In Austria we just learn “Punkt vor Strich”, i.e. “Point before line”, since we write multiplication and division with dots “⋅” and “:” and addition and subtraction with lines “+” and “-“. Precedence of brackets is obvious. At the time when you learn exponents you are advanced enough to make sense of its precedence as well. Left to right is unreliable and should always be made explicit with brackets.
Actually Pemdas is supposed to be a standard in math. The answer is 1. I remember in elementary school they taught us PEMDAS and they would make use follow this from left to right. They never gave any problems where you couldn't use it to figure out the order of computing. However, in pre-algebra class which is 8th grade in the U.S.A. we learned about Juxtaposition. However, they taught us this in one day. However, ,they never told us it's called Juxtaposition. They they would show the brackets as training wheels. Then told us in 9th grade that you have to imagine those brackets to be there without the problem showing them. In 9th grade algebra 1 they had a day that reviewed that concept. People went on did their thing and over time forgot the rule and when they do they default back to PEMDAS and they forget the Juxtaposition and is because they number one don't even know the name or what Juxtaposition is.. because students like me in my school we were never told what it was called. We were just shown what it is and to do that from here on out. Yet, in college some books used PEMDAS with the juxtaposition and some just used PEMDAS. I remember in my finance class I would get answers wrong even following the formula correct. I would spend hours trying to figure out what I did wrong and then found out in the front of the book that it uses PEMDAS and omits Juxtaposition. This wasn't all my finance classes but it depended on the publisher and the author's books. It's just a pain in the ass to have to remember however others use the order of operations. Like you said there's no law or rule of how you do them. However, the standard is PEMDAS with the Juxtaposition.. That is the standard that we are supposed to follow and mathematicians follow it.
"However, ,they never told us it's called Juxtaposition" - well that's because it's not what it's called anyway. The correct name is Terms. Terms are separated by operators and joined by grouping symbols. Therefore, the first 2 steps in PEMDAS are solving Terms, the last 4 is solving operators. "forgot the rule" - yep, just like all the people arguing that the answer isn't 1, and the people making video's like this one - have forgotten the rules of Terms and The Distributive Law. "Yet, in college some books used PEMDAS with the juxtaposition and some just used PEMDAS" - yeah, I've seen a huge issue with university teachers, who haven't done order of operations since they were in high school. We can see this going back at least as far as Lennes, whose complaint came from him not understanding that a Term is a product. He even used the word "product" and completely missed the implications of that! They all just seem to think it's "multiplication", and thus end up with wrong answers. If a=2 and b=3 then axb=2x3 but ab=6. "Like you said there's no law or rule of how you do them" - there is - Terms and The Distributive Law always apply - It's those who have forgotten these rules (and make up the rule of "implicit multiplication" to make up for it, but not entirely) who get wrong answers. Everything you need to know about it can be found in high school Maths textbooks. Students don't get these wrong, only adults who've forgotten the rules do (like the person who made this video for starters).
@@smartmanapps5588 No we are not talking about terms. We are strictly talking about how everyone is missing the part of Juxtaposition.. like ab+cde = f this is the same as (ab)+(cde) it's when multiplication is implied to represent a single term. You have to multiply to get the term it represents before doing anything else.
@@smartmanapps5588 No there's no law on how to do them. Meaning you can do the math however you want. What you failed to understand what I am saying and the person in the video that what you're not understanding is that there's no law or rule that say you must do it this way. The order of operations are nothing but standards that we all agree to use as society to follow for the purpose of communication. So, we can understand each other on how we got our answers.
@@smartmanapps5588 You can use the distribution law which is part of algebra if you do this 6/2(1+2) is 6/ 2*1+2*2 which is 6/ (2+4) which is 6/6 =1 it's this way because 2(1+2) represents a single integer and you need to get to that before doing the division. 6/2(1+2) is the same as 6/(2(1+2)) due to juxtaposition . if you did it the wrong way which would be 6 / 2+4 and then did 6/2 = 3 then 3+4 = 7 this isn't the answer. For the answer to be 9 the problem needs to be written like (6/2)x(1+2) or 6/2 x (1+2) but it didn't. So, the answer is 1.
