The Problem with PEMDAS: Why Calculators Disagree

[@THaWoM]

Having to go through the captions and correct all of KZbin's guesses for "PEJMDAS" has been hilarious. @14:10 "I'm firmly in favor of the pigeon toss". You can quote me on that!

[@justpaulo]

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!

[@Araqius]

What is the inverse of juxtaposition?

[@Araqius]

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.

[@THaWoM]

@@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?

[@Araqius]

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

[@Rising_Pho3nix_23]

I'm a programmer and it doesn't matter how simple the calculation is, I ALWAYS ALWAYS put things in paranthesis. It may be overkill, it it does 2 things. It creates a habit of using good form, and it makes the calculation crystal clear for future editors. An example for percentages is "Percent = (100 / Max) * Val". Even if PEMDAS isn't an issue, I do it anyway. It makes code easier to write, read and debug. It also creates a force of habit that makes PEMDAS bugs impossible and ensures my code operates properly every time.

[@HenryMidfields]

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

[@digiacomtech5589]

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.

[@MostlyPennyCat]

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

[@381delirius]

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

[@MostlyPennyCat]

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 =_

[@brianlara8651]

True Story: I lost a job offer in high school because the test checkers were using a non-scientific calculator. So I was responding to 4 + 4 / 2 as 6 while they marked me wrong because they were getting the answer of 4 ! Anyhow, I got a zero on their "test" and their calculator agreed with them ! Meanwhile, I was actually tutoring Calculus and got a 5 on the Calculus BC AP test. But yeah, there was no way to convince them that they were wrong.

[@deirdre_anne]

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

[@fredashay]

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

[@jensphiliphohmann1876]

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".

[@okaro6595]

Those teachers should be fired.

[@okaro6595]

@@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.

[@X4Alpha4X]

this very problem got brought up in my early highschool math classes. we were told it shouldn't matter what calculator we are using, if we aren't getting the right answer its because we weren't using enough parentheses. That habit has stuck with me and i use them like a maniac, never trusting anything's internal OOP.

[@eekee6034]

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.

[@mikestuart7674]

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

[@alvallac2171]

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

[@deathsheir2035]

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)

[@KEVBOYMUSIC]

@@alvallac2171 🙄

[@cmagdanz6476]

I'm a retired mechanical engineer, and exceptionally flabbergasted I've NEVER heard this before. I have always unconsciously used PEJMDAS (because I didn't know anything different). I simply wasn't aware of this issue, probably because I always used RPN, never entering actual equations into a calculator. I even remember when calculators weren't allowed in school ;)

[@coachmcguirk6297]

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

[@cmagdanz6476]

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

[@mckenziekeith7434]

@@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.

[@SevenTheMisgiven]

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.

[@Andrew-it7fb]

@@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?

[@CamdenBloke]

I would definitely use parentheses in such a scenario. When I use an HP, I use RPN, so such things don't come up. When I tutored math, I always students to error on the side of more parentheses. A lot of modern calculators have formatted entry where you can have things above/below fractions, under root signs, and so on. If I'm using a calculator with that capacity, I make use of it.

[@angeldude101]

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

[@thomasmaughan4798]

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

[@CollinBaillie]

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.

[@angeldude101]

@@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.

[@HenryMidfields]

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.

[@richardward8578]

Thanks for the synopsis of the methods. As a Mathematician myself, I have been presented this problem several times and told each presenter to clarify what they intend. Braces, brackets, and parentheses are all free, so use them. It's is part of your job to define the problem clearly, and you cannot just assume that you are being presented the problem clearly. Always ask questions!

[@alvallac2171]

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

[@realhumanist71]

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

[@louisrobitaille5810]

@@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.

[@realhumanist71]

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

[@troyallen8223]

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

[@johnhaswell6347]

When I first learned algebra as a kid in the early 80's, I was taught if there is no operator between a number and a parenthesis the 'distributive property' must be completed before clearing the parentheses. Therefore, 2(1+2) becomes (2+4) before the parentheses can be completed.

[@emeltea33]

Same!

[@adrianhead6272]

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

[@alvallac2171]

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

[@danieljonsson8095]

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.

[@frederickd.provoncha8671]

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.

[@earthoid]

As an old retired electrical engineer, I have to say that PEMDAS seems so wrong that I'm surprised anyone with higher math experience would promote it.

[@AstralLaVista]

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

[@connorcoultas9629]

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

[@hajkie]

I used PEMDAS because i program. The programming language doesnt know if you want to imply multiplication or not, so you have to define it. Besides, this math problem illustrates the problem. If we have to GUESS what people IMPLY then thats bad math. Hence, use simply rules. Use BRACKETS or PARENTHESIS to define priority. No misunderstandings.

[@ZeYoX-mw7sh]

@@hajkie I never understood why people say PEMDAS has an issue, when its literally solved by using parenthesis correctly and fixes the whole guessing problem. Its as simple as writing 1/(2x) instead of 1/2x.

[@hajkie]

@@ZeYoX-mw7sh Agreed 100%!

[@rocketsurgeon11]

I learned about this while going through my education, but wasn't aware of what exactly was going on. I learned to be VERY specific with how I entered the calculations into the calculator to get the correct answer. Basically I learned "junk in=junk out". Interesting to me that the one Casio you showed does what I learned to do from experience. As another person wrote: I used parentheses almost in an overkill method to ensure that the machine understood my intentions clearly. In the case of writing these mathematical expressions, I think some of the onus is with the author of the expression to make it clear as to what the intended expression is. As with writing out languages, using punctuation is very important to the interpretation of what is trying to be conveyed. Math is a language, so the same can be said here. "Rules" are fine, but if the question is garbage, then the answer will also be garbage.

[@kaboom-zf2bl]

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 ...

[@billybbob18]

As a lifetime lover of math, I see this as a life or death issue worth fighting a war over. There is no excuse for calculators to disagree. It's bad enough that we cant seem to switch fully to metric. I don't care which system we use but it's pointless to have any operating order if we disagree. This is why I use parentheses EVERYWHERE when coding math for a computer. This way I can avoid implicit ordering all together. We need a calculator war and the one who loses gets their calculator destroyed / decommissioned. Kudos to Casio for throwing in the parentheses.

[@BlacksmithTWD]

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

[@neosmagus]

​@@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.

[@BlacksmithTWD]

@@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?

[@samueldeandrade8535]

Silly you.

[@jonathancard4466]

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.

[@billy65bob]

My HP Graphing Calculator from 10 years ago had a rather nifty feature. There was a button to have it draw a proper 'mathematical expression' of whatever I entered, so it would draw horizontal lines, giant square root symbols, logs, and all kinds of other bits of notation. I used it a lot to ensure the expressions I entered matched when I had written down, as it was one of those that automatically inserted explicit multiplication operators.

[@Skank_and_Gutterboy]

I like that with my HP-48, too.

[@billy65bob]

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

[@designerd77]

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

[@VulpeculaJoy]

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

[@Skank_and_Gutterboy]

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

[@jppagetoo]

Order of operations. A tricky subject. I have an applied math degree and I know how mathmeticians and engineers write things down. But one can easily see how it can cause errors. As a professional programmer I always use the parenthesis or break the calculation into smaller parts to make sure it is done with the order of operations I intend. I don't leave it to somebody else to determine that, that is asking for failures.

