit you’re wrong in the general case 😂
We’re not “leaving it out” in front of the 2
High school Maths textbooks, which everyone does, explicitly say it’s there
So far you have not even tried to write down what it would mean for the test to be wrong
What part didn’t you understand in 20 litres is the wrong answer?
I can lay out my definition of “it’s a matter of convention” easily
Because you keep ignoring that they are proven rules Mr. Ostrich 🙄
everything could be done another way
Actually it can’t. Go ahead and try, and you’ll find that out eventually
be consistent with itself and with physical reality
That’s the exact thing which prevents it from being done another way 🙄
You have declined to admit to a simple error you made (that early calculators lacked a stack, and that basic four function calculators all did and still do)
There’s no point having a discussion with someone so stubborn that they can’t admit a single mistake. I’m not sure whether you’re trying to wind people up or just a bit dim, but while it’s fun explaining mathematics - especially parts like this which touch on the formal parts and the distinction between maths itself and mathematical convention - this conversation is like trying to explain something to a particularly stuck-up dog. Except dogs aren’t capable of being snarky.
The real tragedy is that you claim to be out there teaching kids this overcomplicated and false drivel.
Anyway, if you want to continue the discussion - maybe with a whiteboard would be best - I’m quite happy to, but only if you show that you’re not just a troll. You can do that by admitting that you were wrong to say that all calculators have stacks, which shouldn’t be hard if you have a shred of honesty, because I showed you two examples.
Another way you could demonstrate your good faith by admitting a mistake is admitting that when you said, in this post that:
Maths textbooks never use the word “juxtaposition”
you were wrong, and that this screenshot which I believe you first linked demonstrates it. In case that image disappears, it’s from Advanced Algebra by J.V. Collins, pg 6.
On page 3, the concept of juxtaposition is introduced.
So that’s an extra way you could demonstrate your good faith, by admitting to an error on your part not central to your argument that will show you actually are capable of admitting error.
You have declined to admit to a simple error you made
Not me, must be you! 😂
that early calculators lacked a stack,
They didn’t 🙄
that basic four function calculators all did and still do
Have a stack, yes. I have one and it quite happily says that 2+3x4=14, something it can’t do without putting “2+” on the stack while it does the 3x4 first 🙄
There’s no point having a discussion with someone so stubborn that they can’t admit a single mistake.
says someone too stubborn to admit making a mistake 🙄
I’m not sure whether you’re trying to wind people up or just a bit dim
Neither. I’m the one doing fact-checks with actual, you know, facts, like my simple calculator having a stack and correctly evaluating 2+3x4=14. It’s the one I had in Primary school. The one in the first manual works the exact same way
this conversation is like trying to explain something to a particularly stuck-up dog
So maybe start listening to what I’ve been trying to tell you then. 🙄 It’s all there in textbooks, if you just decide to read more than 2 sentences out of them.
The real tragedy is that you claim to be out there teaching kids this overcomplicated and false drivel.
Facts, as per the syllabus and Maths textbooks. Again, you need to read more than 2 sentences to discover that 🙄
only if you show that you’re not just a troll.
says person who has thus far refused to read more than 2 sentences out of the textbook 🙄
You can do that by admitting that you were wrong to say that all calculators have stacks
I wasn’t wrong 🙄 The first manual that was linked to proved it. If you don’t press the +/= button before the multiply then it will put the first part on the stack and evaluate the multiplication first, something it doesn’t do if you press the +/= first to make it evaluate what you have typed in so far. 🙄 Every calculator will evaluate what you have typed in so far if you press the equals button, as pointed out in the first manual
because I showed you two examples
The first of which had a stack 🙄 the second of which was a chain calculator, designed to work that way. You’re the one being dishonest
you were wrong
No I wasn’t
that this screenshot
Which is a 1912 textbook. It also calls Factorising “Collections”, and The Distributive Law “The Law of Distribution”, and Products “Multiplication”. Guess what? The language has changed a little in the last 110 years 🙄
it’s from Advanced Algebra by J.V. Collins, pg 6
Yep, published in 1912
On page 3, the concept of juxtaposition is introduced
And we now call them Products. 🙄 You can see them being called that in Modern Algebra, which was published in 1965. In fact, in Lennes’ infamous 1917 letter, he used the word Product (but didn’t understand, as shown by his letter), so the language had already changed then
admitting to an error on your part
There was no error. The language has changed since 1912 🙄
you actually are capable of admitting error
Of course I am. Doesn’t mean I’m going to “admit” to an error when there is none 🙄
You failed to demonstrate any good faith so this is the end of this conversation. Your reply reveals that you even understand that you were wrong (“it’s designed that way”; “the language changed”) but are so prideful, so averse to ceding ground, that you just… can’t… say it!