No, it doesn't. Distribution is literally the first step in solving brackets. PEMDAS is just a mnemonic to remind you in what order to apply the rules, it's not a set of rules in itself, and it certainly doesn't override any actual rules.
The video is wrong though. Tell him to do a search for me including the phrase order of operations and he'll find a thread with textbook references, proofs, memes, the works - ALL things which are missing from THIS video.
@@smartmanapps5588 i dont know what to say but i'm bad at math that's all,i dont know whats so wrong about this video,and actully you can send him a massage instead order me to tell him about these.
@@aditrizqi941 "i dont know whats so wrong about this video" - he's ignoring the rules of Terms and The Distributive Law, then claiming Maths is ambiguous - it isn't. "you can send him a massage" - how? I don't even know who he is. "instead order me to tell him about these" - I didn't order you. I told you the video is wrong and that I have a thread which you can search for which quotes the actual rules.
So you have 2 degrees in maths so what about it. My Indian teachers in kzn south Africa had no degrees but they did a fantastic job. With your degrees did you realize that there is no subtraction in Pemdas
I'm going to be honest as someone genuinely looking for a reason why people don't use PEMDAS in higher math. You set up a strawman argument of multiplication before division. You acknowledged that most students learned to group multiplication and division and do it in order left to right, but still spent half the video attacking the strawman that we don't know or do that. It is clear to me that this change in the order of operations is not done because one order is comparatively better compared to another but simply because it is the laziest way to write whatever series of equations they need to write. The ease of writing math does not make a order of operations superior. The ease of understanding someone's math is what makes a order operations superior.
I suppose one point is PEMDAS doesn't include implicit notation in it. The variant PEJMDAS does have Juxtaposition included as higher priority (the more academic interpretation which you often see in academic writing and style guides). It lets you not have to write that notation explicitly and still simplify it but that's only really for the academic interpretation and not the more literal/programming interpretation. With PEMDAS you have to apply a notation convention before you start to remove the implicit notation and replace it with explicit notation instead. A lot of these videos argue about the order of operations, but what they really should be focused on are implicit notation conventions. That's the main reason for ambiguity in expressions like these. Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove ambiguity.
Lol I was trying to understand why I never had this issue that you describe where it's "intuitive" that M comes before D in the thing then from that comes a difficulty in understanding the lack of precedence between M and D.... And then it hit me.. we do not learn PEMDAS where I live. I live in chile, here, we speak Spanish, in Spanish, nothing is really taught with these common mnemonics you guys use so frequently in English. I didn't know what PEMDAS was until i saw it on the internet lol.. we were just literally told "there is no priority between multiplication and division. You solve left to right" and that was all 😅
When in doubt, look at what you can test. Let's look at the practice of Dimensional Analysis, units of measurement obey the same algebraic rules as numbers and variables. So Distance divided by circumference is equal to revolutions. There is also a formula for spherical volumes that returns values confirmed by experimental observations. So, beginning with these. D÷2πr≠D/2πr D÷2πr is interpreted as D÷2πr≡D÷[2πr]≡D÷[πd]=revolutions But D/2πr would be evaluated as (D/2)πr≠ revolutions but returns units of area instead, not only that but D/2πr≠D/πd as these both return different incorrect results. Now let's look at the implicit expression for spherical volumes. 4/3πr³≡(4/3)πr³ while 4÷3πr³ would be evaluated as 4÷[3πr³], with units of inverse volume, and numerically not even close to experimental observations. So D÷2πr=correct, 4÷3πr³ is not. And while 4/3πr³ is correct, D/2πr is definitely not. These concepts are stupidly easy to demonstrate experimentally. Likewise D÷2πr≡D÷2π(h+t) when the radius is separated into hub dimensions and tread thickness. Oddly enough, this highlights a video problem. Obviously, if D=2πr=πd Then D÷2πr=D÷[2πr]=D÷[πd]=1 But if h=t=2/π inches, then r=4/π inches. So 2πr evaluates to 8". Leaving D÷2π(h+t)≡D÷2(πh+πt)= πh=πt=2π/π=2" Giving us the equation of 8"÷2(2"+2")=1 revolution. And not 16in² as some profess. Volume divided into spherical flasks can get even more obviously ridiculous. V÷4/3πr³≡V÷[(4/3)πr³]= a dimensionless quantity of filled containers. While any version of V÷4/3πr³ evaluated otherwise will return units in 6 dimensional space or numerically far from observations as accurate. V÷4 -------= dimensionless, but, 3πr³ nothing remotely close to the observations. (V÷4/3)πr³=in³*in³=in⁶ so, no. Not correct. Only by interpretation as V ‐----------= number filled 4/3πr³ Will any calculations match the observational results.
