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 😂
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
Every textbook with an answer key says you’re full of shit
Says person who can't find a Maths textbook that says a(bxc)=(abxac) 🙄
being wrong on purpose is the point
I'm gonna presume that's why you keep claiming a(bxc)=(abxac) 🙄
The answer in either case is shut the fuck up
says person still not doing that 😂
2(n)2 is 2n2
No it isn't! 😂 2xn² is
Anything else is an inane complication nobody else believes in or uses or needs
Except for authors of Maths textbooks 😂
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??
Tell that obvious to over half the population who get this wrong
Way less than half actually. No teachers or students ever get this wrong, only adults who have forgotten the rules, and poll after poll puts this down around 40-45% of adults.
Ok, then explain prefix and postfix, where these conventions don’t apply
The conventions don't apply, the rules still apply. Maths notation and the rules of Maths aren't the same thing.
How can these be rules of math when they didn’t universally apply?
The rules do universally apply 🙄
The order of operations tells us how to interpret an equation without rearranging it
Yep, and you showed you don't know the rules 🙄
When you pick a different convention, you need to rearrange it to get the same answer
Not necessarily, though it makes it easier (but also leads a lot of people to make mistakes with signs, as you found out 😂 )
What you did was rearrange the equation
To show you how to correctly do "Multiplication first". 🙄
which you can only do if you are already following a specific convention
Which you didn't, hence why you ended up with a wrong answer. 🙄 There is no textbook which says put the multiplication in Brackets if doing "Multiplication first", none.
because the conventions are not laws of mathematics, they are conventions
And putting the Multiplication inside Brackets isn't a convention anywhere 🙄
They obey the laws of math. Conventions aren’t laws of math, they’re conventions
Yep, and you ignored both, hence your wrong answer 🙄
And a quick Google search will tell you that not everyone puts juxtaposition at a higher precedent than multiplication
And a quick look in the Google support forum will show you many people telling them that is wrong, and Google just closes the incident 🙄
it’s a convention
No it isn't. It's against the rules. 🙄 Again, you won't find this alleged "convention" in any Maths textbook
As long as people are using the same convention, they’ll agree on an answer and that answer is correct
Unless they disobeyed the rules, in which case they are all wrong 🙄
You can be mean all you like, that doesn’t change the nature of conventions
And you can be as ignorant of the rules and conventions of Maths as much as you want, and it's not going to change that your answer is wrong 🙄
How are you gonna write 2 + 2 ÷ 2 with repeated addition?
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 🙄
That doesn’t mean it has to be expanded first.
Yes it does. Everything has to be expanded before you do the addition and subtraction, or you get wrong answers 🙄
2+3x4=2+3+3+3+3=2+12=14 correct
2+3x4=5x4=5+5+5+5=20 wrong
You could expand 2 + 2 × 3 as (2+2)+(2+2)+(2+2)
Says someone who can't tell the difference between (2+2)x3=12 and 2+2x3=8 🙄
you are unable to tell me what mathematical law prohibits it
The order of operations rules 😂
reverse polish notation wouldn’t work as it does
It works because it treats every operation as bracketed without writing the brackets. Also that's only a Maths notation, not the Maths itself.
In RPL, 2 2 + 3 × is 12
Because the way it calculates that is (2+2)x3, not complicated. Same order of operations rules as other Maths notations - just a different way of writing the same thing
If you had to expand multiplication first, how would it work?
It works because Brackets - 2 2 + = (2+2) - are before Multiplication
The same can be done with prefix notation
Another Maths notation, same rules of Maths
Different programming languages have different orders of operations
Maths doesn't
those languages work just fine
They don't actually. Welcome to most e-calcs give wrong answers because the programmers failed to deal with it correctly.
Your argument amounts to saying that it makes the most sense to do multiplication before addition
No, my argument is it's a universal rule of Maths, as found in Maths textbooks 🙄
that only gives you a convention, not a rule
Left to right is a convention (as is not writing the brackets in RPN). Brackets before Multiplication before Addition are rules.
There's no such convention, given it would violate The Distributive Law 🙄