I’m not sure you have enough theory of mind to understand what that’s like for a normal interlocutor, unfortunately.
The children you really ought to stop teaching are more mature than this. You’re an embarrassment to the profession.
Right, so you cannot derive precedence order from the definition of the operations. Your argument based on the definition of multiplication as repeated addition is wrong.
This is begging the question. We are discussing whether the answers are flat wrong or whether there is a layer of interpretation. Repeating that they are wrong does nothing for this discussion, so there’s no need to bother.
You have nothing to say that I can see about why the different interpretations are impossible, or contradictory, or why they ought to qualify as “wrong” even though maths works regardless; you’ve just heard a school-level maths teacher tell you it’s done one way and believe that’s the highest possible authority. I’m sorry, but lots of things we get taught in high school are wrong, or only partially right. I see from your profile that you are a maths teacher, so it’s actually your job to understand maths at a higher level than the level at which you teach it. It may be easier to to teach high school maths this way, but it’s not a good enough level of understanding for an educator (or for a mathematician).
OK, let’s try a different tack. When I hear the word “rules”, I think you’re talking either about a rule of inference in first order logic or an axiom in a first-order system. But there is no such rule or axiom in, for example, first order Peano arithmetic. So what are you talking about? Can you find somewhere an enumeration of all the rules you’re talking about? Because maybe we’re just talking at cross-purposes: if you deviate from the axioms of Peano arithmetic then we’re fundamentally not doing arithmetic any more. But I contend that you will not find included in any axiomatisation anything which specifies order of operations. This is because from the point of view of the “rules” (i.e. the axioms) the addition and multiplication operations are just function symbols with certain properties. Even the symbols themselves are not really part of the axiomatisation; you could just as well get rid of the + symbol and write A(x, y, z) instead of “x + y = z”; you’d have the exact same arithmetic, the exact same rules.
If you’re able to answer this, we can get away from these vague terms which you keep introducing like “notation definition”, and we can instead think about what it means to be a convention versus whatever it is you mean by “rule”. (For example, Peano arithmetic has a privileged position amongst candidates for arithmetic because it encompasses our intuition about how numbers work: you can’t just take an alternative arithmetic, like say arithmetic modulo 17, and say that’s an “alternative convention” because when you add an apple to a bowl of 16 apples, they don’t all disappear. But there’s no such intuition about how to write mathematics to express a certain thing. I contend that is all convention.)
So, you understand that a notation can evaluate things in a certain order with what you call “treating every operation as bracketed without writing brackets.” What does it mean to be “bracketed without writing brackets”? There are exactly two aspects to brackets:
So what you’re admitting with these phantom brackets is that a notation can evaluate operations in a different order, even though there are no written brackets.
So I can specify these fake brackets to always wrap the left-most operation first:
(x 5 and hey look, this notation now has left-to-right order of evaluation, not the usual multiplication first. If you prefer to think of there being invisible brackets there, go right ahead, but the effect is the same.2 + 3)So, how do we decide whether our usual notation “has bogus brackets” or not? Convention. We could choose one way or the other. Nothing breaks if we choose one or the other. Symmetrically, we could say that left-to-right evaluation is the notation “without bogus brackets” and that BODMAS evaluation is the notation “with bogus brackets”. Which choice we make is entirely arbitrary. That is, unless you can find a compelling reason why one is right and the other wrong, rather than just saying it once again.