[@matthewdaniel1715]

Best comment

[@aditrizqi941]

Please sir,me and my brother are completely beginners in math, please show to us which one is correct pejmdas or pemdas?

[@martinbarringer6808]

For calculators Reverse Polish Notation (RPN) avoids ambiguity. It does require the operator to know what calculation is required rather than relying on the calculator to interpret notation. Once one is familiar with it, it is more efficient to use. It also has the advantage that no-one will want to borrow your calculator because they won't know how to use it! I have been using the HP12C financial calculator for the last 45 years and it is still going strong!

[@eekee6034]

Yeah. I use a RPN coding language. ;)

[@zaqsdk]

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.

[@plektosgaming]

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.

[@LeeFlemingster]

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.

[@NoahSpurrier]

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.

[@DustinSilva]

Thank you a MILLION TIMES OVER for making these videos and explaining everything. Seeing all these "math" peoples saying the answer is 16 has been driving me insane. I took several math classes in college and solving algebraic equations has always stuck with me leading me to an answer of 1.. It really has been making me feel like I was taking crazy pills seeing these people say 16.. As a programmer, we must explicitly define parenthesis as there is no implied multiplication through juxtaposition, so I understand both scenarios. Thank you so much.

[@eliteteamkiller319]

_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.

[@hotjanuary]

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.

[@sbyrstall]

​@@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

[@DustinSilva]

@@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!

[@hotjanuary]

@@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.

[@dperreno]

I found a great explanation from Howard Ludwig on Quora which helps explain why NA teachers were so adamant about using PEMDAS: "No, PEMDAS is not the truly proper order of operations. PEMDAS is an oversimplified set of rules designed to assist students (and teachers who are more education-oriented than mathematics-oriented) in advanced arithmetic and introductory algebra to keep straight the hierarchy of arithmetic operations in a compound arithmetic expression" Note his inclusion of "education-oriented" teachers. In the U.S., many, if not most teachers below the high school level are teachers first and subject matter experts second, if at all. Meaning that many math teachers may not have degrees in mathematics, but all will have degrees in education. So they are more concerned with the process of teaching a method to use than with how things work in the real world. If they were taught PEMDAS, or their books reference PEMDAS, then that is what they think is the way it should be. Rather disappointing.

[@0LoneTech]

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.

[@mhmt1453]

Thank you! I am a 57 year old American, who was taught PEMDAS growing up. However, when I went to university (for sciences), my Texas Instruments calculator routinely gave me problems. As you mentioned, I simply assumed that juxtapositions were prioritized, but did not think to check. One needed calculators for lengthy operations, especially during labs, and while checking my work-and finding discrepancies-I might have assumed the mistake was my own. Never had I considered that the trouble may lie in the calculator’s orders of operation. It was the 80’s; still the age of pencil and paper, and well before the internet. On a lighter note: several years ago I overheard two young people pondering ‘how anyone managed to do college papers before the internet.’ Interrupting, I said, “we went to the library!”

[@NinjaRunningWild]

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

[@TevelDrinkwater]

​@@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.

[@EphemeralPseudonym]

god I wish libraries still existed

[@xx_gamer_xx8315]

​@@EphemeralPseudonymWdym? They still exist

[@plovet]

@@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.

[@Rik77]

For me in the UK, what you call PEJMDAS is the normal rule i was taught. However, in my job where I do write calculations in Power BI or Excel, I tend to write everything out explicitly with many parentheses if necessary. I'm not sure why I started doing that, but I assume it was to avoid this problem, but also just to be exactly clear to someone reviewing the calculation what is intended.

[@mikestuart7674]

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.

[@user-lp3cf5yn5b]

If mathematicians say one thing then why are teachers teaching another thing which only leads to confusion once you go to a real maths class in college?

[@optimist_KMA]

I am from Ukraine. as I recollect my learning and studying years, I was taught to use vertical notation of fractions as the notation of division at all times (since I got acquainted with fractions). and whenever the horizontal notation was needed, it always used as many braces and parentheses as was required to be clear what is really meant. so this video opened my eyes on a problem I have never had even a chance to experience.

[@the_mad_bunnyx9537]

Thank you for taking the time to explain this properly. This really should be the only video on this subject that youtube shows.

[@uni-byte]

Being an engineer by trade and math and physics major by training I agree with you 100%. Most of the people with PEMDAS on the brain have never gone beyond high school math. PEMDAS is something they teach to grade schoolers to keep things simple for the simple math they do. The people I deal with put juxtaposition before explicit multiplication or division and they also assume that everything after a "/" is considered in the denominator. BTW, I use an RPN calculator, so I don't run into this mess.

[@SmallSpoonBrigade]

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.

[@uni-byte]

@@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.

[@phueal]

@@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.

[@garymartin9777]

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

[@garymartin9777]

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

[@privacyvalued4134]

When in doubt, use parenthesis. Never, ever rely on operator precedence. Extra parenthesis is only very slightly slower to parse when compiling code but will result in correct execution of that code every single time regardless of what compiles it. Once you adopt that as a core habit, you will never have issues regardless of what calculator/computing device you use. Software developers learn pretty quickly that different programming languages have different operator precedence and it is better to be a lazy parenthesis fiend than a "clever" operator precedence fiend.

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

[@KanpachiGaming]

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

@@KanpachiGaming they give you zero 😭

[@Vakilando]

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

[@dadestor]

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

[@AFmedic]

What would really help would be if the teacher told their students on day 1 what system they use. Back in the early 80's when I was going to school for electronics my teacher for DC/AC Fundamentals used PEMDAS and my teacher for Solid State Devices (Biasing transistors, etc) used PEJMDAS and we had no clue. After couple weeks we figured it and after some complaining they agreed to both use PEJMDAS and changed our grades on past assignments.

[@franksnyder5754]

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.

[@JoshSmith-db2of]

13:13 With regard to Wolfram|Alpha: Wolfram|Alpha tries to interpret "natural language" inputs, so it's interpreting the input based on what it thinks the user most likely means. Because of this it makes sense that sometimes it would use PEMDAS and others PEJMDAS. If you really wanted to know Wolfram's philosophy on this, you might want to try Wolfram Mathematica (the strictly interpreted language that underlies W|A).

[@BryTee]

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

[@gamerkev30]

Never heard of PEJMDAS in my life until now lol

[@ma3xiu1]

The HP35s is an RPN calculator, which uses postfix notation. When you enter an expression in RPN, there is no ambiguity and no need for order of operations rules. The order you enter the operations determines the order they will be executed. It is therefore up to the human operator to interpret what a written expression means, and then enter the operations in the appropriate order. The HP35s does have an "algebraic mode" as well -- it would be interesting to check it out and see if they implement PEMDAS or PEJMDAS.

[@martinbarringer6808]

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.

[@TihomirVujnovic]

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.

