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 =_

[@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.

[@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?

[@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%!

[@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.

[@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.

[@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.

[@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?

[@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.

[@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'.

[@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.

[@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.

[@fucku2b]

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

[@KanpachiGaming]

@@fucku2b if it was a test question then try supplying them with both working solutions lol

[@fucku2b]

@@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.

[@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.

[@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?

[@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.

[@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.

[@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."🙂

[@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

[@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.

[@Voyager-3]

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

[@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! 🙏🏽

[@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

[@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".

[@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

[@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

[@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.

[@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.

[@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.

[@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.

[@donnellterry9584]

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

[@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.

[@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.

[@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 //)."

[@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.

[@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.

[@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.

[@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

[@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!

[@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. 🙂

[@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.

[@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.

[@butre.]

if I want 6/2(1+2)=1 I'd input 6/(2(1+2)), if I want 6/2(1+2)=9 I'd input (6/2)(1+2). I've always thought of juxtaposition as being part of the parenthesis and I'm not gonna let my calculator get it wrong for me

[@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.

[@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.

[@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

[@ssbmfan4]

It's surprising how many people think PEMDAS is part of math despite it being something we made up. I got into an argument once about this concept over 10 years ago, and I'm still mad about it. Someone sent me one of those "You're a genius if you can solve this" math expression Facebook memes, asking me for my answer. If you're not familiar, it's basically just a bunch of numbers with basic operations thrown between them to look confusing. I responded stating that there is no correct answer because the formula was ambiguously written, and that "the answer was either 9 or 2 depending on if you choose to multiply first or divide first." They prodded me for an answer, so I said 2. They went on to state that you should divide first because it occurs first in the formula. I explained that we made up how math is parsed and there isn't just one way, and cited that different computer systems can give different answers for the same math expression. He responded with "My aunt is a teacher and we're both laughing at you." I didn't even feel foolish or embarrassed, because I knew I was right. I was just annoyed that I spent the time trying to explain parsing to someone who just ignored it and laughed it off. At the time I was unfamiliar with PEJMDAS as an acronym and wasn't too sure how to explain it, so it's partially my fault for not knowing the terms. I know the terms now, so at least I can explain it more clearly if it ever comes up again 🤷‍♂

[@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.

[@nakitsukikuronuma]

I'm just going to save this for the eventual day this becomes relevant again

[@r1234233]

I just thought the same thing lol

[@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.

[@WMHinsch]

Pardon the length of this post, but it is a love story about certain calculators and a brief rant about others. I entered HS in 74. At the beginning of my junior year, at my older brother's recommendation I bought an HP as they were just coming out with the second generation of their scientific calculators. It was an HP 25C (which I had until just a few years ago, and which still worked when I discarded it). IIRC, the manual was a booklet about 115 pages long, spiral bound, and wonderfully detailed. Among other things, it taught RPN (Reverse Polish Notation), which is what all early HP's used. It didn't just teach how to push the buttons, but how to logically work on a problem, working it from the inside out, and juxtaposition done automatically by utilizing a stack. I still think that way. The ability to solve very complex problems with no parentheses and minimal keystrokes has saved my bacon on many occasions. About a decade after getting my 25C, I started an advanced military electronics school aimed at folks in the nuclear field. It was incredibly fast-paced and difficult - we averaged one technical college textbook and an exam every week or two for 26 weeks. The first day, the instructor had written a complex problem that filled the chalkboard, had nested levels of parentheses, and used about every function on a scientific calculator keyboard. He instructed us to use our calculators to solve the problem, write our names on a piece of paper, write the answer if we had time to complete it, fold it, and turn it in at the end of the fifty minutes whether complete or not. I was the first one finished in about ten minutes (and I worked the problem twice to be sure). About 20 more minutes elapsed before the next person turned in their answer, at which point I was beginning to sweat. It didn't help that when I turned it in, the instructor gave me a look that indicated he highly doubted I got it right. At the end of the time limit everyone was finished, with the last couple of folks racing to the desk. I was one of the few that actually got the right answer. It wasn't any particular smarts on my part, but a combination of knowing my calculator and using RPN. During that school, I purchased the HP 15C for its improved display, battery life, and programming capabilities. (I still have that one after more than 35 years, and it works just fine.) Now I use a 15C partial emulator (no programming) on my phone. Those early HPs were beautiful calculators. So well made. Now "schools" force kids to buy crappy $140 TI graphing calculators that cost about $10 to manufacture, which gets an automatic "F" (for "Fraud") in my schools grade book.

[@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.

[@GeirEivindMork]

As an avid programmer even back when I was in high school and university, I always bracketed anything my calculator could mistake. I am quite sure the teacher also told us to do so given I'm sure he have got a few errors himself.

[@alvallac2171]

*so, given *he had gotten

[@GeirEivindMork]

@@alvallac2171 your right (Intentionally misspelled to annoy you)

[@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.

[@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.

[@jonathanrealman8415]

I am from Germany and instead of orders of operation my math teacher just taught me to put everything in parentheses and if someone else doesn't you yell at them.

[@Moriarty70]

Here's my argument. I've never had higher-level math beyond high school, and I still put the juxtaposition higher than basic multi/division. The extra brackets are automatic in my brain. It's a matter of literacy as well as math. Learning how to understand what you're reading and what's being asked is just as important as knowing how to do math. I've always said, the arts classes are as important to understanding the world as technical classes.

[@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.

[@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

[@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.

[@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

[@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.

[@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.

[@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....

[@aylivex]

Finally, someone said that. I have a mathematical education, and multiplication by juxtaposition naturally comes before division, otherwise it breaks lots of conventions. For example, it seems everyone agrees that 6÷2x means 6÷(2×x) rather than (6÷2)×x. But then if x is an expression in parentheses, most people tell you that you should divide first.