What problems does it cause? Are the problems purely that they don’t have the order of operations you expect, and so get different answers if you don’t clarify with brackets? Because that, again, is begging the question.
To re-iterate, you are in a discussion where you’re trying to establish that it’s a fundamental law of maths that you must do multiplication before addition. The fact that you’ve written a post in which you document how some calculators don’t follow this convention and said that they’re wrong is not evidence of that. It’s just your opinion. Indeed, it’s really (weak) evidence that your opinion is wrong, because you’re less of an authority than the manufacturers of calculators.
On calculators, there’s something important you need to realise: basic, non-scientific, non-graphing calculators all have left-to-right order of operations. You can test this with e.g. windows calculator in “standard” mode by typing 2, +, 3, x, 5 (it will give you 25, not 17). Switch it to “scientific” mode and it will give you 17.
Why is it different? Because “standard” mode is emulating a basic calculator which has a single accumulator and performs operations on that accumulated value. When you type “x 2” you are multiplying the accumulator by 2; the calculator has already forgotten everything that you typed to get the accumulator. This was done in the early days of calculators because it was more practical when memory looked like this:
Now, you can go on about your bogus brackets until you’re blue in the face, but the fact is that this isn’t “wrong”. It has a different convention for a sensible reason and if you expect something different then it is you who are using the device wrong.
From your other comment, since having two threads seems pointless:
What do you mean “we don’t”? I just made the definition. It exists. This is why terms like “notation definition” are not actually helpful IMO, so let’s be precise and use terms that are either plain english (like “convention”) or mathematical (like “axiom”, “definition”, etc).
Yes you can. I’m not sure what you’re not understanding about Division before Addition 😂
No it isn’t! 😂
Flat wrong, as per the rules Of Maths 🙄
So stop doing wrong things and I can stop saying you’re doing it wrong 😂
If you have 1 2 litre bottle of milk, and 4 3 litre bottles of milk, even a 3rd grader can count up and tell you how many litres there are, and that any other answer is wrong. 🙄 2+3x4=2+3+3+3+3=14 correct 2+3x4=5x4=5+5+5+5=20 wrong See how the Maths doesn’t work regardless? 😂
Nope, I’ve proven it myself - that’s the beauty of Maths, that anyone at all can try it for themselves and find out. I’m guessing that you didn’t try it yourself 😂
says person failing to give a single such example 🙄
No it isn’t. I’m required to to get the Masters degree which is required to be a teacher here, and that’s the end of it.
The correct way, yes 😂
Nope, neither.
What don’t you understand about 20 being a wrong answer for 2+3x4??
Thing which results in wrong answers if disobeyed - like 2+3x4=20 - not complicated. This is what we teach to students - if you always obey all the rules then you will always get the correct answer.
Of course not, just a different function of Maths, that doesn’t involve Arithmetic at all (other than the steps along the way in doing the long division), unlike 2+3x4 🙄
Nope! Just a different rule to Arithmetic 🙄
Same thing as we’re adding the 2 in 2+3 without writing a plus (or a zero) in front of the 2 - all Arithmetic starts from zero on the number-line. Maths textbooks explicitly teach this, that we can leave the sign omitted at the start if it’s a plus.
Just like we aren’t writing the plus sign in 2+3 🙄
Nope. Same order as though we did write it in a notation using Brackets, same as we always start with adding the 2 even though we didn’t write a plus sign in 2+3.