Good grief why is this still a thing! I’m currently tangled up in yet another battle on Facebook. Of course the PEMDAS or die crowd is insisting that I am wrong, and all these videos on RU-vid are wrong. They are so entrenched in PEMDAS they can’t see the mistake they are making. And they refuse to even research what the problem is. Thank you for this video as it very effectively illustrates the issues in my opinion. I’m going to direct some of these wingnuts to your video but I’m sure they won’t even watch it.
"And they refuse to even research what the problem is. Thank you for this video as it very effectively illustrates the issues in my opinion" - OMG. I'm not sure you have any idea how much this is dripping in irony, given the presenter of this video DID NO RESEARCH (and apparently neither did you, since you apparently agree with him). Did you notice the complete lack of MATHS TEXTBOOKS in this video? Yeah, them. Ditto for the (highly-acclaimed by him) video by the red-headed woman (who is a PHYSICS major, NOT even a Maths major!). Do some research on Terms and The Distributive Law - those 2 things that NONE of these people ever mention - and you'll find why there's no ambiguity whatsoever in PEMDAS or any of the other mnemonics. i.e. 2(1+2) is a single, factorised Term, to be solved as part of solving Brackets. i.e. BEFORE Division. There is no such thing as "implicit multiplication" - since it's not even Multiplication (it's a Term) - it's a made-up rule by people who have forgotten the actual rules, and can't be found in any Maths textbook (which is why no-one who ever talks about it mentions textbooks - they haven't looked at any textbooks, just Google and Wikipedia). Equally ironic is when these people say "they should teach this in Maths"... having not actually consulted ANY Maths teachers on how it's done or on what we teach! Yes, we teach that - students have no trouble answering these - adults who've forgotten the rules does NOT mean it wasn't taught to them (and you can see it was by looking in Maths textbooks - any Year 7-8 Maths textbooks, any era).
@@smartmanapps5588 Except that implicit multiplication is what is taught to 99% of students. sure, your specific class may have learned it as terms, but almost every other person has grown up being explicitly told that ab means a times b, and that writing ab is just a useful shorthand. This is also told in many textbooks.
@@kemcolian2001 "Except that implicit multiplication is what is taught to 99% of students" - no it isn't taught to ANY of them. They're taught Terms and The Distributive Law - just look in a Year 7 Maths textbook - that's why students don't get these questions wrong. "implicit multiplication" is a made-up rule by adults who've forgotten the actual rules. "ab means a times b" in brackets. i.e. ab=(axb). e.g. 6/2(3)=6/(2x3). "This is also told in many textbooks" - what I've said is in many textbooks, yes. I know because I teach Maths and have literally dozens of them. I guarantee you can't find a textbook that says "implicit multiplication".
@@smartmanapps5588 Mate, I'M a student right now. I've seen with my own eyes me and all my classmates being taught that a(b) = a * b. I notice you used the term "year", so maybe in the british system they do things differently, but i can say for certain in the IB system it's taught as implicit multiplication.
