You can’t keep your own horseshit straight
No idea what you're talking about, again, I've been saying the same thing the whole time
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You insist they’re not the same. How?
Not difficult, I already did in another post. If a=2 and b=3...
1/ab=1/(axb)=1/(2x3)=1/6
1/axb=1/2x3=3/2=1.5
Convention saying 1/a(b+c)2 is 1/(a(b+c)2)
There's no such convention, given it would violate The Distributive Law 🙄
By all means, humiliate yourself by splitting that hair
I'll take that as an admission that you're wrong then, given you can't defend your wrong interpretation of it (which you would know is wrong if you had read more than 1 paragraph of the book!) 😂
They’re more than equal
They're not equal at all 🙄
If a=2, b=3...
1/ab=1/(2x3)=1/6
1/axb=1/2x3=3/2=1.5
It’s an identity, which you’d understand
Nope! axb==ab is an identity, which is NOT how it's written, "illiterate fraud" as per your other comment
if you weren’t lying about being a teacher
says person who is lying about what the textbook says 🙄
Illiterate fraud
says person who thinks "means" and "equals" mean the same thing 😂
“a X b is written ab and means a times b.”
Notice that it doesn't say equals, speaking of Illiterate fraud, as per your other comment 🙄
So b * c, which is a product of the variables b and c
Nope. bc is the product of b and c. bxc is Multiplication of the 2 Terms b and c.
according to this textbook
Says person who clearly didn't read more than 2 sentences out of it 🙄
none of the examples on this particular page feature the multiplication symbol ×
and why do you think that is? Do explain. We're all waiting 😂 Spoiler alert: if you had read more than 2 sentences you would know why
That means that the expression bc is just another way of writing b×c;
No it doesn't. it means bxc is Multiplication, and bc is the product 🙄 Again you would've already known this is you had read more than 2 sentences of the book.
it is treated the same other than requiring fewer strokes of the pen
No it isn't, and again you would already know this if you had read more than 2 sentences. If a=2 and b=3, then...
1/ab=1/(2x3)=1/6
1/axb=1/2x3=3/2
this is just a custom
Nope, an actual rule of Maths. If you meant 1/axb, but wrote 1/ab, you've gonna get a different answer 🙄
That should clear up your confusion in interpreting this textbook
says person who only read 2 sentences out of it 🙄
though really, the language is clear:
It sure is when the read the rest of the page 🙄
you don’t dispute that b×c - or b * c - are products, do you
What don't you understand about only ab is the product of a and b?
Elsewhere in this thread you are clearly confused about what brackets mean
Not me, must be you! 😂
They are explained on page 20 of your textbook, where it says that you evaluate the expression inside the (innermost) brackets before doing anything else.
Until all brackets have been removed. on the very next page. 🙄 See what happens when you read more than 2 sentences out of a textbook? Who would've thought you need to read more than 2 sentences! 😂
the “distributive law” is not mentioned, because the distributive law has nothing to do with brackets
And yet, right there on Page 21, they Distribute in the last step of removing Brackets, 🙄 5(17)=85, and throughout the whole rest of the book they write Products in that form, a(b) (or just ab as the case may be).
is not an operation
Brackets aren't an operator, they are grouping symbols, and solving grouping symbols is done in the first 2 steps of order of operations, then we solve the operators.
Thus the expression 3 × (2 + 4) can be evaluated by first performing the summation inside the brackets to get 3 × 6 and then the product to get 1
3x6 isn't a Product, it's a Multiplication, done in the Multiplication step of order of operations.
The textbook then says that it is customary to omit the multiplication symbol and instead write 3(2+4)
It says you omit the multiplication sign if it's a Product, and 3x6 is not a Product. I'm not sure how many times you need to be told that 🙄
again indicating that these expressions are merely different ways of writing the same thing
Nope, completely different giving different answers
1/3x(2+4)=1/3x6=6/3=2
1/3(2+4)=1/3(6)=1/18
You have suggested that you must evaluate this as (2a+2b)² because you must “do brackets first”
Yep
this is not what “doing brackets” means.
Yes it is! 😂
Not what is outside the brackets.
Yes it is! 😂 Until all Brackets have been removed, which they can't be if you haven't Distributed yet. Again, last step of the working out...
Distributing 2 over a+b is not “doing brackets”;
Yes it is! 😂 Until all Brackets have been removed
it is multiplication and comes afterwards
Nope, it's Distribution, done in the Brackets step, before doing anything else, as per Page 21
following your textbook’s instruction to do what is inside the brackets first, this is equal to 2(4)²
Which, when you finish doing the brackets, is 8²
The next highest-priority operation is the exponent
After you have finished the Brackets 🙄
giving us 2×16
Nope. Giving us 8²=64
we now must write the × because it is an expression purely in numerals
Nope! If you write it at all, which you don't actually need to (the textbook never does), then you write (2x4)², per The Distributive Law, where you cannot remove the brackets if you haven't Distributed yet. There's no such rule as the one you just made up
The fact that these two answers are different is because
You disobeyed The Distributive Law in the second case, and the fact that you got a different answer should've been a clue to you that you did it wrong 🙄
what it means to “do brackets” and the distributive law are wrong
No, that would be your understanding is wrong, the person who only read 2 sentences 🙄 I'm not sure what you think the rest of the chapter is about.
Since I’m working off the textbook you gave
Says person who only read 2 sentences out of it 🙄
I referred liberally to things in that textbook
Yep, ignoring all the parts that prove you are wrong 🙄
I’m sure if you still disagree you will be able to back up your interpretations with reference to it
Exact same reference! 😂
it does rather seem like this rule is one established not by the fundamental laws of mathematics but by agreement as they say
You know Mathematicians tend to agree when something has been proven, right? 😂
Care to comment?
Yep, read the whole chapter 🙄
Yeah, you clearly don’t even know what a convention is, and what are math conventions and math “rules” as you put it
Says person who actually doesn't know the difference, as per Maths textbooks
You’re wrong
oh no! you better start contacting all the textbook publishers and tell them that all Maths textbooks are wrong 😂
even a 2 minute Google search would show you that and explain why
Even a 2 minute Google search will bring up Maths textbooks which prove that Google is wrong 🙄
I’m done being Google for you
Maths teachers don't use Google - that's what Maths textbooks are for
when you’re not willing to Google it yourself
says person who was unwilling to use Google to find Maths textbooks 🙄
a*b and ab are both the product of a and b,
Nope. Only ab is the product of a and b. axb is Multiplication of 2 terms
As explained by the textbook you chose
If you had read more than 2 sentences of it, you would discover that you cannot use axb to show the product, only ab 🙄
a*b2 is ab2
No it isn't 😂 1/axb²=b²/a. 1/ab²=1/ab². Welcome to why we teach students about Terms 🙄
No textbook you’re grasping for contains your made-up exception
Law is the word you're looking for, and I posted dozens of them here in this post which you keep ignoring Mr. Ostrich
They all show what I’m rubbing your nose in. You’re just full of shit.
Nope, they all show you are full of shit Mr. Ostrich. See previous link
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
There's dozens here - knock yourself out! 😂
There is no “definition of multiplication” here
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 don't know what Maths textbooks say because you were too poor to go to school? I'm sorry to hear that