No you can’t, because you get a wrong answer 🙄
No it doesn’t, Multiplication before Addition 🙄
You know we were writing this without brackets for several centuries before we started using brackets in Maths, right?? 😂
Nope. proven rules 🙄
No we can’t. Even a 3rd grader who is counting up can tell you that 🙄
Yes it does. Again ask the 3rd grader how many litres we have, and then try doing Addition first to get that answer 😂
No we can’t. Ask the 3rd grader, or even try it yourself with Cuisenaire rods
Nope. proven rules 🙄
Count up how many litres we have 🙄
wrong answers 😂
As per Maths textbooks 😂
rule
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
Nope! proven rules as found in Maths textbooks 🙄
says person ignoring the Maths textbooks I quoted and the actual calculators giving the correct answer 🙄
Demonstrably not 😂
No they don’t! 😂
The Windows calculator is an e-calc which was written by a programmer who didn’t check that their Maths was correct. 🙄 Now try it with any actual calculator 🙄
Written by a different programmer, but one who didn’t know The Distributive Law, so even in Scientific mode it gives wrong answers to 8/2(1+3) 🙄
No it isn’t. All basic calculators obey Multiplication before Addition, 🙄 and if the programmer had tried it then they would’ve found that out
Instead of using the stack, to store the Multiplication first, like all actual calculators do 🙄
No, the dumb programmer is. All actual calculators did the Multiplication first and put the result on the stack
But actual calculators have put that result on the stack 🙄
No it wasn’t. All calculators “in the early days” used the stack
Nope, it’s just disobeying the rules of Maths because dumb programmer didn’t check their Maths was correct 🙄
And even then the stack existed 🙄
Yes it is! 😂 Again, ask the 3rd grader to count up and tell you the correct answer
knows the rules of Maths 🙄
What don’t you understand about “we don’t”?
Of the notation, not the rules 🙄
And this notation says to do paired operations first, same as if they were in Brackets. You so nearly had it 🙄
says person who keeps calling the rules “convention” 🙄
You know we have Mathematical definitions of the difference between conventions and rules, right??
Your habit of snipping replies into tiny segments and replying shortly to each makes the discussion much harder to follow. Try and collect your thoughts into something coherent, if you can.
So this is the most interesting thing you’ve said.
In mathematical notation with BODMAS order of operations, you can write your 14 litres of milk as 2 + 3 x 4, sure. But if you had right-to-left order of operations you could just write 2 + (3 x 4). So why is 2 + 3 x 4 the correct way to describe the situation? Writing out a real-life situation in mathematical notation is a question of correctly using the notational conventions to express reality.
Consider another scenario where you have two three litre bottles of milk and two three litre bottles of orange juice - how much liquid do you have in total? With BODMAS order, you could not write this as 2 + 2 x 3 = 8 litres; you’d have to insert brackets: (2 + 2) x 3 = 12 litres. But with left-to-right order you could write this as 2 + 2 x 3 = 12.
So what we have are two scenarios, where one translates readily to BODMAS order without brackets, and the other translates readily to L2R order without brackets. Neither tells you which is the superior or correct order. Neither leads to a contradiction, or problems, or incorrect results, as long as it is interpreted correctly. Yes, if you incorrectly translate my scenario as 2 + 2 x 3 with BODMAS order, you get the wrong answer. But the problem is that you translated the problem into mathematical notation using L2R order, then evaluated the expression using BODMAS order.
I’ll certainly agree that translating the problem with one convention then evaluating that with another is wrong! It leads to answers that don’t reflect reality! But of course, if you translate the problem into mathematical notation with L2R order, then evaluate it with L2R order, you get the right answer, and all is fine.
This should be easy for you to verify: pick your axiomatisation and write the proof! Or link it; that’s fine too. But you’ll have a struggle given that order of operations is about notation and that is not a (first-order) mathematical concept.
Unfortunately I suspect you think that your scenario above constitutes a proof. It does not. Here is the mathematical definition of a proof in a first order theory: It is a finite sequence of formulae in the theory, where each formula in the sequence is either an axiom of the theory or follows from one or more previous formulae by some rule of inference. The proof is then said to prove the last formula in the sequence.
There is no room for milk and bottles in a proof, unless they have first order definitions in the language of your theory. But the language of arithmetic only has the symbols for addition, multiplication, successor and zero, plus the logical symbols (quantifiers, and, or, brackets).
You’re trying to establish that it’s wrong. You’re still begging the question. Maybe you’re referring back to the milk, in which case, see above. Either way though, this is an example of a pointless comment; it’s adding nothing beyond restating what you’re already saying.
Your imaginary third-grader would be quite capable of looking at the milk and orange juice and writing down 2 + 2 x 3 = 12 and get the correct answer, if you taught him or her the right-to-left convention.
