The Problem with PEMDAS: Why Calculators Disagree 

Teacher to student: why do you bring a pigeon to the exam? Student to teacher: well, if I’ll face any implicit multiplication, I’ll jut toss the pigeon to figure which calculator I should use!

What is the inverse of juxtaposition?

www.jstor.org/stable/2972726?seq=2#metadata_info_tab_contents In case an indicated product follow the ÷ sign the whole product is always used as a divisor, ................ Let's say a product is in front of the ² sign. The whole product is to be powered by 2, right? Does 2(4)^2 = (2(4))^2? Now, let's take a look a the product of u and v when v = 1/u^2 The product of uv is 1/u. So, 8/1/u = 8/(1/u), is that right? The product is a result of multiplication. So it is a result of repetitive addition. So it is a result of addition. That means it is also a sum. "In case a sum follow the ÷ sign the whole sum is always used as a divisor" Now, what is the sum of u and v? Its' (u + v), right? If a sum is inside a parenthesis, so must the product. "He (Chrystal) overcomes the difficulty by never using the sign ÷ with a product after it." That is because Chrystal know the problem.

@@Araqius Since it's a notation for multiplication, the inverse operation would be division. Some people have argued that the reason multiplication and division should always have equal priority is that division is simply multiplication by the reciprocal. Is that what you're getting at?

​@@THaWoM Power - Root = inverse = same order Addition - Subtraction = inverse = same order Multiplication - division = inverse = same order Juxtaposition - division = inverse = different order?

Bingo. That's probably the *real* solution to the above question.

As a programmer myself I totally agree! As ones code can be copied from system to system, language to language, complier to compiler, you can never be 100% sure how your code is going to be interpreted/parsed on the destination systems (same as we see in calculators). As stated in my comment below, it's all about precision, and leaving out parentheses is just as lazy as leaving out operators.

Yes, this is what I teach all my junior programmers as they join our ranks

yea i realised i did the same thing in c++ computer graphics. yComponent = sin(3.14 * (trigAngle/180))

Are any of you old enough to have gone through school _before_ everybody had these "algebra" or "graphical" calculators? I'm _just_ old enough to have done, we had scientific calculators with parentheses, the answer was pre-calculated each time you entered _), ², ³, √, ^, ×, ÷, +, - or =_

Their calculator must've been "simplifying" '4+4' to 8 the moment the '/' was entered. 🤦🏻‍♀️

You dodged a bullet by not getting a job as a mathematician at a company full of people who don't know math.

This is an utter mess, since it's not even PEMDAS but these guys - and their calculator - interpreted 4+4÷2 as (4+4)÷2 which doesn't even follow the rule we Germans call "Punktrechnung vor Strichrechnung" ("dot operation before stroke operation") which orders multiplication and division before addition and substraction. In German, we normally use '∙' instead of '×' for simple multiplication ('×' is used for the cross product of vectors) and mostly ':' instead of '÷', thus we call them "dot operations".

Those teachers should be fired.

@@deirdre_anne That is how all four function calculators work. Wit them you need to first calculate 4/2, store it in memory and then add 4 to the memory.

I've been getting told by teachers in America that I'm dumb for using juxtaposed multiplication, told that I'll at best get a job at Maccies for not understanding maths, thanks for providing me some clarity that we've just been taught slightly different order of opporations

TBF a lot of lower level math teachers don’t practice advanced math like at all…

You had never heard of the order of operations as a mechanical engineer and just subconsciously were doing things in the correct order?

@@coachmcguirk6297 incorrect, I knew only one order, which I now know is called PEJMDAS.

@@coachmcguirk6297 I am sure they mean they have never heard of the fact that there are two different systems used by calculators. I was unaware of that, too, and probably for the same reason (I used RPN also). I am an EE not an ME. All engineers use PEJMDAS mentally when evaluating expressions written out on paper. Strict PEMDAS is stupid. The solution for calculators is probably to disallow multiplication by juxtaposition. You should throw the operator in there to make it unambiguous. Like excel. 2pi() is a syntax error.

Calculators are often still stated to be against the rules on many tests but I have literally been at an entrance exam to a financial education at university where *all* the other students brought their calculators and were using them and despite me pointing out to the observer that it was specifically stated to be against the rules he let it be because otherwise he had to fail every single one of them. And I was the big loser because I had not even brought mine. The big lesson being that you should *always* bring a calculator regardless of what the rules say. I even had to discuss my maths scores with the intaker because it was the only part I hadn't really done that well at but the general scores were really good so he didn't understand.

@@mckenziekeith7434 all engineers do not use PEJMDAS. Where did you get that idea? I only found out that some people prioritize multiplication by juxtaposition recently. It was never taught in any of my math or engineering classes, so why would I assume that it has priority?

15 years ago, I would have disagreed. Now, I agree 100%. I've been bitten too many times by tiny details of syntax and semantics to trust any coding language unless I know the compiler like the back of my hand.

Must have been nice to have calculators to fall back on. We only had slide rules.

*high school (two words, not one, just like "elementary school" and "middle school")

That makes sense. Writing out a math problem on paper or blackboard, allows us to visually separate each portion of a math problem from another, using very few symbols, to keep it clean. I have a lovely phrase, for this visual representation we can do in writing, that we cannot do with calculators. So using the example in the video above: 1 ___ 2sqrt(3)

@@alvallac2171 🙄

Good ol' RPN. No need to worry about order of operations when there are no infix operators.

"When I use an HP, I use RPN, so such things don't come up" Hooray for RPN!

This is great, because you already understand implied parentheses, and you're aware of the limitations of the melted sand (silicon) The crux of these videos is that too many people haven't learned about implied parentheses, and they use melted sand which is programmed to match the same broken learning, so they feel they're justified in their confusion. They also don't like hurt feelings, so you better not point out they're lacking. YOU should be making all the changes so they don't have to.

@@CollinBaillie The first step to learning is accepting that what you think you know can be wrong. If you're not used to accepting that concept, then you _will_ get your feelings hurt, or you will remain in willful ignorance. This particular problem is with how math is initially taught, but being able to accept corrections is an important skill across nearly every field of life.

This is ideally what higher-level education teachers should be acknowledging and instructing. And universities should update their writing style manuals to include maths operation, whether they are adopting PEMDAS, PEJMDAS or whatever.