@@kemcolian2001 Well, I've heard there's an issue with how it's taught in the U.S., but the U.S. textbooks I've seen teach it correctly (i.e. you have to expand the brackets BEFORE you remove them), so if your teacher is leaving out the brackets prematurely then that's just wrong, simple. You can't remove brackets unless there is only 1 term left inside. This is the way that all hand-held calculators, except for Texas Instruments, do it (and T.I. claims that U.S. teachers asked them to do that, but I'm a bit sceptical given the textbooks teach it correctly).
Multiplication and Division are the opposite of each other (inverse operations). That means they both share the same spot on the operations order. If it so, the rule from LEFT to RIGHT kicks in. 6÷2(1+2)=9 You added an extra set of parentheses that aren't there. Multiplication and division are interchangable in the order, they apply in order that they appear from left to right.
Brackets isn't "multiplication". There's been no extra brackets added. Brackets are always solved first, which includes Distribution. a(b+c)=(ab+ac) 6/2(1+2)=6/(2+4)=6/6=1.
It's grade school ARITHMETIC using grade school notation. The correct answer is 9 6 -------(1+2)= 6÷2(1+2)= 6/2(1+2)=9 2 6 --------- = 6÷(2(1+2))=6/(2(1+2))=1 2(1+2)
@@Questiala124 whenever being taught inline infix notation and the basic rules and principles of math in school AND when dealing with inline infix notation in places like social media where Latex support isn't available and you're unable to use a vinculum accept in a very clumsy way and you're forced to use inline infix notation for division... 6 -------(1+2) I had to use three terraces of 2. space to type that fraction... 6 ------(1+2) = 6÷2(1+2) in inline infix notation 2 6 ----------- = 6÷(2(1+2))= 1 2(1+2) in inline infix notation...
I don’t like the new math system. It’s restricting, and it makes me hate math. PEMDAS and the gazillian ways they try to reach you math is confusing asl. Just teach me one way, and stick with it please! I’m going on yt and learning this way because the videos have instructors that aren’t like my teacher, who I can’t swap out for another teacher/have her teach me one way, but people who explain it in one simple concept that’s easy to understand. Plus, I can’t ‘rewatch’ an in person lesson. But I definitely can when it comes to a well thought out math video about pre-algebra, measures, and IQR.
P... (1+2)=3 E... none M&D equally left to right as they appear 6÷2=3 & 3×3=9 correct answer A&S equally left to right as they appear None except for what was INSIDE the parentheses. It don't get any easier than that...
While I agree with notion of this video, as a mathematician I am not a fan of using ab/c to mean (ab)/c and I would encourage my maths colleagues to not abuse the notation. Just because you know what you mean and everybody in your extremely small field knows what you meant it does not mean that anybody else (including other mathematicians) will know.
I always loved the video from The How And Why Of Mathematics about this subject. TI still doesn't sell PEJMDAS calculators, so I'm always buying Casio, after being a TI 83/84 user for years. Ofcourse I always check the manuals or inquire about Order of Operations before buying!
You fail to understand the Order of Operations. When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended you get the only correct answer 9 not 1
One action that I think should be taken first is conversion. Specifically, convert all of the subtractions to additions, all of the divisions to multiplication, and then any fractions to decimals if it is practical (in the case of 1/3 it is likely easier to leave it as a fraction). So in the case of "4/2" you would change it to "4*0.5", and in the case of "5-2" you would convert it to "5 + (-2)". The advantage of this is that due to the communitive property the order of the operands in addition and multiplication doesn't matter. In the case of (4*0.5) it is the same as (0.5*4). This is not the case with subtraction and division. By doing this, it simplifies the process of working the equation and eliminates some of the confusion concerning PEMDAS since now it just becomes doing parentheses, exponents, multiplication and addition.