Wow, 8 separate replies from you all expressing the exact same thing, and all confidently incorrect.
https://en.wikipedia.org/wiki/Order_of_operations#Calculators
Note especially the phrase: “Many simple calculators without a stack”
https://en.wikipedia.org/wiki/Calculator_input_methods#Chain
Here is an example of a calculator manual from the 70s showing (in Example 6) that the order of operations is left-to-right: https://www.wass.net/manuals/Sinclair Executive.pdf
And the successor, one of the first affordable pocket calculators (bottom of page 8): https://www.wass.net/manuals/Sinclair Cambridge Scientific.pdf
I have no idea how you have forgotten these old, basic calculators.
So, now we’ve established that you’re confidently incorrect about “all actual calculators” having a stack, and about Windows calculator being “wrong” in its emulation of stackless calculators, let’s bring this back to the point: calculators are perfectly usable even though their order of operations is left-to-right. As I said before: it had a different convention for a sensible reason, and if you expect something different it is you who are using the device wrong. How to use the device is written in the manual, so every user of it can use it correctly.
By the way, if you want to continue this discussion, please acknowledge that you were wrong about this. This is a simple, verifiable matter of fact that you’ve been shown to be wrong about, and if you can’t cough to that then you certainly won’t cough to something more nebulous.
So, as above, the different calculators have different conventions. But let’s stick with textbooks. Because you are saying all through this that order of operations is not merely a convention, but a rule. So, it’s not actually about textbooks, is it? Yet they are, in fact, the best resource you have: your spilled milk establishes the opposite of what you want it to, so textbooks are all you have.
So consider, if all the textbooks were edited overnight to teach L2R order of operations, what would happen? Children would learn that to add 2 litres of milk to 3 bottles of 4 litres, they ought to write 2 + (3 x 4), which they would calculate and get 14. They would learn that to add the volume of two three litre bottles to two three litre bottles you would write 2 + 2 x 3 and get 12.
The textbooks are, in fact, how you can see that this is just a convention. If the textbooks changed, only what people write would change. The answers would stay the same.
But the textbooks you’ve been linking haven’t been about order of operations, but about the “primitive meaning of multiplication”. Yet, here are the axioms of arithmetic:
There is no “definition of multiplication” here because when you get down to it, definitions are things for human beings, not mathematics. Axiom 6 no more (partly) “defines multiplication” than it (partly) “defines addition.”
There’s a mathematical definition of an axiom in a first order theory, but there’s certainly no mathematical definition of a convention, because a convention is a social construct.
The definition exists. Saying “we don’t have it” doesn’t make sense. I’ve told it to you, so now you have it; you can choose to ignore it, but that’s just making the choice of convention I’m saying you’re making.
So… what is it then?
In first-order arithmetic, the + symbol is a binary operation. We’re not “leaving it out” in front of the 2, because it would make no sense to put it there.
Your repeated talk of “wrong answers” makes it sound like you’re a slave to the test. A test has right and wrong answers, after all, and if you read 2 + 2 x 3 on a test and answer 12 you’d be marked wrong. But your job is to establish not that the answer is wrong in this situation, but that if you changed the test then it would be wrong. How are you going to do that? So far you have not even tried to write down what it would mean for the test to be wrong. But I can lay out my definition of “it’s a matter of convention” easily: It’s a matter of convention because humans have agreed to do it one way even though all of maths, all totalling of milk and orange juice, everything could be done another way, and be consistent with itself and with physical reality. Maybe you can say what you find defective with that.
Says person who did it in a random order, and included stuff that wasn’t even in this thread to begin with, thus making it impossible to follow 🙄
You on the other hand haven’t said anything interesting, so do us all a favour and give it a rest
You “can” write it the way it’s always been written, yes 😂
Which we don’t 🙄
Right, you would write 3x2+3x2 😂
Or you just write it correctly to begin with, then Factorise
No you can’t. As you already pointed out 2+2x3=8. 😂 Have you forgotten that we already do evaluate left to right??