[@normfromga]

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."🙂

[@franksnyder5754]

Your two videos on PEMDAS/PEJMDAS are the highly creditable examples of how one should address attempt to resolve controversies such as this viral "shit-storm". I appreciate the fact that you didn't straddle the fence and try and pass it off as ambiguous. Your real-world examples and documentation that corroborate your perspective are considerably more creditable than just restating what PEMDAS or PEJMDAS is. I haven't seen anybody in PEMDAS camp provide any creditable documentation to support their perspective. I did find 2 online expression calculators and put them in Algebra mode (MathPapa, Symbolab). Both came up with the same answers when using the solidus (1) and both came up with the same answer when using the obelus (16), This would indicate that these Algebra calculators parse the solidus and obelus differently. I couldn't find any creditable source on the internet to justify these interpretations.

[@abstractapproach634]

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?

[@RavenMobile]

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).

[@yjo0]

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

[@subaction]

@@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.

[@0LoneTech]

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

[@VampireBuddha]

When I started learning to program, I quickly figured out that I should ignore order of operations and use brackets liberally. This seems to be common practice even among professionals.

[@marsrevolutionary]

I always thought that the P in PEMDAS isn't just for calculation, it's also for assignment. Formulas should be linearized before calcuation and that requires assigning parentheses to place the numerator and denominator portions inline. I think that this is best done when converting division and subtraction to multiplcation by inversion [ x/y = x(1/y) ] and addition of negatives [ x-y = x+(-y) ] in order to reliably ensure accurate calcuation. This is mandatory for programming.

[@alvallac2171]

*multiplication

[@chrismcquiggan9742]

I was always taught that multiplying out the brackets was part of the first stage (p or b depending on the abbreviation that you use) ie before evaluating any explicit multiply or divide signs so it was never an issue. We used the explicit operators to break the equation into evaluation checks, basically meaning that anything in between +-x/ was treated as being inside brackets. Also every school/University in the UK (as far as I'm aware) forces students to use "correct" calculators like the Casio one

[@CFWhitman]

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.

[@Wimblefish]

That's how I was taught in the UK during the 90s. The number directly before the brackets has always been part of the sum, and is always multiplied by the answer from inside the bracket. It wasn't until it became a thing on social media that I even realised some people didn't know that the multiplication was automatically implied

[@IgorRockt]

I don't know how it is in other countries, but in Germany, I learned at school right from the start (over 40 years ago already) that "omitted" or "implied" operators always have a higher priority (actually, that they replace the parenthesis around the two factors) than the ones being written down (that's why they are omitted in the first place, to save space instead of using parenthesis for everything and to illustrate what the nominator and denominator is, since the factors "stick together"). So the nominator "6 / 2(1 + 2)" is "6", while the denominator is "2(1 +2)", and as such can ONLY resolved in this order: 1. Parenthesis: (1+2) = 3 -> 6/2(3) 2. Omitted/implicit "times": 2(3) = 6 -> 6/6 3. Multiplikation/Division: 6/6 = 1 Or, to say it in another way: replacing the implied operator in "6 / 2(1 + 2)" gives you "6 / (2 * (1 + 2))", which is different from "6 / 2 * (1 + 2)" (without the implied operator). The formula "6 / 2 * (1 + 2)" would mean that you have a fraction (6 being the numerator and 2 the denominator) multiplied by "(1 + 2)", which resolves to "3 * 3 = 9". It's the same with other formulas, like the ones with "a" and "b", so "2ab / 2a" would be the "2ab" as the numerator (because the implicit multiplication has a higher priority), and "2a" as the denominator (again because the implicit multiplication has a higher priority), which can of course then be reduced to just "b", since "2a / 2a" equals 1, which means you get "1b / 1", and as such "b". I never even heard of doing it in any other way in my life before I immigrated to North America. Btw: I love the reverse Polish Notation (as e.g. used in the programming language "Forth") for this when writing down formulas - there is no way to get to the wrong result with that one, since you would write the original one as "6 2 1 2 + * /", while the second one would be "6 2 / 1 2 + *" 😎 PS: As a long time programmer, I always use parenthesis anyways (even if I think they are not needed), since most programming languages (well, at least all of the two dozen or so I know myself 😉) simply would throw an error if you used something like "2(1 + 3)" in a formula - they don't know anything about implicit operators. Edit: fixed a few typos and changed the wording a bit to be more precise

[@alvallac2171]

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

[@TreeDancingCloud]

Voice to Texas... @@alvallac2171

[@endxofxeternity]

This exactly

[@janbudaj2173]

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 :)

[@VilemBenjaminLiepelt]

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.

[@zelandakhniteblade5436]

As a TI calculator user from the 90s, here's a little tip: If you ever have a very complicated fractional formula, calculate the denominator first and then press the 1/x key. Multiply this result by the numerator and only then add/subtract any secondary terms. It will save you a ton of trouble over relying on the calculator blindly to get it all right.

[@ewthmatth]

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

[@alext8828]

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

[@a1smith]

Or get a different calculator make

[@Herr_Bone]

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?

[@hoppetosse8]

It's really weird that there are two different ways to calculate and that even calculators use not all the same system. Especially in school their shold be an international agreement how to handle this "problem". I can't believe that there is this mess. Imagine being a student in a new school, high school, in another country etc and having the "wrong" calculator / convention and therefore messing up a test. :D

[@RS-fg5mf]

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...

[@neilmorgan2273]

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.

[@RS-fg5mf]

@@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

[@frabjousfrank]

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

[@gregstunts347]

@@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.

[@kennethmcgee8795]

Hello. I went to high school in the U.S. in the early to mid '90s and learned the order of operations with the mnemonic 'Please Excuse My Dear Aunt Sally'. Our math teachers taught us that multiplication, whether by juxtaposition or otherwise, was to be done before division, in order as listed, not in the order as they appear in the problem. I guess what the way they taught us would now be called PEJMDAS.

[@fewwiggle]

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

[@Emilamlom]

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.

[@fewwiggle]

@@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)

[@fewwiggle]

@@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.

[@garymartin9777]

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

[@reubensmith4836]

Very good video. I noticed the same thing on my Casio. I was taught in electronics school to always add implicit outer parentheses for the numerator and denominator. If you do this step first, you will get the same answer on any calculator. It seems to be a forgotten rule.

[@okaro6595]

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.

[@thewhitefalcon8539]

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

[@jurgnobs1308]

so, in swiss highschool we learned that while something like 2xy would be a term in most mathematical usages. that term happens to correspond to the value of 2*x*y. still, this slight difference if perspective on it made it so we never really questioned that this implied multiplication is done before normal multiplication and division. maybe this different perspective to the very strict PEMDAS rules that north american teachers use explains why mostly north american teachers asked Casio to change the system.

[@markprange2430]

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

[@tommy8290]

Thank you for doing this video and taking the time to reference many different materials on the subject. It's by far the most comprehensive source of information I've found on this particular topic. As a physicist I'm glad your preferred method matches mine!

[@RS-fg5mf]

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

[@tommy8290]

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

[@RS-fg5mf]

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

[@junkfoodguy]

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

[@RS-fg5mf]

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

[@Voyager-3]

Your explanation, historical research, and summary of calculators is the best I've seen. Thanks.

[@zzzyxad]

To me, the whole disagreement is very abstract because I haven't used the divide sign since I was 14. After you learn to treat division as multiplication by an inverse, all ambiguity goes away.

[@DimkaTsv]

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

[@zzzyxad]

@@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.

[@DimkaTsv]

@@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.