"The advantage of this is that due to the communitive property the order of the operands in addition and multiplication doesn't matter" - it already doesn't matter even if you don't do that. "This is not the case with subtraction and division" - +3-2=-2+3
@@dkstudiosQC Laugh but implicit multiplication is a convention that doesn't trump the basic rules and principles of math... People confuse and conflate two different types of Implicit multiplication .... One without a delimiter and one with a delimiter.. Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed i.e. juxstaposed WITHOUT a delimiter and forms a composite quantity by Algebraic Convention... Example 2y or BC This type of Implicit Multiplication is given priority over Division and most other operations but not all other operations... This can be seen in most Algebra text books or Physics book. Physics uses this type of Implicit Multiplication quite heavily.. Type 2... Implicit Multiplication between a TERM and a Parenthetical value that have been juxstaposed without an explicit operator but WITH a delimiter...The parentheses serve to delimit the two sub-expressions.. Parenthetical implicit multiplication. The act of placing a constant, variable or TERM next to parentheses without a physical operator. The multiplication SYMBOL is implicit, implied though not plainly expressed, meaning you multiply the constant, variable or TERM with the value of the parentheses or across each TERM within the parenthetical sub-expression. Parentheses group and give priority to operations WITHIN the symbol of INCLUSION not outside the symbol. Terms are separated by addition and subtraction not multiplication or division. The axiom for the Distributive Property is a(b+c)= ab+ac but what most people fail to understand is that each of those variables represents a constant value OR a set of operations that represent a constant value... A single TERM expression like 6÷2(1+2) has two sub-expressions. The single TERM sub-expression 6÷2 juxstaposed to the two TERM parenthetical sub-expression 1+2. The lack of an explicit operator implies multiplication between the TERM or TERM value outside the parentheses and the parenthetical value or across each TERM within the parenthetical sub-expression... The parentheses DELIMIT the TERM 6÷2 from the two TERMS 1+2 maintaining comparison and contrast between the two elements... Implicit multiplication is always by juxstaposition but not all juxstaposition is Implicit multiplication. Example 2½ = 2.5 not 2 times ½... There is “implicit multiplication” WITH delimiters and there is “implicit multiplication WITHOUT delimiters. Two different types of Implicit multiplication and mathematically different. 6÷2y the 2y has no delimiter.... 6÷2y=3÷y by Algebraic Convention. 6÷2(a+b) has a delimiter... 6÷2(a+b)= 3a+3b by the Distributive Property... 6y÷2y = 6y÷(2y) = 6y÷(2*y) 6y÷2(y)= (6y÷2)(y)= 6y÷2*y 6y÷2y(y)= (6y÷(2y))(y)= 6y÷(2y)*y= 6y÷(2*y)*y ÷2y the denominator is 2y ÷2(y) the denominator is 2
@@dkstudiosQC The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division. The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them... The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication... The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral... A variable can represent a single value, a set of operations that represent a TERM that represents a single value or a solution set... 6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property...Parentheses removed... 6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property... Inner parentheses REMOVED This can be further demonstrated using the vinculum.... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ------------ = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.... ________ 2(1+2) how many grouping symbols do you see grouping OPERATIONS within the denominator?? ÷(2(1+2)) how many grouping symbols do you see grouping OPERATIONS within the denominator?? That over bar (vinculum) is a grouping symbol __________ 2×1+2×2 how many grouping symbols do you see grouping OPERATIONS within the denominator AND what was REMOVED?? ÷(2×1+2×2) how many grouping symbols do you see grouping OPERATIONS within the denominator AND what was REMOVED?? Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation... 6. 6 -------- = 6÷(2(1+2)) = 6÷(2×1+2×2) = -------------- 2(1+2) 2×1+2×2 If you choose to Distribute the 2 into the parentheses by itself you have to do one of two things. Either take the division symbol with it, as division is right side Distributive or change the division to multiplication by the reciprocal... ÷2= ×0.5 So... 6÷2(1+2)= 6(1÷2+2÷2) still equals 9 Or... 6÷2(1+2)= 6(0.5×1+0.5×2) still equals 9 Variables can represent more than just a numeral and it's important to understand that when you replace a variable with a constant value or a set of operations that represent a constant value that you apply grouping symbols where called for by the Order of Operations and the basic rules and principles of math... example 6÷a does not have parentheses BUT a= 2+4 so 6÷a = 6÷(2+4) not 6÷2+4. BUT if a=2×3 and we have a÷2 we can write 2×3÷2 because we evaluate Multiplication and Division equally from left to right... a(b+c)... a=12÷3, b= 2×3, c= 2^2 we have... 12÷3(2×3+2^2) = 4(6+4)= 4(10)= 40 ab+ac = 12÷3×2×3+12÷3×2^2= 4×2×3+4×4= 8×3+16= 24+16= 40. <<< same answer What most people don't understand is that you can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression... 6÷2×1+6÷2×2+6÷2×3^2-6÷2×4= 6÷2(1+2+3^2-4) as the LIKE TERM 6÷2 was factored out of the expanded expression. I hope this helps you understand the issue a little better...