The sad part is, PEMDAS is *bad* for teaching mathematics. It doesn't describe negation at all, and the left to right order only serves to enforce similar solution steps, not understanding. The fact that addition is commutative should be the first observation after long addition, and yet PEMDAS attempts to bludgeon that out of existence. Worse yet, those who do remember PEMDAS tend to forget that MD and AS are paired up. It's a mantra, not a useful guide.

Best comment

*It's part (or "It is part")

@@alvallac2171 Dude. This isn't a grammar channel. Chances are this was an innocent typo. Stop showing off.

@@realhumanist71The only thing being shown off here is your lack of discipline. Writing properly without mistakes should be a given, not something only "elites" or nerds do.

@@louisrobitaille5810 Ouch. I deserved that. I'm sorry for the comment about showing off.

​@@realhumanist71more like being an ass at this point especially when it's obvious

Same!

Same here. Never heard of PEDMAS/BOMDAS, just got taught mathematics!

*'80s (apostrophe goes before the decade, taking the place of the omitted millennium and century)

Because that's the actually correct way. There's actually quite a lot of tiny rules like that in math that for some reason people either ignore or forget.

Strange. I was taught parentheses ALWAYS comes first. That's the whole point of parentheses, to force you to calculate what's in them first.

I like that with my HP-48, too.

my calculator is an HP-39gs if you wanted to know

I believe it was called "Pretty Print" or maybe that is Ti Specific

Casio also has it. It's pretty much been my standard calculator for the last 10 years.

@@designerd77 I think that is TI-specific. On my HP-48 it called 'Equation Writer'.

punctuation ... always makes me think of the disgruntled dine and dash panda .... Panda eats shoots and leaves .... but yes order does matter ... there is a reason we learned to count first then add then subtract then multiply the divide .. you needed the one before to do the next one ... teachers forgot to mention that one as they have been doing it institutionally all along ... the order you need to learn it in is the reverse order you calculate it in by the pairs ...

My calculus teacher would typically remark upon studying new topics in calculus "It's not rocket science, it's just algebra". I was unaware of the acronym PEMDAS, "implied multiplication", or "multiplication by juxtaposition" until a year ago. For the past 6.2734 decades algebraic shorthand taught in Algebra 1 was sufficient for me in the determining the order of operations and the resolution of expressions like 6/2(1+2) = 1.

That's why most engineers use HP stack based calculators; 48, 50, etc. All RPN.

​@@NinjaRunningWildRPN, yeah baby! The HP 35s that wasn't available was the last HP calculator that supported RPN mode. It does have an algebraic mode, but I don't think I've ever used it in algebraic. The calculator app Inuse on my phone uses RPN. I fear RPN has been consigned to the dinosaurs though. But I do recall every Engineering student had an HP 48g when I was in school.

god I wish libraries still existed

​@@EphemeralPseudonymWdym? They still exist

@@NinjaRunningWild That is what I was thinking. I never trusted 'fancy' enter an equation functions. Just write down your formula and with a little practice RPN becomes quite intuitive. RPN is a also very good for helping you write down partial answers, which greatly helped when looking for errors. I still use RPN, just as a mini-program on my computer.

I agree with most of what you said, I just don't think we should depend that much on calculators.

​@@BlacksmithTWD life is too short to do everything manually. You realize computers exist solely because of this? like nobody every really solves problems like 6 - 2(2+1) in real life except in facebook comment wars. It's one thing for the average person to be able to calculate 10% tax on their stuff when at the shop, or simple head calculations like that... but actual real world maths is too complicated to waste time trying to figure it out on paper. Engineers and accountants need reliable calculators that give them the correct answers as quickly as possible.

@@neosmagus I disagree with every sentence you just uttered as they are all based on faulty premises. Are you able to think for yourself and correct yourself here or do I need to spell it all out for you?

Silly you.

I use metric when I design devices myself, but whenever I see a post like this, I have an immediate desire to say that computers would prefer imperial units, because fractions over a power of 2 can be expressed with floating point numbers precisely, whereas a base-10 system has to be approximated. It's probably a character flaw.

Yeah. I use a RPN coding language. ;)

I'm actually surprised "Reverse Polish Notation" isn't mentioned more in the comments. I suspect the HP calculators she mentions where she can't find the order of operations might be because they're RPN! or ? It's probably also why HP calculators are all over the place. The engineers have just thought if people were bothered about order of operations, they should just get a scientific calculator with RPN.

The HP35s that the said wasn't in stock uses this. They've been making that model since the 80s. I know of a LOT of engineers and scientists that still use one as it's impossible to get incorrect answers as it'll give you a syntax error.

I use a RPN calculator. I wish I'd known about it at school. Even though I used a normal scientific calculator I would never put a whole equation into it. Using the example above I would have entered 3, √, *, 2, =, 1/x.

I use RPN, but that has nothing to do with this problem because you’re still the one interpreting the order of operations. If you interpret an equation wrong then RPN won’t fix that.

Your father was a smart man! Even at $395 each, those RPN calculators would always get the right answer if the user knew how to do math!😊

With respect to " i have seen one by a former Microsoft calculator programmer who was adamant that the left to right was a strict rule, and everybody else was wrong." I worked at Microsoft Research, and I fortunately, worked with a number of individuals with vastly superior intellect than the aforementioned "former Microsoft calculator programmer".

_This isn't a math problem._ Whoever told you it was lied to you. This is an ENGLISH problem. Math only happens when the intended audience is aware of the convention being used by the person communicating. This isn't math and it's never been math. If the author wants to avoid doing English instead of math when they intended to communicate math, they should _use brackets_ at every possible place where there may be a different convention with respect to the operators and their implied grouping.

Thing is, whoever is teaching these people BEDMAS or equivalent dropped the ball in explaining how to carry out operations involving brackets. I never had a problem using BEDMAS. I used to be a math tutor. Here’s how I would teach it. 6 : 2(2+1) Do what’s INSIDE the brackets. Remember to leave the brackets alone. = 6 : 2(3) NOW you must deal with the BRACKETS first before continuing. = 6 : 6 There. Simple. Easy to understand.

​@@hotjanuary Let's home the students you tutored got a second lesson. The parentheses parts deal with values INSIDE the parentheses, not if one value is in one. The time you got to 6 : 2(3) you then move to multiplication and division LEFT to RIGHT. Next step would be the (6 : 2) * 3 = 3 * 3 = 9

@@sbyrstall the distributive property would disagree with you. If they didnt want you to use the distributive property on 2(3) they would have written it 2*(3), but since that multiplication symbol was left out, u have to distribute 2 inside the parenthesis before removing them...and u cant just replace the parenthesis with a * and then go do something else somewhere first because that's not how it was written!