[@taoliu3949]

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

[@MadocComadrin]

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

[@adrianhead6272]

The problem is - most people seemingly can't do mathematics without a calculator (including those that program the calculators)... I was taught during the 80s, and there was no BODMAS/PEMDAS... we were just taught how to perform the functions for addition, subtraction, multiplication and division... AND were very clearly told that IMPLIED multiplication took precedence. Somewhere down the line someone got lazy, missed a "rule", thus rewriting the "rules" incorrectly. Maths HASN' T changed, the cultural "mis-"understanding of it has! Thank you for this video clarifying this.

[@JimFikes]

Thank you. I teach computer science in a high school in Texas. One issue in programming is that few languages allow implied multiplication (I think only MATLAB and Mathematica can accept such formulas). However, our students come to my classes having been taught strict PEMDAS. When we see a formula such as 3 / 2(1+2), we have to stop and find out what the author of that formula intended. Coding requires explicit notation -- and I'm thankful for it. But this does make a great teaching point in my classes.

[@sbyrstall]

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).

[@pulsar22]

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

[@jiminverness]

_"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 //)."

[@Matticitt]

10:00 that's because of the acronym itself. PEMDAS isn't known anywhere else. When I was taught math in school we simply learned "parenthesis > mult + div > add + sub". The acronym causes problems like implying you need to do multiplication before division because it comes first in the acronym. When it comes to things like 5(2+4) I was taught to do 5*2 + 5*4 = 30 so essentially to include the 5 with "do parenthesis first". This has the same result as 5 * 8 = 30 but you get less confused on the order of operations with longer equasions.

[@angeldude101]

That's the distributive property and is largely unconnected to order of operations. It's an actual mathematical rule rather than just a notational convention.

[@glasstransport]

Thank you so much for posting the PEMDAS/PEJMDAS videos!!! I feel vindicated! PEJMDAS (though we weren't taught by acronym) is what I learned in school in the 60s/70s. These younger people kept telling me I was wrong until they had me believing it. I knew there was something weird about their answer, but I couldn't put my finger on it. Now I know! THANKS AGAIN!!!!

[@CiscoWes]

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! 😂

[@justingeorge7910]

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

[@Epic_C]

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.

[@Zyphera]

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

[@derorje2035]

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.

[@GraemePayne1967Marine]

Thank you very much for this video! I went through my education before electonic calculators existed. (I am in my 70's) For complex math I sometimes used a slide rule. In later years, when _low-cost_ (under $100) calculators became available, have used those from several manufacurers. I rarely had two so i did not notice this problem until later in my working life. I discovered, while taking or proctoring professional certification exams, that people would get differing results from time to time. This eventually resulted in at least one professional society specifying which models of calculator were suitable for the exam. Whenever I personally had a question about a result, I would go back to the old way I learned in school - write it out the long way and do each step - and accept that as the answer. I also remember when my father (an engineer) first got a HP scientific calculator. After using it for a week, he did two things. First, he ordered one for each of the engineers who worked for him. Then after they were delivered, he canceled the mainframe computer time-share service he had been using for many years. Looking back, I now realize that I also rarely ever saw him using a slide rule after that.

[@normfromga]

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!😊

[@FosukeLordOfError]

Someone else said this in a reply and it fits for me too. Pedmas is taught in elementary school and then you never see the division sign again in your higher level math courses because they just put denominator under the numerator. Juxtaposition only comes up when writing complex formulas on a single line.

[@nirfz]

That video may be already 4 years old, but it is the best one on the "issue" i have come across. the last one before this, 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. As an engineer, we always learned to use brackets to be sure of the intention, so with the calculations presented in the "facebook maths problem stuf" i tend to say there are 2 possible solutions and in a maths exam (thankfully i don't have those anymore for decades) i would write down both results and both ways expresed with brackets. It's the problem of people relying too much on technical devices and such a device can only be programmed to use one way of solving the calculation. The human brain can do more, but people don't want to use their brain, and don't get trained to use it.

[@franksnyder5754]

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".

[@SteveRowe]

Interesting. My (American public school in the midwest) education said that juxtaposition was identical to multiplication. So I would absolutely get this wrong, and probably will continue to get it wrong for the rest of my life. Also, I work as a professional engineer, so the ramifications of having these two rulesets will certainly have problems beyond inconveniencing a maths student.

[@shikeridoo]

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.

[@shikeridoo]

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.

[@geminirox8635]

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

[@justinvzu01]

​@@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.

[@Pajo25ify]

In Finland we are technically taught pemdas, but split into P E MD AS, and within MD juxtaposition takes precedence over normal multiplication and division, so technically we are taught pejmdas, just worded as pemdas.

[@josephbenjamin6426]

OMG!!! I LOOOOOOOVE this video! It will LITERALLY vindicate my (several) rants and arguments I’ve had with literal MATH TEACHERS over their “over simplification” of higher-level math practices. THANKS! 🙏🏽

[@jamesglendening5180]

All of my middle school and high school math courses were in America in the 1990s. We were taught the order of operations, but never an acronym for it. We were taught that multiplication always comes before division, which usually functioned the same as your description as PEJMDAS. My teachers always insisted that we should never use our calculators to solve more than one operation at a time. When the math courses became more advanced and the restrictions were no longer enforced, several students complained to the teachers about calculators getting the answers wrong when combining multiplication and division. The teacher's response was always that all the math teachers were worried about students relying too much on the calculators doing their thinking for them so they got together and told the calculator manufacturers that they had to make sure that calculators gave the wrong answer when combining multiplication and division. I don't know if my teachers were just making up the reason for it, but it was the same story from multiple teachers and it does seem like the type of thing American teachers would do.

[@mitchbogart8094]

People aren't going to like this, but I feel like saying, "Serves you right for not getting behind RPN (Reverse Polish Notation), which not only uses far fewer keystrokes, but also has none of this PEMDAS or PEJMDAS hilarious nonsense. The problem is essentially translating multi-line algebraic notation into a linear sequence of keystrokes. Both my HP42S calculator and my Windows HP42S calculator app neither have nor need any parenthesis keys! Think about that. 50 years ago (in high school) I felt sure that HP and RPN would win out and become the standard because it is so much better! Sadly, it didn't work out that way. Yet, we holdouts still feel that we won! RPN requires only a little bit of meeting the calculator halfway. The advantages are immense. One also gets to immediately view the intermediate results of all calculations. Here is 6/2(1+2) in RPN. When I key it in, I simply choose which I mean: If I mean (6/2)(1+2) = 9, I simply key in: 6 Enter 2 / 1 Enter 2 + * and I get 9 (9 keystrokes) If I mean 6/(2(1+2)) = 1, I simply key in: 1 Enter 2 + 2 * 6 Swap / and I get 1 (9 keystrokes) I could have put the 6 on the stack first and it would eliminate the Swap making 8 keystrokes.

[@Dweller777]

I went to school in the 80s and early 90s... I was using Facebook for *years* before I ever heard of "PEMDAS." We were just taught to do parenthesis first, then implied multiplication, and to always replace the (ambiguous) "divided-by" symbol and write the equation in the form of a fraction... Then solve.

[@i_a_r_n_a]

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

[@markprange2430]

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

[@DorFuchs]

Great video! Information tracked down to some actual sources and also links to everything in the description. Thank you.