8:31 small correction: teach "operator precedence", not "order of operations". Precedence is a property of the operator, not the operation. To resolve operators (e.g. "juxtaposition") to operations ("scalar multiplication"), operand types need to be inferred first, which is an issue of semantics - something you want to do after you have parsed the expression and obtained its syntactic tree. However, the syntactic tree will be different based on which operator is evaluated first, making syntax inappropriately dependent on semantics if we try to assign precedence to operations instead. The following expression shows the problem with "operation precedence": f(x) = 6/x(1+2) The type of "x" is not known in advance. If we take f(2), we get the viral problem - and since the operation in this case will be multiplication, it would inherit the precedence level of multiplication, resulting in a different parse tree than f(sin), which makes juxtaposition resolve to function application instead. By teaching that precedence is a property of the operator, both cases f(2) and f(sin) can now be analysed as having the juxtaposition operator, which has a higher precedence than multiplication, and so we can construct a parse tree without needing a priori type information, enabling syntax-aware type inference. Thus, teaching operator precedence aids, rather than hinders, subsequent education into higher math.
People confuse and conflate two different types of Implicit multiplication .... One without a delimiter and one with a delimiter.. Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed i.e. juxstaposed WITHOUT a delimiter and forms a composite quantity by Algebraic Convention... Example 2y or BC This type of Implicit Multiplication is given priority over Division and most other operations but not all other operations... This can be seen in most Algebra text books or Physics book. Physics uses this type of Implicit Multiplication quite heavily.. Type 2... Implicit Multiplication between a TERM and a Parenthetical value that have been juxstaposed without an explicit operator but WITH a delimiter...The parentheses serve to delimit the two sub-expressions.. Parenthetical implicit multiplication. The act of placing a constant, variable or TERM next to parentheses without a physical operator. The multiplication SYMBOL is implicit, implied though not plainly expressed, meaning you multiply the constant, variable or TERM with the value of the parentheses or across each TERM within the parenthetical sub-expression. Parentheses group and give priority to operations WITHIN the symbol of INCLUSION not outside the symbol. Terms are separated by addition and subtraction not multiplication or division. The axiom for the Distributive Property is a(b+c)= ab+ac but what most people fail to understand is that each of those variables represents a constant value OR a set of operations that represent a constant value... A single TERM expression like 6÷2(1+2) has two sub-expressions. The single TERM sub-expression 6÷2 juxstaposed to the two TERM parenthetical sub-expression 1+2. The lack of an explicit operator implies multiplication between the TERM or TERM value outside the parentheses and the parenthetical value or across each TERM within the parenthetical sub-expression... The parentheses DELIMIT the TERM 6÷2 from the two TERMS 1+2 maintaining comparison and contrast between the two elements... Implicit multiplication is always by juxstaposition but not all juxstaposition is Implicit multiplication. Example 2½ = 2.5 not 2 times ½... There is “implicit multiplication” WITH delimiters and there is “implicit multiplication WITHOUT delimiters. Two different types of Implicit multiplication and mathematically different. 6÷2y the 2y has no delimiter.... 6÷2y=3÷y by Algebraic Convention. 6÷2(a+b) has a delimiter... 6÷2(a+b)= 3a+3b by the Distributive Property... 6y÷2y = 6y÷(2y) = 6y÷(2*y) 6y÷2(y)= (6y÷2)(y)= 6y÷2*y 6y÷2y(y)= (6y÷(2y))(y)= 6y÷(2y)*y= 6y÷(2*y)*y ÷2y the denominator is 2y ÷2(y) the denominator is 2