@@sbyrstall dude. That’s not how brackets work. You’re completely ignoring the distribution law that would give you the same answer. That’s why you deal with brackets FIRST. Here’s using the distribution law in action. 6 : 2 (2 +1) = 6 : (4 +2) = 6 : 6 = 1 The notation is telling you that the question looks like this 6 ------ 2(2+1) 2(2+1) = denominator. 6 = numerator.

Yes we were also taught pejmdas in the 60's but they didn't have the acronym for it. It was just how you did things.

Yeah, but you'll always have a phone in your pocket with access to dozens of calculators each capable of generating wildly incongruous results.

Yes, you can always use parenthesis but the whole idea of precedence is to avoid them. you could declared that everything is leg to right unless indicated by parenthesis but for convenience one has for example given multiplication higher precedence.

Implicit parentheses is probably not a good way to teach it, but if it works for you, then okay

So on more advanced calculators if I say var = 5, is it then v×a×r=5 or do I have a variable var set to 5?. Do you introduce a whitespace operator?

Just letting you know, the word you're looking for is "credible" (a reliable source of information) not "creditable" (something that can be credited back).

@@RavenMobile As used above, creditable ~= praiseworthy, which seems perfectly apt.

@@abstractapproach634 Calculators (and programming languages) that allow multi-letter variable names will parse that expression as "var" set to 5, while calculators which only allow single letter variables will simply throw an error. No calculator or programming language will interpret that statement as the equation "v*a*r = 5". That syntax is never used to assign a value to a variable or to solve for one variable.

​@@subaction I contend that more will parse it as a comparison, and use a different symbol for assignments, such as an arrow.

well, you can't change the question (adding brackets) they give you 🙄

@ if it was a test question then try supplying them with both working solutions lol

@@KanpachiGaming they give you zero 😭

100% this. in school it's just anoying but in real life this can cause severe trouble.

In compiled languages there is no speed difference at all, it is just a matter of good taste

That's the way I was taught in upstate New York, USA as well. Apparently, though, there are places in North America that are/were not taught that. I've never seen a textbook for Algebra or more advanced that didn't use that method, though. Perhaps I've led a sheltered life.

"order as listed" or "order as they appear" Hmmmm, I'm not sure how you are interpreting those as two distinct things

Same here. Honestly, I don't see how someone could misinterpret PEMDAS to somehow mean that juxtaposition is on the same priority as division. Juxtaposition is already covered by the multiplication step because, it's multiplication. It's just weird to see everyone in the comments have the take-away that PEMDAS is wrong. It's not the mnemonic's fault that some people got confused and used it to justify incorrect order of operations.

@@Emilamlom "the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction" Multiplication and Division have the same precedence in 'pure' PEDMAS (as do addition and subtraction)

@@Feroce "where multiplication take priority over division," That's not PEDMAS -- in PEDMAS, multiplication and division have the same precedence. You need another name for what you want to do.

A scheme which is not supported by computer languages. An operator must appear between factors and terms.

I was in school at the same time and we got taught the term PEMDAS but taught PEJMDAS by example , that is, we had to use PEJMDAS to pass tests, though I don't recall the difference ever being called out -- they treated PEJMDAS as how you actually did PEMDAS

Keying an expression verbatim into an electronic calculator is not always correct. Rewriting is often needed.

*multiplication

See 9:56 in the video. Maybe Wolfram website needs to determine the country of origin of the user. Or since North American businesses and scientists use PEJMDAS, then maybe if the user is coming from a North American school then use PEMDAS, but everyone else PEJMDAS

Never heard of PEJMDAS in my life until now lol

Agreed that RPN is the way to avoid ambiguity in calculators. I have used the HP12 financial calculator for the last 45 years; more efficient as well as unambiguous.

PEMDAS in algebraic mode. The reason why HP calculators interpret differently is because HP rebrands or outsources calculators to other manufacturers. I guess the HP from the video is actually rebranded Casio and HP 35S is Kinpo Electronics.

The original hp35 came out 50 years ago without brackets or equal signs. We never had any of these problems, but, then again, our education system was "different."🙂

I’ve been tangled up in debates every time this 6/2(1+2) comes up on Facebook. Long before I’ve discovered these videos, I always came up with 1 as the answer. The majority of the comments will come up with 9. My point was your math instructor in college would write the problems as 6 over 2(1+2) and have you simplify the 2(1+2) first (following the juxtaposition first rule) and the answer would be 1 of course. I’ve even sent this problem to my son and he and his friends would come up with 1 as the answer. As you said, I feel vindicated as well. You and I both were going thru the same thing! 😂

Same. Its what I was taught in the 90s and still how I would teach it today.

I was taught by the way it was written, especially when it comes to fractions. If it was visually put as a fraction, the top and the bottom parts are all interpreted as parentheses to be done first, with the fraction divide done after. If there was a number to the left, it meant a whole number + the fraction. If there was a number or variable after, it was interpreted the answer to the whole fraction be multiplied by it. It is just all confusing.

There is no wrong or right. Just two different dialects.

I never thought about PEMDAS or PEJMDAS. I've never been taught either of them (I'm from continental Europe) I just thought that ":" (division) is adifferent symbol than "/" (fraction) when talking about the facebook discussions. The division is equal to multiplication and the fraction is the term with higher priority. That is why I thought she used the "wrong" key in 12:58. Casio has a key for fractions (second row, first key) to minder miscommunications.

In the US, math ability has been hindered by faith in calculators.

Really? Let's test your resolve by thumbnail of this exact video. 6÷2(1+2) By your idea of turning division into inverse multiplication what would you get? And why? [Also division sign is still there as invertion assumes such] 1. 6×1/2×(1+2) = 6÷2×(1+2) = 6÷2×3 = 2. 6×1/((2+1)×2) = 6÷(2(1+2)) = 6÷(2×(1+2)) = 6÷(2×3) You don't get out of interpretation ambiguity of already existing and written equation, if it has been written wrong at very beginning. Because to do inverse multiplication you must properly understand equation in the first place! To be fair, by rules it should always be 6÷2×(1+2)=9

@@DimkaTsv Oh, I absolutely agree with you: if the equation has been written ambiguously no trick will help, you need to know which notation the author used. My comment was too brief. What I meant by "I haven't used (...) since 14" was nobody in high school or university used it. Neither students, nor teachers, not even textbooks. If the thumbnail was an exercise in a textbook that division would be written as either "multiply by a half" or "six over two times parenthesis" depending on what was meant. As a side note, I think PEJMDAS sounds very sensible, but this video made me realize that we never cared about the distinction. It seems that where I live people decided to circumvent the whole problem by avoiding divide sign completely.