[@THaWoM]

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?

[@RS-fg5mf]

@@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....

[@Araqius]

@@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.

[@cronnosli]

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?

[@jamey7003]

@@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.

[@aikiwolfie]

Interesting video. When I was at school we were taught BODMAS. Brackets, Ordinals, Division, Multiplication, Addition and Subtraction. We would always do the juxtaposed multiplication after the Brackets.

[@Mercadian]

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.

[@thewhitefalcon8539]

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)

[@armor2009]

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.

[@trevorritchie2575]

My teachers lied to all of us😢

[@nocturnal6863]

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.

[@technerd9655]

The solution to this calculator problem, that I was taught in school is to not rely on your calculator to do all the work, you use the calculator to do the individual calculations. Also, if you avoid linear notation, then this isn't an issue.

[@tedunguent156]

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

[@SianaGearz]

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.

[@DiggitySlice]

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

[@deirdre_anne]

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.

[@hustler3of4culture3]

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

[@robertdeland3390]

Bottom line is when you originate a problem always parenthesize to eliminate ambiguity. When solving some one else's written problem write down parentheses to show your choice. Also, for sure follow what the instructor says.

[@ramonfreire4860]

it assumes 2(2+1) are two terms but when an algabraic expression appears defines 2a as a single term. treats numbers and algebraical expression differently

[@RS-fg5mf]

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

[@alvallac2171]

*algebraic

[@jiminverness]

Just to add to the video, From 9:00 you can read what CASIO wrote to her, and here is most of it transcribed: CASIO describe their PEJMDAS as: " take following process about calculations because they said that it is natural to calculate inside parenthesis and multiplication with abbreviated multiplicative mark right before parenthesis as top priority, after that to calculate the divisions at the both side: 6÷2(1+2) =6÷6 =1 "Since 2005 we have launched our calculator fx-ES series to the market ... we adopted an idea supported mainly in North America. They say it is natural to calculate a formula with abbreviated multiplicative sign just the same as a formula without multiplicative one. Based on our hearing result at that moment and under the calculation process below: 6÷2(1+2) =6÷2x3=3x3 =9 "After launching out calculator FX-ES PLUS series, such as fx-95SG PLUS ... since 2008, our calculator returned to the same specification on the calculation order in a formula as the unit like FX-570MS, base on latest hearing result; we adopted the way of thinking to calculate inside parenthesis and multiplication with abbreviated multiplication sign right before parenthesis as top priority, after that to calculate the divisions at the both side. And this way of thinking is to be kept using as a specification at Casio future calculator products, such as fx scientific series."

[@okaro6595]

You missed the 2 in the first one.

[@jiminverness]

@@okaro6595 You're right! Thanks. 🙂

[@D2SProductions]

I think the problem we have in North America is that grade school teachers aren't experts in math, they simply have a master's degree in teaching, so it's assumed that they know enough about math to teach it to students. When I was in high school I wanted to take the pre-college math class they offered, but the class was full, so I was stuck having to take business math instead, so I didn't get to learn PEMDAS until I was in college, but in college I was taught that when you see an expression written as a fraction to treat it like the things on both sides of the, "/," are in parentheses.

[@robertjenkins6132]

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).

[@Dhalin]

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.

[@somewhatfunnyguyy]

@@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.

[@Dhalin]

@@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.

[@somewhatfunnyguyy]

@@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.

[@StephenBoothUK]

I’m in the process, as a back-burner project, of writing a maths textbook, current title is “Why Does Maths Have To Be So Hard”. I will definitely be talking about PEJMDAS, and warning about the dangers of dogmatic adherence to PEMDAS. I was taught BODMAS in secondary school (UK, 1980s) but even then it was just understood that implied multiplication (aka multiplication by juxtaposition, although I don’t think I’d have been able to spell juxtaposition back then) was a special case and higher priority. Certainly by the time I reached university and was studying Electronic Engineering (which is basically all maths at that level) and Engineering Mathematics (an IEEE accredited diploma course required to complete the first year of the Electronics degree) implied multiplication being higher priority was just taken as understood.

[@jacquesmalan5950]

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

[@StephenBoothUK]

@@jacquesmalan5950 I don’t think anyone really thought about it, it was so ingrained that it just went without saying. I saw a KZbin 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.

[@gwpcs]

In the UK I was always taught BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction) which puts all division above multiplication implied or not. This could be part of the issue here

[@jakemccoy]

That is same as PEDMAS (instead of PEMDAS).

[@JohnRunyon]

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.

[@jakemccoy]

@@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.

[@RnRnR]

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

[@Orchestration1983]

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.

[@nbecnbec]

One pretty good solution is to avoid the ÷ and / symbols in your notation. Always use fractions. In college so far I've only rarely encountered the ÷ symbol and only occasionally encounter a /. And I've never encountered either of them in an ambiguous situation.

[@nejchrovat2685]

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

[@okaro6595]

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.

[@bigpod]

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

[@fernandaroig2964]

PEMDAS has always felt completely arbitrary to me, I always use parentheses to eliminate any possible ambiguity. It’s a habit from using tons of Excel in uni, but it just feels like it makes it so much easier to avoid any mistakes. Idk why people don’t just do that.

[@rightwingsafetysquad9872]

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.

[@lio1234234]

@@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...

[@rightwingsafetysquad9872]

@@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

[@MeToob]

@@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).

[@MadocComadrin]

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

[@Daemon6437]

I was just starting to learn Algebra in the U.S. in the early 2000's, and I remember being discouraged from using the multiplication operator at all except in the simplest of situations. I was taught to make liberal use of parenthases both on paper and with calculators, and to try to write all multiplication through juxtaposition. This was also the height of the oft-memed "You're not always going to have a calculator in your pocket" philosophy, so our textbooks made a point of formulating problems so that a calculator could rarely ever be used anyway.

[@franksnyder5754]

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

[@blackbeardsghost6588]

I like to take advantage of the differences. Let's say I have 6 beers. I want to share with my friends. I have two groups of friends, composed of 1 man and 2 women each (1+2). I divide the beers between the two groups and find that each person gets . . . 9 beers! That's why I'm so popular!

[@ivanmelendez1206]

Interesting.... Pemdas was probably done by some educator trying to oversimplify things.... it is not a mathematical rule. Now, I am interested in what you think about this: 45÷(9+6)3 With Pemdas it should be 9, and with me Pejmdas it should be 1, correct? I put it in a Sharp calculator, and also on a TI-84. The TI-84 did give 9, however the Sharp gave a syntax error. Can you explain that?

[@THaWoM]

Calculators don't usually recognize it as multiplication if the number is after the parentheses, so 45÷3(9+6) would likely give 1 on the Sharp but 45÷(9+6)3 is a syntax error. I can't find my Sharp right now, but that's what my HP and both the Casios do. I think it's probably in case someone types (9+6)3 when they meant (9+6)³ - they want to give an error rather than letting the user think it's calculating something it's not. However, you can probably store 3 in a variable, and type e.g. 45÷(9+6)A to get 1. That works on my Casio.