Dating back many centuries before we even started using brackets in Maths 😂
Umm, it’s the same one 😂
Welcome to the order of operations rules - so glad you could finally join us
What you mean is you get the wrong answer, having written it out wrongly to begin with 🙄
They’re the same order 😂
No it isn’t! 😂 All conventions give the same answer. Disobeying the rules on the other hand…
Umm, there’s no axioms involved, and I already showed you the proof 🙄
Nope. It’s about rules. That’s why everyone the world over gets the same answers regardless of the notation they use in the different countries
says someone revealing they only know about the two types of proof, not all the others ones as well 🙄
Which is one type of proof 🙄
There’s room for Cuisenaire rods though. Welcome to even a 3rd grader can prove it 😂
I already proved it’s wrong 🙄
And yet, you keep ignoring that it’s been proven correct Mr. Ostrich, hence I need to keep repeating it 🙄
I can assure you that they aren’t imaginary! 😂
Ah, nope! They would write 3x2+3x2
We taught them first how to use Cuisenaire rods, then the order of operations rules, which follows on logically from there 🙄
says person about to prove that they are the one who is confidently incorrect… 😂
Note the lack of a reference 🙄
No it isn’t. It’s only employed by calculators designed to use chain calculations, which is another specialist, niche market, like RPN calculators. Note again the lack of a reference
BWAHAHAHAHAHAHA! No it doesn’t! 🤣🤣🤣 It shows you to press the +/= button after the bracketed part in order to evaluate that first, because, if you don’t, it will evaluate the Multiplication first, as per the order of operations rules, which it will use the stack for. 😂 When you press the x button, the parser know you meant the previous button press to be used as an equals and not as addition. You need to work on your reading/comprehension skills dude
A chain calculator, so this is just you rehashing your RPN argument with a different, niche notation
says person who forgot to check that the manual agrees before posting it, leading to proof that they are the ones who have forgotten how they work! 🤣🤣🤣
No, we’ve established that you are the one who is confidently incorrect 😂
We’ve established that isn’t what it’s doing, given it’s not called Chain mode, it’s called Standard mode, which it most definitely isn’t! 😂
Yep, that point being that simple calculators, like the first one, will say 2+3x4=14. To get 20 you have to do 2+3=x4 😂
only chain calculators do it left to right. You’re making a false equivalence argument, just like RPN was a false equivalence argument
Which you just proved the first one doesn’t have a “different convention”. 😂 The second one does, but again that’s a false equivalence argument to all other calculators (same for RPN)
You proved they both do exactly what I expect 😂
Which you didn’t read carefully 🤣🤣🤣
As I have been, the whole time
Except you just proved that you were the one who was wrong about this! 🤣🤣🤣 I expect you are now going to acknowledge that you were wrong about this, because otherwise you’re exposing yourself as a hypocrite
Nope, you were shown to be wrong 🤣🤣🤣
As above, only niche calculators like RPN and Chain have different conventions, and it’s right there in their manual, that you didn’t read carefully
Which part didn’t you understand about the rules can be found in Maths textbooks?
Umm, no it doesn’t. It establishes that there is only one correct answer to 2+3x4, that being 14
and calculators, and Cuisenaire rods, and counting up, and proofs 😂
They already do teach left to right! 😂
No, they would learn the same thing they learn now 2+3x4. You know they haven’t been taught about brackets yet, right? They don’t learn about brackets until Year 5
No, Cuisenaire rods show that this is a rule. 🙄 That’s why kids are shown how to use them before they first learn how to multiply
Because notations change but the rules don’t 🙄
There’s dozens here - knock yourself out! 😂
In other words, not the right tool for the job. Glad you finally worked that out! 😂
And the rules aren’t 🙄
In your mind maybe, not in Maths textbooks, as I would’ve told you at the time (wherever it was - you’re now referring to something that isn’t even in this thread originally, so I don’t even know what you’re talking about anymore)
And I still don’t know where you’re having trouble in understanding that
And I told you that we don’t have that definition 🙄
And I told you that you were wrong 🙄
I’ve been talking about rules the whole time Mr. Ostrich
Proof by disproof 🙄
So now you’re resorting to the minority of the population that has studied that at University. Way to admit you’re wrong in the general case 😂
High school Maths textbooks, which everyone does, explicitly say it’s there
What part didn’t you understand in 20 litres is the wrong answer?