@@zzzyxad well, for me if you want small division - use division sign. If you want long one though, either do it as 2 lines, or use parentheses om both sides (unless one of them is just one number), that way you won't break any calculator. And that's what i do for my calculator, when i use it.

​​@@DimkaTsvhere is no such "rules". AIP and AMS gives precedence for implied multiplication. Different conventions do things differently.

Multiplicative inverses are a luxury you don't always have.

*parentheses (plural) parenthesis = singular, meaning just a single "(" or ")" by itself. "Nominator" should be "numerator." *multiplication

Voice to Texas... @@alvallac2171

This exactly

I was explaining this to people - if you do algebra, you are using the "shortened" or "implicit" form even if you dont realize it (eg 2ab/4b etc). It's always funny to me how people who don't use math on daily, weekly or even monthly basis argue about this with people who do. In the end - if people who I need to share my math with understand and can calculate correctly, I do not really care what anyone else thinks :)

If PEMDAS was applied such that all multiplications bind tighter than all divisions (instead of doing them at the same precedence level from left to right), this would be a non-issue. Seems more elegant to me.

That's on the non-graphing calculators, right?

@@ewthmatth I have a non-graphing calculator. What would happen on a graphing calculator?

Or get a different calculator make

I was already wondering why young US citizens don‘t know anything about geography outside of the US, and now I learned that even in mathematics their education is not teaching right. Together with their weird measuring units using body parts, it seems to me they want to keep the normal people dumb. Is that by purpose?

And you used texas instruments in school?! I remember studying in Germany and hating on TI. Not only did they look worse and have a worse UI than Casios but they also made these weird mistakes she shows in this video. And I never understood how calculators can mess up basic maths when that‘s the only thing they were built for.

Now that I‘ve seen more of the video, this seems to be on the same level as pounds, ounces, inches, feet etc. vs mili/centi/kilograms. Americans sticking to a nonsensical system the planet doesn‘t use, and not even their own mathematicians use.

Likely not. Practical applications of math tend to be less ambiguous than "puzzle" expressions.

​@@shikeridooLuckily the TI-84 CET is really nice. Here in the Netherlands our school demanded we use them. Nice graphing calculators luckily don't make this mistake. Cheap calculators do it wrong because it's too expensive to change the program to include it, because they use hardware to execute the calculations instead of software.

There are a lot of seriously confused people when it comes to the basic rules and principles of math... Please read carefully.... 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 When you actually understand that TERMS are seperated by addition and subtraction not multiplication or division and that Multiplication can be moved around ANYWHERE within a TERM as long as you do not affect the denominator of a division operation you will understand the only correct answer is 9 When you actually understand that division is nothing more than multiplication by the reciprocal and that ÷2 is equal to *2⁻¹ then you will understand that the only correct answer is 9 When you actually understand that GROUPING SYMBOLS only group and give priority to operations WITHIN the symbol of INCLUSION as a priority and that the 2 is not WITHIN the symbol of INCLUSION and there is no math book that states "with the exception of " then you will understand the only correct answer is 9 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... ________ 2(1+2) and (2(1+2)) both have two grouping symbols ________ __________ 2(1+2) = 2×1+2×2. Distributive Property. Parentheses REMOVED. One grouping symbol... (2(1+2))= (2×1+2×2) Distributive Property. Inner parentheses REMOVED. One grouping symbol 6÷2(1+2) does not equal 6÷(2×1+2×2) as you have not REMOVED any parentheses and you still have the same number of grouping symbols.... 6. 6. 6 -------(1+2) = ------- ×1 + ----------×2 2. 2. 2 The same as 6÷2(1+2)= 6÷2×1+6÷2×2 There are two types of implicit multiplication and they are not mathematically the same.... Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed and form a composite quantity by Algebraic Convention... Type 2... Implicit Multiplication between a TERM and a Parenthetical value or across each TERM within the parenthetical expression... Terms are separated by addition and subtraction not multiplication or division.... 6/2(1+2) is a single TERM with two TERMS inside the parentheses. In the axiom A(B+C)= AB+AC the A represents the TERM or TERM value i.e. monomial factor of the TERM outside the parentheses.... The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y. This is an inaccurate comparison... These two expressions utilize two DIFFERENT types of Implicit multiplication... 6÷2y = 6÷(2y)= 3/y by Algebraic Convention 6÷2(a+b)= (6÷2)(a+b)= 3a+3b by the Distributive Property... All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. There are no coefficients in the expression 6÷2(1+2)... 6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b is 3 not 2 Correlation does not imply Causation. Just because both expressions utilize implicit multiplication doesn't inherently mean they are treated in the same manner... The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. For people who argue 6÷2(1+2) and 6÷2y should be evaluated the same way, their argument is circular and is an informal fallacy that is flawed in the substance of their argument... 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(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression...

Well there used to be international agreement and when it comes to algebra there still is but Americans decided to go their own way. Apart from America the answer is accepted as 1.

@@neilmorgan2273 WRONG... I know several Engineers, Mathematicians and University Professors with PhDs from different parts of the world who disagree with you... Sounds like you're Xenophobic to me.... SMDH Your statement that apart from America the answer is 1 is blatantly false and extremely misleading... F. Kingdom

@@neilmorgan2273 This is a classic case of "American Exceptionalism". America(ns) is(are) always right even if the rest of the world disagrees.

@@RS-fg5mf Yes, the answer is technically 9 due to the rules of BODMAS. However, this is does not match the common conventions of maths and science, where 2(1+2) is treated as it’s own term. To me, it really doesn’t matter about which answer is “right”, but instead which answer would be more useful under different conventions. Following strict rules is a lot less important if it isn’t as useful. If 6/2(1+2) is treated as 9, you may as well write it out as 6(1+2)/2, creating redundancies when writing out expressions/equations. That’s why it’s always treated as 1 by physicists, mathematicians, etc.