[@ivanmelendez1206]

@@THaWoM Thank you for your reply. I simply put the 3 in parenthesis and it gave me 1. 45÷(9+6)(3)

[@GanonTEK]

Depends on the calculator but here are some that give one or the other: These give 1: Casio FX 83GTX Casio FX 85GT Plus Casio FX 50FH Casio 991ES Plus Casio 991MS Sharp EL-546X Sharp EL-520X TI 85 These give 9: Casio FX 570MS Casio FX 83ES Casio 991ES TI 86 TI 84 TI 83 Plus If we just follow international guidelines, like ISO-80000-1 for the case of division on one line with multiplication or division directly after, and use the required number of brackets to remove ambiguity then all calculators would agree. (6/2)(1+2) or 6/(2(1+2)) Problem solved.

[@RS-fg5mf]

@@ivanmelendez1206 45 --------- ×3 = 45÷(9+3)×3 or 45÷(9+3)3 or 9+3. 45÷(9+3)(3)= 9 NOT 1 45 ----------- = 45÷((9+3)3) = 1 (9+3)3 A vinculum (horizontal fraction bar) is a grouping symbol and 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.... ______ 9+3. = (9+3) _______ (9+3)3. = ((9+3)3) ________ 3(9+3) = (3(9+3)) _________ 3×9+3×3. = (3×9+3×3) _________ 9×3+3×3. = (9×3+3×3) When you remove the grouping power of the vinculum you must replace it with the grouping power of parentheses in order to maintain the denominator....

[@ZapAndersson]

This is amazing, and I especially love how you figure out Wolfram Alpha is not only WRONG here, but gloriously INCONSISTEN!! I just wish I could get this video in front of the eyes of Dave at Dave's Garage, who just made a super-embarrasing PEMDAS video. That guy used to be my hero, but there he totally jumped the shark. I also loved how you formally said - as I've always said - "PEMDAS" isn't a "RULE", it's mnemonic for small children.

[@shmackydoodRon]

I’m an English teacher in North America. You should see the stupid things we do with spelling, commas, and quotation.

[@crimfan]

One thing I was taught in my post-secondary and graduate technical education, including a graduate degree in mathematical statistics, was ALWAYS use groupings and intermediate variable definitions in a computer or calculator.

[@gregshonle2072]

Having been a firm "believer" in PEMDAS, and having just stumbled across your videos, your videos have been rather enlightening! So, it would seem the real problem of PEMDAS vs PEJMDAS is due to shortcuts taken to simplify typesetting in research papers and textbooks (most likely to save on typing and/or space on a page). From comments I've seen on another video of yours on this topic, I think some would agree. However, I'm sure math/physics writers and publishers would object to the extra effort. E.g. writing x = e^1/2pi as either x = e^1/(2pi) or putting a long division bar between 1 and 2pi (and the latter would probably need a parenthesis around the block, to make sure what is being raised to the e power). So, hmmm... Further, as a software engineer, I always use parenthesis in math and logical expressions, to avoid any possible ambiguities -- not all computer languages agree on order of evaluation...

[@Moleculor]

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.

[@pulsar22]

@@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)

[@Moleculor]

@@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.

[@dannydonnelly8027]

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…

[@MadocComadrin]

​@@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.

[@ToddLuvsGolf]

By the way...excellent video with spot on analysis and explanation as to why so many modern educators haven’t a clue of PEJMDAS’ existence. The entire physics world would be wrong on newly every expression describing the world around us. That’s scary.

[@meganwilliams2962]

Not to mention population genetics and epidemiology calculations.

[@ToddLuvsGolf]

@@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.

[@RS-fg5mf]

@@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

[@tommy8290]

@@RS-fg5mf you are just embarrassing yourself now

[@RS-fg5mf]

@@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.

[@MartinSlucutt]

To me 6/2(2+1) and 6/2*(2+1) have always meant two different things, especially when you write them with spaces included: 6 / 2(2 + 1) and 6 / 2 * (2 + 1) or have a variable involved: 6 / y(2 + 1) and 6 / y * (2 + 1). And because of what they represent when written in full with division taking two lines, for the former it becomes obvious the 2(2 + 1) is all under the "dividing line", i.e. being the denominator. In the latter (without parenthesises) only the 2 is the divisor of the 6. Evaluating them using PEJMDAS gives two different answers whereas evaluating them using PEMDAS gives the same answer. PEJMDAS is therefore more expressive and whilst brackets should always be used to clarify (and document) intent (especially in programming instead of relying on precedence rules), PEJMDAS avoids the unnecessary use of them through the simple use of implicit vs explicit multiplication to differentiate the two. Where the use of the explicit multiplication: 6 / 2 * (2 + 1) can be used instead of having to resort to (6 / 2)(2 + 1) or (6 / 2) * (2 + 1). And with the use of implicit multiplication where 6 / 2(2 + 1) can be used instead of having to resort to 6 / (2(2 + 1)) or 6 / (2 * (2 + 1)).

[@timothygunckel7162]

Your video is the first I've seen that explains this senario. Very well done. You could almost tell when someone was born by the answers they were giving to the problems.

[@MrStevemur]

The psychology of this is interesting, especially the textbooks that stated PEMDAS as the rule but didn’t follow it themselves. For myself, I’m not very conscious of multiplication when it’s implied. If it were written explicitly, more people might think about where it should fall in the order of operations.

[@eekee6034]

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.

[@flummer7]

@@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.

[@marccarney7]

I love your explanation. It's the best yet. I've been debating this online for over two years, and I'm from UK, and it's true that we always treat juxtaposition as taking priority, so my solution has always been 1(not 9). This video deserves an award, I can't really thank you enough for your hard work and research 😘👍🥂

[@adamwalker8777]

only 9!!!!

[@MuffinsAPlenty]

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

[@RS-fg5mf]

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

[@RS-fg5mf]

@@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...

[@marccarney7]

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

[@0LoneTech]

There are quite a few things I like about Qalculate! Here's an example: > 1/2sqrt3 Please select interpretation of expressions with implicit multiplication. 0 = Adaptive 1/2x = 1/(2x); 1/2 x = (1/2)x; 5 m/2 s = (5 m)/(2 s) 1 = Implicit multiplication first 1/2x = 1/(2x); 5 m/2 s = (5 m)/(2 s) 2 = Conventional 1/2x = (1/2)x; 5 m/2 s = (5 m/2)s Parsing mode: 0 1 / (2 × sqrt(3)) = 1 / (2 × √(3)) ≈ 0.2886751346 > 1/2 sqrt3 (1 / 2) × sqrt(3) ≈ 0.8660254038 Yep, it explained the issue and asked which interpretation to use. It also notes the result is inexact, how it was interpreted, supports intervals (error propagation) and units.

[@donnellterry9584]

When using a calculator, I always put parenthesis to make sure I get the intended outcome.

[@analholes77]

In Germany in primary school (grade 1-4) we learned "Punkt vor Strich", which translates to points (•,÷) before lines (-,+). At that age we don't know about fractions or parentheses. Then in 5th grade first we learn that parenthesis always need to be calculated first. 2(3+2)=6+4. After that we learn about fractions and got told that ÷ is just a sometimes more convenient way to use a fraction bar. Problem solved, no calculators needed.