Because you keep ignoring that they are proven rules Mr. Ostrich 🙄
Actually it can’t. Go ahead and try, and you’ll find that out eventually
That’s the exact thing which prevents it from being done another way 🙄
You have declined to admit to a simple error you made (that early calculators lacked a stack, and that basic four function calculators all did and still do)
There’s no point having a discussion with someone so stubborn that they can’t admit a single mistake. I’m not sure whether you’re trying to wind people up or just a bit dim, but while it’s fun explaining mathematics - especially parts like this which touch on the formal parts and the distinction between maths itself and mathematical convention - this conversation is like trying to explain something to a particularly stuck-up dog. Except dogs aren’t capable of being snarky.
The real tragedy is that you claim to be out there teaching kids this overcomplicated and false drivel.
Anyway, if you want to continue the discussion - maybe with a whiteboard would be best - I’m quite happy to, but only if you show that you’re not just a troll. You can do that by admitting that you were wrong to say that all calculators have stacks, which shouldn’t be hard if you have a shred of honesty, because I showed you two examples.
Another way you could demonstrate your good faith by admitting a mistake is admitting that when you said, in this post that:
you were wrong, and that this screenshot which I believe you first linked demonstrates it. In case that image disappears, it’s from Advanced Algebra by J.V. Collins, pg 6.
On page 3, the concept of juxtaposition is introduced.
So that’s an extra way you could demonstrate your good faith, by admitting to an error on your part not central to your argument that will show you actually are capable of admitting error.
Not me, must be you! 😂
They didn’t 🙄
Have a stack, yes. I have one and it quite happily says that 2+3x4=14, something it can’t do without putting “2+” on the stack while it does the 3x4 first 🙄
says someone too stubborn to admit making a mistake 🙄
Neither. I’m the one doing fact-checks with actual, you know, facts, like my simple calculator having a stack and correctly evaluating 2+3x4=14. It’s the one I had in Primary school. The one in the first manual works the exact same way
So maybe start listening to what I’ve been trying to tell you then. 🙄 It’s all there in textbooks, if you just decide to read more than 2 sentences out of them.
Facts, as per the syllabus and Maths textbooks. Again, you need to read more than 2 sentences to discover that 🙄
says person who has thus far refused to read more than 2 sentences out of the textbook 🙄
I wasn’t wrong 🙄 The first manual that was linked to proved it. If you don’t press the +/= button before the multiply then it will put the first part on the stack and evaluate the multiplication first, something it doesn’t do if you press the +/= first to make it evaluate what you have typed in so far. 🙄 Every calculator will evaluate what you have typed in so far if you press the equals button, as pointed out in the first manual
The first of which had a stack 🙄 the second of which was a chain calculator, designed to work that way. You’re the one being dishonest
No I wasn’t
Which is a 1912 textbook. It also calls Factorising “Collections”, and The Distributive Law “The Law of Distribution”, and Products “Multiplication”. Guess what? The language has changed a little in the last 110 years 🙄
Yep, published in 1912
And we now call them Products. 🙄 You can see them being called that in Modern Algebra, which was published in 1965. In fact, in Lennes’ infamous 1917 letter, he used the word Product (but didn’t understand, as shown by his letter), so the language had already changed then
There was no error. The language has changed since 1912 🙄
Of course I am. Doesn’t mean I’m going to “admit” to an error when there is none 🙄
You failed to demonstrate any good faith so this is the end of this conversation. Your reply reveals that you even understand that you were wrong (“it’s designed that way”; “the language changed”) but are so prideful, so averse to ceding ground, that you just… can’t… say it!
I’m not sure you have enough theory of mind to understand what that’s like for a normal interlocutor, unfortunately.
The children you really ought to stop teaching are more mature than this. You’re an embarrassment to the profession.