Only it's complete NONSENSE because it is in direct conflict with the Order of Operations and the various properties and axioms of math... Just because it supports your ignorance doesn't make it right... The Distributive Property is a PROPERTY of Multiplication and as such has the same priority as Multiplication... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9

@@RS-fg5mf it is a little advanced for you isn't it? maybe ask an adult

@@tommy8290 go back to your color book and crayons until you're ready to understand the facts. Dump Azz

@@RS-fg5mf go back so call order of operration nonsense because bemdas rules over pemdas!

@@junkfoodguy BEMDAS PEMDAS BODMAS PEDMAS PODMAS BOMDAS etc. All mean the same thing

Which is sloppy math and can cause problems if somebody comes along that doesn't adhere to that convention. This is the 21st century there's no need for that assumption. Either carry the fraction bar far enough to cover everything that's intended to be included, or use a different grouping symbol. Then you don't run that risk. The order of operations is the order of operations, it's just that it's probably best described as the priority of operations as you typically are just looking at portions of a problem after you've been at it for a while.

@@SmallSpoonBrigade The only people that would not adhere to the convention (BTW, math is ALL about convention - like the order of operations) are those not properly educated. Seems to be an ever increasing failure of the 21st century. But having said that I agree with you. You should be as explicit as possible so that errors in assumptions do not occur.

@@SmallSpoonBrigadethe reason juxtaposition should take priority is that it is clearly intended to group the juxtaposed terms as a single unit in the equation. It is counterintuitive and unhelpful to artificially split a juxtaposed term away from the rest of its unit in order to rigidly apply the priority of operations.

now try programming that in a computer language and you will see why pemdas rules.

@@phueal most (if any) computer languages do not support juxtaposition. An operator must appear between factors and terms.

Correct. Anyone who say there's "implied" math needs to go back to middle school and learn math. Math is logical and direct, not floating (except for decimal points).

As a computer programmer myself, I seldom memorize the priority of operations of any particular language. So, to avoid confusion, either I use parenthesis a lot, or separate each individual calculation to its own line/statement. You can never trust the interpreter to get it right. :D

_"One issue in programming is that few languages allow implied multiplication (I think only MATLAB and Mathematica can accept such formulas)."_ The Julia Programming Language: "A numeric literal placed directly before an identifier or parentheses, e.g. 2x or 2(x+y), is treated as a multiplication, except with higher precedence than other binary operations." and "The precedence of numeric literal coefficients used for implicit multiplication is higher than other binary operators such as multiplication (*), and division (/, \, and //)."

Yeah, I learned the same. Juxtaposed multiplication was considered part of the brackets. 6 / 2(A+B) was pretty much interpreted as "solve 2A + 2B" as part of the B, and then do the whole thing.

BODMAS and PEDMAS and BEDMAS are the same with different words. DM is actually the same as MD, because MD is taught as multiplication and division at the same time left to right, not multiplication then division, and it always works out the same as doing division first. (multiplication first is different)

I was taught BODMAS as well back in the 80’s… and it was taught first do inside brackets then outside brackets ( for juxtaposed)….. then the rest as usual.

My teachers lied to all of us😢

In school the text book was full of wrong answers. We just took that as a given. The popular theory was, it was intentional to catch people copying the answers out of the back.

it's interesting that implied multiplication is nowhere to be found in BODMAS

@@jacquesmalan5950 I don’t think anyone really thought about it, it was so ingrained that it just went without saying. I saw a RU-vid video on the history of PEMDAS and it was pointed out there that even the textbooks that popularised PEMDAS in the early 20th century also used implicit multiplication at a higher priority than division without comment.

Hi, Johann, right? I'm curious about your perspective on this as a German. Do you think most Germans see 6/2(1+2) as 1?

@@Jtzkb FALSE.. There is no rule in math that says you have to open, clear, remove or take off brackets. The rule is to evaluate OPERATIONS inside the brackets and nothing more... This lady like so many other individuals is confusing an algebraic convention given to variables where there are no brackets neccessary (coefficient/variable compound quantities) to something different (parenthetical implicit multiplication) NOT the same thing... ALL variables have coefficients and real numbers (constants) can be coefficients but real numbers do not have coefficients... The variable a has a coefficient of 1 whether it is written or not... a/a = 1a/1a by algebraic convention... BUT 1a/1(a) = 1a/1(1a) or 1a/1*a= a^2 Any seperation between a constant and a variable by an explicit multiplication sign or by a grouping symbol seperates the coefficient/variable bond... 6a/2a= 3 6a/2(a)= 6a/2(1a)= 6a/2*1a= 3a^2 Algebraic convention of implicit multiplication between a constant and a variable does not equate to parenthetical implicit multiplication... This lady has done a nice job in misleading the viewer... 6a(1+2)= 6a*1+6a*2 TRUE... 6a*1+6a*2= 6/a⁻¹*1+6/a⁻¹*2. ALSO TRUE SOOO... 6a(1+2) = 6/a⁻¹(1+2) 6a(1+2)=9... then a=? AND a⁻¹=? 6/a⁻¹(1+2)=?? The Order of Operations does not support parenthetical implicit multiplication.... 6/2(1+2)=9 The Order of Operations not to be confused with the simplistic memory tool PEMDAS supports 9 The Commutative Property supports 9 The Distributive Property supports 9 The Operational inverse of multiplication and division by the reciprocal supports 9 The proper use of grouping symbols supports 9 Algebra supports 9 They took a convention applied only to variables and only in algebraic expression/equations and incorrectly applied it to parenthetical implicit multiplication... 6/2a where a= 3 would be written as 6/(2×3) not 6/2(3) just as if we had 20/a where a=5+5 would be written as 20/10 or 20/(5+5) BUT never 20/5+5 6 -----(3) = 6/2(3)=9 2 6 ----- = 6/(2(3))=1 2(3) When you remove the vinculum which is a grouping symbol and groups operations within the denominator and rewrite the expression in an inline fornat extra brackets are required to maintain the grouping of operations within the denominator....

@@Jtzkb Your teacher is a complete idiot. scholar.harvard.edu/files/contrastingcases/files/chapter_1.pdf 1. First, simplify expressions ***in*** parentheses. 2. Second, apply exponents. 3. Third, do all multiplication and division from left to right. 4. Fourth, do all addition and subtraction from left to right.