The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division. The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them... The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication... The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral... A variable can represent a single value, a set of operations that represent a TERM that represents a single value or a solution set... 6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property...Parentheses removed... 6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property... Inner parentheses REMOVED This can be further demonstrated using the vinculum.... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ------------ = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.... ________ 2(1+2) how many grouping symbols do you see grouping OPERATIONS within the denominator?? ÷(2(1+2)) how many grouping symbols do you see grouping OPERATIONS within the denominator?? That over bar (vinculum) is a grouping symbol __________ 2×1+2×2 how many grouping symbols do you see grouping OPERATIONS within the denominator AND what was REMOVED?? ÷(2×1+2×2) how many grouping symbols do you see grouping OPERATIONS within the denominator AND what was REMOVED?? Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation... 6. 6 -------- = 6÷(2(1+2)) = 6÷(2×1+2×2) = -------------- 2(1+2) 2×1+2×2 If you choose to Distribute the 2 into the parentheses by itself you have to do one of two things. Either take the division symbol with it, as division is right side Distributive or change the division to multiplication by the reciprocal... ÷2= ×0.5 So... 6÷2(1+2)= 6(1÷2+2÷2) still equals 9 Or... 6÷2(1+2)= 6(0.5×1+0.5×2) still equals 9 Variables can represent more than just a numeral and it's important to understand that when you replace a variable with a constant value or a set of operations that represent a constant value that you apply grouping symbols where called for by the Order of Operations and the basic rules and principles of math... example 6÷a does not have parentheses BUT a= 2+4 so 6÷a = 6÷(2+4) not 6÷2+4. BUT if a=2×3 and we have a÷2 we can write 2×3÷2 because we evaluate Multiplication and Division equally from left to right... a(b+c)... a=12÷3, b= 2×3, c= 2^2 we have... 12÷3(2×3+2^2) = 4(6+4)= 4(10)= 40 ab+ac = 12÷3×2×3+12÷3×2^2= 4×2×3+4×4= 8×3+16= 24+16= 40. <<< same answer What most people don't understand is that you can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression... 6÷2×1+6÷2×2+6÷2×3^2-6÷2×4= 6÷2(1+2+3^2-4) as the LIKE TERM 6÷2 was factored out of the expanded expression.
Schools like PEMDAS and GEMS so much because schools were originally built to train factory workers (AKA memorization and repetition), at least in the USA. This is also why art, music and other programs struggle to get funding and why so many art teachers (at least trad art teachers), buy so many materials out of pocket. Schools don’t care about kid’s creativity or need to function in the real world (notice the lack of classes teaching about sex ed, taxes, financial management, loans, etc), they just want to create good little worker bees to go into the workforce or military.
The problem here is that these acronyms are used to teach the order of operations to children for basic numeracy purposes. The issue with grouping numbers or variables will only likely turn up at the algebra stage by which time these acronyms should no longer be required. As we learn more advanced Maths, we learn how to interpret it and when implied brackets should apply. There is no acronym that accounts for all expressions (where does tetration fit in) nor could there be as ab^2 will be interpreted by mathematicians and engineers as a×(b^2) whereas 2^ab will be interpreted as 2^(a×b). Calculators should interpret expressions in the same way as the people using them and overrule acronyms when appropriate. Casio calculators seem to do this well. Wolfram Alpha uses the basic numeracy acronym method and as such will produce results other than wished for when such expressions are used.
No such thing as implied grouping SYMBOLS... Total nonsense The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended is 9 PEMDAS works because of the basic rules and principles of math not in spite of them...