[@EphemeralPseudonym]

In the US we learn fractions before parentheses

[@estebanod]

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

[@sethwilliamson]

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. 👍

[@jonselleck9635]

The switch to ignore the multiplication by juxtaposition is the prefect representation of the dumbing down of society. There are many visual representative advantages to prioritizing the implied multiplication. If I don't want the a/(bc) in that order without the parenthesis and instead i wanted (a/b)c then i would just write a / b * c. a / bc should always be a / (bc). The world is degrading every day. Thank you for your efforts in preserving some sanity on this issue.

[@skaeggo]

Or, better: if you want (a/b)c, just write ca/b.

[@valdir7426]

so basically strict enforcement of 'PEMDAS' is gaslighting from North American school teachers (incredible how some of them will not accept the reality of the usage and just tell advanced mathematicians they're just wrong). a bit of intellectual curiosity should help in this matter.

[@oliverknill631]

This is a great compilation. Some examples I have never seen. It should be seen by any teacher of algebra. Thanks!

[@RS-fg5mf]

@@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...

[@RS-fg5mf]

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

[@jamey7003]

@@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.

[@RS-fg5mf]

@@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

[@RS-fg5mf]

@@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....

[@RupertMDoc]

Fantastic topic covered very well. This also comes up in coding (I teach new hires SQL, SAS, and R). I always state the order of operations should be obvious when a person reads the code, which typically means adding more parentheses. Even if the expression is "right" for the language, ambiguity will create confusion if the code ever needs to be read/bug hunted/modified in the future.

[@ozok17]

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).

[@ozok17]

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

[@anon_y_mousse]

I never really gave it much thought, but I just instinctively group implicit multiplications with parens when I convert algorithms into code. Didn't know some calculators got it wrong, so I'll have to be careful with that in the future if I ever use a physical calculator again, instead of bc. Although, obviously I only use bc for prototyping because it's slower than molasses in a freezer.

[@rightwingsafetysquad9872]

I had never looked nearly this deeply at the topic, but I had observed that every textbook or paper and in my own unconscious usage, juxtaposition implied paratheses. In much the same way that fractional notation implies parentheses when converted to a single line for entering into a calculator.

[@timehunter9467]

I’ve heard PEMDAS, PEDMAS, BODMAS and BIDMAS, I group division and multiplication together and do left to right, then group addition subtraction together the same. Very interesting video.

[@gonnfishy2987]

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

[@timehunter9467]

@@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.

[@gonnfishy2987]

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

[@zlcoolboy]

The explains when my replacement TI calc was worse than my broken casio despite being more expensive. I couldn't fathom why it felt different when I entered equations. Its because my new TI is pemdas, but my casio is pejmdas. Despite this, I think its better to just always specify what you want the calculator to do with perenthesis (takes a little bit more time, but you don't want to be wrong when you're doing an exam).

[@sqlexp]

Thank you and this video very much for telling us that TI calculators are trash.

[@JohnnyKimchi]

Thank you for easing my mind that I've learned PEJMDAS my entire life and was correct. I've been seeing lately that on different platforms that PEMDAS where the famous FB answer is 16 instead of 1 and it was driving me crazy. Thank you for reminding me that I did in fact learn this correctly!!

[@RS-fg5mf]

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

[@tommy8290]

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

[@RS-fg5mf]

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

[@nickmcginley4570]

@@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.

[@nobodyknowsforsure]

@@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

[@eyeball226]

Hmmm, I've never heard the rule called PEMDAS before. In the UK I was taught it as BODMAS (brackets, orders, you can figure out the rest). I work in education and I mostly hear teachers use BIDMAS now, which swaps orders for indices. In any case, none of them are any better than PEMDAS because they don't specify juxtaposition.

[@jamey7003]

@The How and Why of Mathematics That is how we were taught (pejmdas) even though we were not taught pemdas. It was brackets/parenthesis then operation in the order it appeared. It is algebra that shows us that 2(a+b) is always one not separate operations. If it applies here, it must apply at all times is my understanding🤷‍♀️ Thanks for your video. The best, most informative I've seen🙂

[@RS-fg5mf]

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

[@tommy8290]

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!

[@RS-fg5mf]

@@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.

[@jamey7003]

@@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.

[@jamey7003]

@@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.

[@calebfuller4713]

That the Japanese calculators all follow PEJMDAS and only newer American calculators went with simple PEMDAS should be your first clue... 🤣🤣🤣

[@chitlitlah]

In my American school system in the 80s and 90s, we used the division symbol in elementary school arithmetic and multiplication was always explicit. Then as we started pre-algebra, implied multiplication entered the picture and the division symbol was completely tossed out in favor of writing division as a fraction in which case there's no ambiguity when the division should be done. So to me, it's like you're writing half the words in a sentence in English and half of them in French and then debating over whether you should put adjectives before or after nouns. You can't mix two different syntaxes or languages and have a meaningful debate over which one is correct for one particular element of the grammar.

[@shiinondogewalker2809]

the example she use in the video isn't two different syntaxes though. It maybe not be on either of the two forms you mentioned but that doesn't mean much

[@chitlitlah]

@@shiinondogewalker2809 She (or rather the person who came up with this problem) mixed implicit multiplication with the classic division symbol. I have never seen that done except in this very problem and apparently most other people haven't either given how much confusion and debate there is over it.

[@shiinondogewalker2809]

@@chitlitlah now this I agree with

[@PrezVeto]

I'm shocked-SHOCKED, I tell you-that most American primary and secondary school teachers are in favor of dumbing down the curriculum. 😒 What a disgusting state of affairs.

[@metalchemik]

I have watched this video and your previous one about problems with pemdas and yet I can't understand one thing. After such a detailed explanation with clear examples and historical brief review, why would anyone still arguing about this things?

[@RS-fg5mf]

Because the information she supplies is subjective and wrong... The confusion comes from the misconceptions and conflation of 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 6/2(a+b)= 3a+3b 6/(2(a+b))= 6/(2a+2b)= 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

[@tommy8290]

@@RS-fg5mf yawn. no one is listening are they?! must be very frustrating for you

[@RS-fg5mf]

@@tommy8290 you're an ID it. Must be very frustrating for you... I think I here your mommy calling... Go play little boy.

[@nickmcginley4570]

@@tommy8290 Don;t you get it Johnny? Everyone is being subjective, except RS. He is 100% objective. THE arbiter of math. Why, he is pretty much Math God.