To the people that are freezed in the PEMDAS thing, do you really thinks that phisicists and engineers from all the globe are all wrong?

@@cronnosli the books I'm presently using to teach my children as well say 2(a+b) is one, not separate parts of an equation. Funny those who are saying she's wrong when she's showing how calculators are being reprogrammed to reflect pejmdas as the proper mode of calculation.

I'm *horrified* to hear there are textbooks which state a rule and don't follow it. That never would have happened when I was a kid; not in math.

@@eekee6034 Well if they always considered juxtaposition as a grouping symbol, as they probably were, then they are following the rule, since grouping symbols is on same level as paranthesis.

That is same as PEDMAS (instead of PEMDAS).

6/2*3 is always 9, not 1, BIDMAS or BIMDAS or PEMDAS or PEDMAS. Neither division nor multiplication is "above" the other just like neither addition nor subtraction is above the other. You do it left to right.

@@JohnRunyon Think again. Using PEMDAS for the problem, you have to know the rule of "left to right for operations of equal status". Otherwise, you get 1 as the (wrong) answer. A long time ago, you got the wrong answer for that problem, and then you learned the rule that gives you the correct answer. But you are too arrogant now to remember or admit you got the wrong answer back in the day. In contrast, with PEDMAS (not PEMDAS), a student does not have to learn a special rule.

⁠@@JohnRunyon Are you thinking of 6/2(1+2)? The answer can be both 9 and 1.

I am also from the UK. I learnt maths in the 90's and I was taught BODMAS. I was also taught division has a higher priority than multiplication. However I was also taught that numbers attached have priority. So in 6÷2(1+2) I would interpret the 2(1+2) as being a single number, in this case that would equal 6. So I would then do 6÷6 = 1. However if the equation were written as 6 ÷ 2 x (1+2), I would get 6 ÷ 2 x 3. I would then put the division higher than multiplication and end up with 3 x 3 = 9. The problem is the ambiguity which means the question is written incorrectly and should have more brackets for clarity.

You missed the 2 in the first one.

@@okaro6595 You're right! Thanks. 🙂

Not to mention population genetics and epidemiology calculations.

@@meganwilliams2962 Exactly. Not understanding this basic concept leaves me stunned. How the stupidity of a few could change how math equations are taught, proves the Peter Principle is true.

@@ToddLuvsGolf more ignorance from the peanut gallery... You are confusing and conflating an ALGEBRAIC CONVENTION given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing.... 6/2y = 3/y by ALGEBRAIC CONVENTION. 3y is a directly prefixed coefficient and variable that forms a composite quantity. No parentheses involved or required BUT 6/2(y) The TERM value outside the parentheses is the coefficient of the parenthetical value inside the parentheses. 6/2(y)= 3y 6/2(a+b)= 3a+3b NOT 3/(a+b) You are part of the problem spreading misinformation and arguing bad math. The correct answer is and always has been 9 NOT 1 .... PEJMDAS is BS

@@RS-fg5mf you are just embarrassing yourself now

@@tommy8290 you're projecting. The embarrassment here is you... 🙄🙄🙄 Your opinion of me is about as irrelevant as your opinion of this math expression. You're completely clueless.

obvious can sometimes be a misleading term, when one expression seem to obviously mean something to one person and to obviously mean something else to another person, seeming so strongly for each person that each thinks no further clarification is needed... until, of course, they encounter each other (and then sometimes just decide the other person is obviously wrong for interpreting differently).

...that is, it might be useful to consider using a more-specific term than obvious, such as (maximally, or perhaps sufficiently) unambiguous.

@@ak47tetris84 keep being a jackass... PEMDAS is an acronym a memory tool for the Order of Operations. A very simplistic approach that is not complete... The Order of Operations is more detailed when you bring into play the Associative/Commutative/Distributive properties as well as the Operational inverse of multiplication and division by the reciprocal... Sad that so many people want to trash these properties and the proper use of grouping symbols just to incorrectly support the wrong answers... Other than the algebraic misconceptions that lead to FALSE pretense the rules of math do not support parenthetical implicit multiplication...

Oliver Knell True or False 6a(1+2)= 6a*1+6a*2 ??

@@RS-fg5mf which would be 6a+12a= 18a. But 6÷ a(1+2)= 6÷ a+2a=6÷3a So 6÷2(1+2)= 6÷2(3)= 6÷6=1 Algebraic order of calculations 2a is always seen as one number, not separate.

@@jamey7003 2a doesn't require parentheses as it is a directly prefixed coefficient and variable. 6/2a is not the same as 6/2(a). 6/2a= 3/a BUT 6/2(a)= 3a 6/2(a+b)= 3a+3b NOT 3/(a+b) The Distributive Property is a PROPERTY of Multiplication and as such has the same priority as Multiplication... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9

@@jamey7003 BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = (1/2) 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....

Yes! Great reply. Only use the calculator to do the individual calculations. Don't rely on it for the proper order of operations.

Simple sci calculators from mid 80s like Sharp 506P don't have an entry method which would be ambiguous, and because you don't end up having to enter much markup, you actually operate it much faster than modern expression-based two-line ones. Also their metal fascia was so sexy. I want one of those again, a clone perhaps. I got a faithful clone in the 90s, but i wore it to death.

YES thank you. The actual correct answer is to stop writing equation ambiguously.

That method won't help you when so many texts exist that print the equations linearly, and showing them in the other form is not always easy unless you have LaTeX available and know how to use it or are able to simply handwrite the equations and scan them. There has to be general understanding of linear notation. They are written this way in hundreds of texts and in engineering work, not to mention most calculator screens. There's no reason to conform to a rule that never was.

Yup linear notation is weird in most cases. And that's really where the problems lay.

As a programmer myself, when taking a course or two of physics in college I came to the belief that these shortcuts of leaving off operators was harmful in another way: It meant that a variable tended to be strictly limited to only a single character (ignoring things like subscript and superscript notation). What this means is that you have to memorize absurd associations such as how h represents the Planck constant, and you *certainly* can't just write planckConstant * lightFrequency, because someone might think you're multiplying two dozen variables together! 🤦‍♂ Which means that someone sitting there trying to learn physics is going to struggle to follow along as the professor throws four letters onto the screen with an equal sign somewhere in the mix (for example T - μN = ma) and expect it to be a comprehensible concept at first glance to the people seeing it for the very first time.