In Germany we have ONE word for that. It's called "Punktrechnung", literally "dot calculation", because multiplication and division are written in dots (2•2=4; 2:2=1). There is no confusion about the equality about multiplication and division. At least in Germany. I think I will add that confusion in PEMDAS to my list of "this is why I don't get, why German was replaced in Science and math by english" Maybe English is easier, but German is a precise language and less vulnerable to misunderstandings and confusion.
You haven't shown a single example where PEMDAS was ambiguous. For some reason you just have difficulties applying multiplication before division. No clue what is so hard about that.
An anecdote to emphasize your last point: My friend in High school was very good at math classes. Because he could follow instructions, socialize with teachers and turned in homework. He constantly reiterated that "its just memorization, following instructions and order of operations". He went to college and dropped out of his BA program after failing business calculus 5 times in a row citing "When am I ever going to use any of this?!" I say either teach math or get rid of the requirement.
So bedmas isnt ambiguas at all to be fair. Following the rules, brackets, exponenets, d/m left to right, add/subtract left to right gives us a clear order. The problem is a few things. 1 - people remmeber their stupid acronym (pemdas or bedmas) and forget the elft to right portion and argue about it 2 - assuming anyone who posts a math problem actually answer is (if they answer it not following pemdas) then they wrote it wrong. Eg, 6÷2(1+2) and 6÷[2(1+2)] are 2 complete different equasion.
Most computers use the pemdas algorithm. You have to be very accurate with your parentheses in programs like excel and even the calculator on your smartphones.
Basic rules for BEDMAS (we call it this here in Canada with the B standing for brackets- easier to say and spell than parentheses!) work much of the time and are especially helpful for kids learning math in younger grades. Yes, there are times when the math gets harder and the rules are tweaked but for the majority- especially those who will just use basic math skills in their daily living- BEDMAS is definitely an asset to know. I used it yesterday when calculating something. I cringe when I see comments on those click bait math equation memes when people say: "I was taught a different way". or "we used to do it another way, why did they change it?" Nothing changed, folks, you always did multiplication or division first! I really hope they are just mis-remembering and not that they all had horrible math teachers.
6÷2(1+2) = 6÷2×3. I hope everyone is happy with that. The problem comes with order. Bodmas says divide first. pemdas says multiply first. So let's use Bodmas. 6÷2=3×3 =9. Now let's use pemdas. 6×3=18÷2=9. But let's forget about Bodmas, pemdas or Thomas let's use our own as. The rule for division is × And invert the divisor. 6×1/2= 3×3=9. So don't say pemdas is wrong. Just use grade 5 maths. All comments are welcome. I am in grade 5 but I like to learn from great people. Thanks
It's simple.don't think of P and E as different things. They are the same, they are functions and should be done together left to right. Multiplication and division are the same thing too since division is multiplying by the reciprocal. The same goes with addition and subtraction (Subtraction is adding the opposite) So think of all the operations as separate and group them as pairs all done from left to right. Now you can handle anything as the rules are always the same 5 x (3 + 6)^4 - 7 fine what do you do first PE from left to right (3 + 6) first then the exponent. etc. But lets say you have a function works the same way 5 x ln(3+6)^4 - 7 do the ln of what's in the parentheses first and then exponent. if it was written like this 5 x ln(3+6^4) - 7 you do the ln of what's in the parentheses. What's in the parentheses 3 + 6^4 so you do 6^4 first and then + 4 OK what about -4^2 well -4 is not subtraction it's a number so it's -4 squared. If you wanted the negative of 4 squared that would be written -(4^2) The whole problem with how PEMDAS is taught is that for no reason parentheses and exponents are done separately. If they were treated as a pair like MD and AS then no problem, one rule for every pair.
PEDMAS is a quick guide for primary school kids who do not understand the rules of expansions and factorization. expansion and factorization are secondary school topics. PEDMAS does not get everything correct. It will work in most cases for math in Primary School. Most of the time. An Adult who uses PEDMAS and forgoes the rules of expansion and factorization, should NOT BE A MATH TEACHER.
It just comes down to deconstructing each operation into the bunch of +s that they really are. Whichever can be deconstructed to +s in the least steps gets done first _beside_ their respective inverses.