[@7654321220]

@@RS-fg5mf How much mental gymnastics is required to pull out two sets of convention for "algebraic" (juxtaposition) and "parenthetical" implicit multiplication and put so much emphasis on the bracket existing or not? And please tell me how does your holy PEMDAS interpret cos2π/x by slipping in all those lovely parentheses until the order of operation is obvious without need to refer to any convention Sorry everyone else but I love to troll trolls

[@Drgn8DDragonsDungeon]

(sorry for spontaneous wall of text!) Modern calculators are why I now always manually insert the brackets to force the juxtaposition, because I cannot trust them anymore. My phone also does pemdas instead of pejmdas, it's extremely frustrating. In my opinion, also a result of very lazy coding, or inept coding. More, when I replace every item in the example equations with variables and reverse-engineer the question (ie, a/b, where b= cd, where d=x+y), people still vehemently argue that it's fair to separate the value of c to become part of value a. I've had people argue that *I'm* the one changing the equation by doing this, asking me if I understand basic algebra, etc., and it... it just drives me nuts that this is the way math is being taught these days. Kids being told they can insert a normal multiplication symbol to the first bracket and that's it, they've resolved the brackets (they just drop the brackets entirely). I've even tried explaining it as when you take say, 2+4, and pull out the factor 2, you now have 2 sets of (1+2), that's why you can't make it 6 over 2, times 3. But... they've not been taught this I guess? I'm entirely dumbfounded at this point, that the implied multiplication by juxtaposition has been utterly lost on at least the youngest 2 generations in North America, so now modifying the equation is the expected result :( Thank you for your incredibly thorough and informative videos

[@nwgverified]

Yeah same. I am just paranoid that it will be wrong, so I just put a bunch of brackets to ensure im not gonna lose marks for no reason

[@THaWoM]

Yeah, I would usually type something like 1/(AB) into the calculator as 1/A/B. I know I could type 1/AB on my casio but it takes longer to go through the mental checks ("am I sure this isn't actually my phone calculator?" etc).

[@Drgn8DDragonsDungeon]

@@nwgverified I'm thankful that I'm well out of school years now so I don't have to worry about this coming up for me personally, but I'm increasingly concerned for my nephews who are in grade school at this point. Maybe it's time for me to take this into my own hands...! A zillion brackets for clarity never hurts, you can never be too precise in mathematics, as far as I'm concerned.

[@Drgn8DDragonsDungeon]

@@THaWoM I guess, in a way, it doesn't hurt to reinforce the habit of adding the () structure for explicit notation, either. It's just such a shame that it even had to come to that point where we can't trust our tools implicitly to do their expected job. =\ (I look forward to more of your interesting explanations on obscure topics, you've made sense of so many concepts that weren't explained properly before!)

[@nwgverified]

@@Drgn8DDragonsDungeon I have trials next week :(. This is like the 2nd most major exam in Australia for high school students rip

[@meganwilliams2962]

My 2018 Casio fx 82AU PLUS II Still applied PEJMDAS. Why just Nth American teachers? Maybe the same reason that the USA doesn't use the Metric system?

[@RS-fg5mf]

It's not just North American teachers... AND CASIO has admitted to programming different models for different markets based on popularity and opinion not the rule of math. The Order of Operations and the various properties and axioms of math when correctly applied give you 9 not 1

[@jiminverness]

@@RS-fg5mf Not true. Casio's take on it is in the video. From 9:00 you can read what CASIO wrote to her, and here is most of it transcribed: CASIO describe their PEJMDAS as: " take following process about calculations because they said that it is natural to calculate inside parenthesis and multiplication with abbreviated multiplicative mark right before parenthesis as top priority, after that to calculate the divisions at the both side: 6÷(1+2) =6÷6 =1 "Since 2005 we have launched our calculator fx-ES series to the market ... we adopted an idea supported mainly in North America. They say it is natural to calculate a formula with abbreviated multiplicative sign just the same as a formula without multiplicative one. Based on our hearing result at that moment and under the calculation process below: 6÷2(1+2) =6÷2x3=3x3 =9 "After launching out calculator FX-ES PLUS series, such as fx-95SG PLUS ... since 2008, our calculator returned to the same specification on the calculation order in a formula as the unit like FX-570MS, base on latest hearing result; we adopted the way of thinking to calculate inside parenthesis and multiplication with abbreviated multiplication sign right before parenthesis as top priority, after that to calculate the divisions at the both side. And this way of thinking is to be kept using as a specification at Casio future calculator products, such as fx scientific series."

[@percival413]

i know thst its probably just from trying to keep quiet/not disturb roommates or neighbors but your voice is so calm and so clearly articulated and it really helps me understand what youre saying

[@Matt-qi5ff]

this problem exists because we try to fit mathematical writing into latin orthography, which produces ambiguity. they're different systems, so it could be fixed if there was a way to write them separately on computers or if there was an official/standard adaptation rule

[@haphazard1342]

Right. There is absolutely no confusion when using "full" division symbology with horizontal bar. Then the hierarchy is clear and juxtaposition is obviously first. But compressing the expressions into single-high or "shorthand" gives us the ambiguous division operator and thus order of operations.

[@salvatronprime9882]

PEJMDAS is the way I learned in American school in the 90's. In Texas, no less. The answer is 1, yes.

[@RS-fg5mf]

FAIL

[@sonnybunny4279]

Hello. My original answer to 6÷2(1+2) was 9, however, upon watching your videos, I want to switch over to 1. I still believe that in a literal format, the answer would be 9; however, the more logical answer would be 1. It's like asking what the answer is to 2y/2y would be. The more logical answer is 1, or y, but following the left to right rule, you'd get the answer to be y². So how would people end up with a squared variable? As you stated, a majority of people, and historically all people, would interpret a/bc as a ------- (b*c) In algebra, you learn that two like-terms can be divided by each other, which also means that multiplication by juxtaposition is already implied (the coefficient and the variable). But when people don't follow this rule, that's how you end up with a squared variable. I personally believe that we (the teachers and educators in North America) do need to teach kids that when we "combine like-terms," we are also doing multiplication by juxtaposition. But because 2(1+2) are not like-terms, we don't understand to do juxtaposition first, we are taught early on to go left to right, and that 2(3) is the same as saying 2x3. I believe because no variables are present in the equation 6÷2(1+2), we should follow the left to right rule. We are more likely to see this problem in elementary (primary) grade levels than in more advanced levels. So should we teach younger kids that the answer is 9, or should we teach them that later on you will need to learn juxtaposition to understand more algebraic equations, and show them how to get the more acceptable answer, which is 1?

[@mateussousa6606]

9 makes more sense at all. If you think multiplication and division are in fact the opposite operation you'll have 6×0.5(1+2). This problem you only have 9 in the solution. But I think when you have variables in the problem it's simpler to understand if they follow PEJMDAS, and it's easier to write in just on line. With PEMDAS you'll need 2 lines.

[@RS-fg5mf]

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 = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b)

[@salvatronprime9882]

@@RS-fg5mf 6/(2a+2b) gives the correct answer, which s 1. 9 is simply the wrong answer.

[@RS-fg5mf]

@@salvatronprime9882 6/(2a+2b) does not equal 6/2(a+b) you can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol... 6/(2a+2b)= 6/(2(a+b)) 6/2(a+b)= 6/2*a+6/2*b The correct answer is ABSOLUTELY 9

[@salvatronprime9882]

​@@RS-fg5mf Yes you can, it would be the same as if you cross multiplied by 2(1+2). 6/2(1+2) = Y 6 = Y(2(1+2)) 6 = Y(2)(1) + Y(2)(2) 6 = 6Y

[@ScienceTalkwithJimMassa]

And if one follows BIDMAS, it gets more confusing! PEMDAS - do multiplication first, then division; BIDMAS, do division them multiplication. It is crazy confusing. I like what that calculator you showed did with inserting the brackets. I am a firm subscriber to use parentheses, so that there is no confusion. Example you had 6/2(1+2) and the result is either 9 or 1. I would write: (6/2)(1+2)=9 or 6/[2(1+2)]=1 (as that calculator did). This removes any confusion that may result.