@@Moleculor LOL. I am a programmer too but it is easy to distinguish between what a mathematical equation is and what a computer program is. If you need the actual equation in your program then make comments to explain the variables you are using and their meaning. Or you can stick with long descriptive variable names (and leave a warning FOR COMPUTER PROGRAMMERS EYES ONLY)

@@pulsar22 The point is not about the contents of computer programs. The point I was (apparently failing?) to communicate is the difficulty in human-language-parsing the contents of a chalkboard in a classroom when you're taking mathematical shortcuts that discourage anything more than single letter variables. For a really egregious example, hand-write out a w and a ω on a chalkboard or equivalent, and then tell me how to quickly differentiate between them. And then note how W can be weight. Or Work. And ω can be rotational velocity OR angular frequency. Or notice how s is *typically* distance, and S is entropy. And S is also a poynting vector. And if you're staring at an equation containing one of these letters, how do you quickly determine which of the multiple dissimilar concepts it's representing? Seeing these for the first, or second, or even eighth time, even in context with other single-letter variables, still means you have to sit there slowly translating the individual letter's meaning into English before you can understand what v = v₀ + at even is. But resultantVelocity = initialVelocity + acceleration * time is much more quickly parseable. Computing has the RIGHT idea: force operator usage to enable multi-letter variables.

Just while casually reading your comment, I naturally interpreted “x = e^1/2pi” to mean that e is being raised to the power of “one over two pi”, but I got really confused for a second, bc my natural interpretation of “x = e^1/(2pi)” was e to the power of 1 all over 2pi… (so basically x = e/2pi) which is very different. This obviously isn’t even a PEMDAS issue specifically, it’s a “how do you know for sure without explicitly marking it where an exponent ends and the reset of the equation resumes when equations are written on a single line” issue. I think in my time studying physics in school I learned to parse the ways that equations are conventionally written, but once you start adding some parenthesis beyond those that would be conventionally used, it actually introduces more potential confusion, rather than less…

​@@MoleculorAs a computer scientist, I program the way you describe (aside from some edge cases), but I'd never in a million years typeset an equation like that nor would want to read an equation typeset like that. It's a lot easier to learn what the variables represent than it is to break an equation over the massive number of lines it would take to do so.

You didn't learn it correctly and a video that supports your delusion doesn't make it right. It makes you willfully ignorant... What many people do is confuse and conflate an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing... 6/2y = 3/y by ALGEBRAIC CONVENTION 6/2(y)= 3y by the Distributive Property... The Distributive Property is a PROPERTY of Multiplication and as such has the same priority as Multiplication... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9

R S (also posts as Neoicon Mint) is just a troll and fails to let this go. He's posted arguments most days for over a year! So far he's: -Tried to argue with Dr Trefor Bazett (maths professor at University of Cincinnati) on one of his videos on this topic. -Tried to argue with Dr Oliver Knill (maths lecturer at Harvard University) on a post made by Oliver on this video. Guess who came out of that looking stupid! He has a few issues

@@tommy8290 you're a clueless troll... Someday you may realize your ignorance but I doubt it....

@@tommy8290 Johnny, if social media and such has taught us anything, is that we must be like Mr T. We gotsta pity a lot of fools.

@@RS-fg5mf If A=6÷2 it would be like this (6÷2) not 6÷2 the two are not the same in context to the sum

Delusions feeding the delusional... This video is total crap...

@@RS-fg5mf By definition, if everyone in the world is crazy but you, you are the one who is crazy. The herd is always sane. The rogue is the insane one.

And I mean that in the best way possible! :-)

@@nickmcginley4570 good thing about math is that it's based on rules not popularity or personal opinion... No matter how ignorant the populace is... The majority of people once thought the earth was flat too

@@RS-fg5mf said: _"The majority of people once thought the earth was flat too"_ And when was this mythical "once" you're referring to then?

In the US we learn fractions before parentheses

What you described is just PEMDAS, everyone learn about it • is the same as × or * ÷ is the same as / or the horizontal fraction bar

I appreciate that you used the distributive rule on your 5th grade example. In the simplified example, it gives the same answer that you'd get if you evaluated the addition inside the parenthesis first, but it is important to know that 2(a+b) is evaluated as 2a+2b when things start to get more complex! Well done German schools. 👍

The Distributive Property is a PROPERTY of Multiplication and as such has the same priority as Multiplication... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9

R S (also posts as neoicon mint) has been trolling on this and similar videos most days for over a year! You'd think he'd learn something in that time!

@@tommy8290 you would think you would learn not to be an ID it... I'm NOT Neoicon and you're just being a jack azz comparing me to him... and I can't help it that you're not as smart as your average 10 year old when it comes to math... You're a troll and a Maroon. I'm starting to think you like me as many times as you keep bringing up my name.

@@RS-fg5mf 2(a+b) is ALWAYS treated as one in operation. I've never seen D ÷ a(b+c) calculated as (d/a)(b) + (d/a)(c) but instead D ÷ (ab +ac) or D ------- ab+ac It is true that the original could be seen as ambiguous unless you apply the standard of equating algebraic expressions. This is the manner we were taught in school, this is the manner the books teach, therefore the easiest would be make it the standard of equating. I believe the issue with those who say 9 is the answer is that they don't remember algebraic computation.

@@RS-fg5mf also, my 16 year old is PRESENTLY learning algebra and worked the problem as he has been taught and answered 1 in under 30 secs.

BODMAS never got me anything but correct answers. Still using that iteration

@@gonnfishy2987 BODMAS was what I was taught. O being “powers of” and (as I said) grouping DM and AS has always worked. Hasn’t failed me yet lol.

@@timehunter9467 i always had it in my head. ORDINALS. same effect.

But normally in handwritten math such an expression will be written vertically, which is impossible to type. In LaTex you can type e.g. \frac{ab}{xy} which will render as a vertical fraction equivalent to (a*b)/(x*y); the latter is how you would have to type it in most any programming language, C++, Java, whatever... If you try typing ab/xy in C it will give an error, something like "variable 'ab' not declared"; if you try typing a*b/x*y it will not throw an error (assuming you defined the variables) but the code will evaluate as ((a*b)/x)*y which is not the same as \frac{ab}{xy} = (a*b)/(x*y).

I was blessed with a highschool math teacher that taught that 2(3) means the 2 is *attached* to the parenthesis and is counted with parenthesis in PEMDAS, and you cannot separate it from the parenthesis without properly evaluating it. He went on to say that PEMDAS is actually "Inside Parenthesis, Outside Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction". Also, he would have given you an F if you used the division symbol instead of writing it like a fraction. He said division symbols are for gradeschoolers and nobody who knows algebra should ever use one, for any reason. If you'd write this problem as a fraction as "6 over 2(1+3)", all the confusion would immediately end. 1 is the only possible answer you could get when you write it like that. As for calculators, they simply need to be programmed to treat x(y) as needing to be solved before any other multiplication or division.

@@DhalinIf the 2 in 2(3) is counted in the parentheses of pemdas, that means that 2(3)^2 = 6^2= 36 instead of 2(3)^2 = 2(9) = 18???? I’m not sure you meant to say it like that because every source I’ve checked says that exponents are done before implied multiplication.

@@somewhatfunnyguyy I have never seen an expression written 2(3)^2, lol. That's rather ambiguous to start with, as that doesn't explain what exactly is squared. However, if I were to take a guess, I would assume that problem started out as 2(1+2)^2 and it would make more sense if it was written (2(1+2))^2 in which case it would be 36. If they wanted an answer of 18, it would be instead originally written as 2((1+2)^2). At the end of the day, in actual application in Math, the story behind the equation you're trying to solve would tell you what exactly is happening, and you'd be able to correctly know how to write the problem before attempting to solve it. If the answer was 36, I could imagine a problem like this: "If Joe decided to make a square room whose walls are 1 foot long, and then changed his mind and decided to add another two feet and decided that the room was way too small and decided to double the size of the room, what was the resulting area of the room?" That would make it clear that the entire thing was to be squared. But, for example, if the correct answer was 18, I could envision something like this: "If Joe originally measured 2 feet and decided to add another 1 foot to a wall of his square room, and Tom also measured 1 foot and decided to add 2 feet to his square room, what would the square feet be for the two rooms total?" Since we are working with 2 guys measuring out 2 areas, you would multiply _after_ the squaring.

@@Dhalin Yes but the problem is that your(or your teacher’s) order of operations gives incorrect answers for problems written poorly, this is the same problem as pemdas with problems shown in the video. You can’t yell at pemdas for giving incorrect answers to weirdly written questions when your alternative also gives incorrect answers to weirdly written questions.

It's not arbitrary, there is a logic proof for it. I don't remember what it is, but when only about 15 of 200 students showed up the morning after St. Patrick's day my Calc III professor went off topic on it.

@@rightwingsafetysquad9872 Except that the multiplication and division, and addition and subtraction are meant to be horizontal to one another (equally weighted). MD doesn't mean multiplication then division, it means or...

@@lio1234234 A, I think you’re responding to the wrong comment. B, check out this paper written by a grad student, it’s not the proof my prof demonstrated, but he arrives at basically the same conclusion. Multiply-Divide are only different operations when notation is ambiguous. Likewise with add-subtract. The symbols for multiply and divide from elementary arithmetic should be avoided when at all possible in favor of juxtaposition and fractional notation. When programming or using calculators you probably need to just use a lot of parenthesis (or better yet break your operation into many commands). math.berkeley.edu/~wu/order5.pdf

@@lio1234234 My thoughts exactly.. I was taught BEDMAS (brackets, exponents, div, mult...). Additionally we were taught that after exponents, you work left to right solving multiplication OR division. Changing it to a fractional expression however, is like putting everything in the denominator in brackets (parentheses).

​@@rightwingsafetysquad9872There is no proof of PEMDAS because PEMDAS is a convention, not a mathematical proposition.

This two simbols are the same to me. Both mean division.

This is not always practical as if you write scientific article the space is not unlimited. The elementary school division symbol is not used at higher levels.

Well do not confuse fraction with division. just like implied multiplication when we write these stuff out we as humans pretend they mean the same things just like we pretend 2(2+1) and 2*(2+1) mean the same thing while in reality depending on how you resolve things can mean wildly different things

All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. This does in fact make Numerals and Variables different. When you replace a variable with a constant value you must use proper grouping symbols where required... A numeral is a numeral and nothing more but a variable can represent a TERM or TERMS or a single value... 6/2y the 2 is the coefficient of y. Directly prefixed and forms a composite quantity by Algebraic Convention... 6/2y = 3/y BUT 6/2(y) = 3y by the Distributive Property. A(B+C) = AB+AC where A equals the TERM or TERM value outside the parentheses. B and C equal the two TERMS inside the parentheses ... A= 6÷2 B=1 C=2 6÷2(1+2)= 6÷2×1+6÷2×2 6 is a single TERM 6÷2 is a single TERM 6÷2×3 is a single TERM 1+2 is two TERMS 6÷2(1+2) is a single TERM with two TERMS inside the parentheses. 6÷2(1+2) is mathematically the same as 6÷2×3

*algebraic

only 9!!!!

@@Jtzkb This is why I reject the notion that math is a universal language :P Well, one of the reasons.

I know math teachers from the U.K. who would say you're lying... The correct answer is 9

@@Jtzkb the basic rules and principles are there, you just have some people who refuse to understand or accept the basic rules and principles of math as intended...

@@RS-fg5mf Do you disagree with the video explanation too? There seems to be a lot of evidence that the answer is 1.

There is a standard. It's just that most people don't understand it and argue. Division is a fraction. Remember how to do fractions. Add, subtract, and multiply them. 6 ÷ 2(1+2) 6÷2 is a fraction, which is 6/2 So, 6 ÷ 2(1+2) Is 6/2 × (1+2) Or 6/2 × (1+2)/1 6/2 × (1+2) 3 x 3 = 9 Or 6/2 × (1+2)/1 Numerator 6 × (1+2) = 6 × 3 = 18 Denominator 2 × 1 = 2 So you get 18/2 = 9 Not once did I use Pemdas to get the right answer.

Let's change it 6 × 2 ÷ (1+2) Pemdas 6×2÷3 Multiplication first 6 × 2 ÷ 3 12 ÷ 3 = 4 Division first 6 × 2 ÷ 3 6 × 0.6666666667 = 4.0000002 Nah -zzzzzzz No pemdas 6 × 2 ÷ (1+2) 6/1 × 2/(1+2) 6/1 × 2/3 N = 6 × 2 = 12 D = 1 × 3 = 3 So, 12/3 = 4