[@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.

[@ilovemonkeyos]

Math teacher: “Numbers don’t lie.” Me: DOUBT.

[@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.

[@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

[@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.

[@thoughtsengineer]

In South America we also use PEMDAS system (kinda). Well basically the rule is to always put parentheses or write proper fractions just to avoid confusion.

[@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.

[@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)).

[@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.

[@christopherstafford1788]

I'm an American, and I was taught to distribute the multiplicand across the parenthesis to get rid of the parenthesis. If you are doing anything else while the parenthesis is still there, you get the wrong answer. The answer is 1.

[@whatsinaname7289]

This was a very interesting video about a problem I didn't even realize existed. I love deep mathematical discussions and I'm very glad I found your channel.

[@SimpsonDG]

What you SHOULD do is get a calculator that uses Reverse Polish Notation (RPN), like the "classic" HP calculators or today's SwissMicros calculators. Then you don't have this issue at all. You do: 3 SQRT 2 x 1/x and get 0.2887.

[@FineBakedPastry]

I loved my HP48GX, but most people are not going to be able to use that. If they are at a level where they can use RPN, then they probably will not be confused by implied multiplication.

[@angeldude101]

I don't have any traditional calculators, but I use RealCalc on my phone which has an RPN mode that I use exclusively. Also, is that 1/x a division by one followed by pushing "x", or is it really a RECIP?

[@costakeith9048]

There's an efficient binary algorithm for taking the reciprocal of a number, so like square root, it doesn't actually require an additional entry on the stack.

[@chrisjfox8715]

I've always found it fascinating that languages, math included, are able to evolve towards a civilization-wide consensus on the meaning of things. Even more fascinating is that Mathematics has something as simple as this as its achilles heel, regarding a convention that we haven't managed to settle one. For us engineers, programmers, and mathematicians, it's easy enough for us to understand this unfortunate history/present and simply deal with it accordingly moving forward (i.e., use of parentheses, consulting manuals and documentation when in doubt, etc). But what's truly unfortunate are the everyday know-it-alls that can't be told otherwise because "Algebra was my favorite subject! You don't know math." Or the types that blindly take their calculator as gospel, which speaks to our future of people getting so used to trusting automation that they have no clue how anything works...and even worse, have no interest in questioning it. But given this contradictory convention, the idea that anyone on either side of it would claim that "no *this* is the proper way" is doing a disservice to it ever being resolved. The existence of the disagreement over the convention is proof that there is no convention yet, and we have to acknowledge that if we ever expect to arrive at a convention that works for everyone. That said, the genie may be too far out of the bottle at this point...maybe.

[@Pajo25ify]

The worst part is american math teachers, especially algebra teachers, have decided to simplify math, and then we have calculators programmed to fit math teachers. What the american math teachers have decided goes against mathematicians, physicians, engineers and international math teachers, but still calculator manufacturers make calculators to fit the math teachers specifics, because it's their biggest clientel. You have a situation where math teachers and calculator manufacturers create a circle of confirmation, confirming what is false.

[@jon9509]

This video needs to be linked in every one of those 'mindyourdecisions' clickbait videos and others like them. Great video.

[@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

[@philipmorse-fortier5499]

Thank you for this, the example of 5a / 2b was really illuminative and helpful for showing that even people who think PEMDAS are often doing PEJMDAS even if we don't realize it. Great video!

[@haphazard1342]

The missing piece here seems to be the typical context of juxtaposition: you can only use it on un-like values. The most common use case is numbers with variables, e.g. 2a. Obviously you can't use juxtaposition with two numbers: is 22 juxtaposition or a number? And with variables it is prioritized over multi-letter variables. This reveals the purpose of subscripts (and primes, in some cases) which exist because making the variable multiple characters would conflict with juxtaposition. When using traditional/stacked division, it's clear by relative position what juxtaposition means. Same for other expressions like integration, roots, etc. The problem really comes in using shorthand division and parentheses, both of which come from using single-height or "inline" forms of expressions instead of traditional full-size ones.

[@MrGreensweightHist]

They were also wrong.

[@calebfuller4713]

5a / 2b should clearly be done as (5×a) / (2×b). However, researching this online I came across writing from extremist devotees of the PEMDAS cult, even at university level, who insist juxtaposition isn't a thing and it should be ((5×a)/2) × b which is to me insane and basically breaks algebra.

[@MrGreensweightHist]

@@calebfuller4713 "which is to me insane and basically breaks algebra." Not our fault you don't understand basic algebra. 5a / 2b = (5×a) / (2×b) This is correctg. "who insist juxtaposition isn't a thing and it should be ((5×a)/2) × b" Literally nobody is saying that. making your claim a strawman argument. The problem with this video is... 5a/2b is not analogous to the problem at hand. 5/2b would be closer, and would come out to 5/(2*b) but that still isn't the problem at hand's format. The format we are looking at is... 5/2(b) And because ( ) are not juxtaposition THAT is (5/2)*b The original problem... 6/2(1+2) 6/2(3)

[@calebfuller4713]

@@MrGreensweightHist it's not my fault you don't understand basic algebra. If you can't substitute a bracketed entity for a variable and still process the equation the same way, it is you who have broken the system.

[@giovannyarguello2645]

Take this POV from someone who learned Math in South America when scientific calculators back in times when those devices were rare and unaffordable; It isn't about what "rules about the hierarchy of operations you learned in school. It's really all about clearly defining the required operation's order using as many parentheses (or brackets) as needed. That's all.

[@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

[@dbutcher84]

This was very interesting. I believe I was taught pemdas through school since I remember all multiplication being on the same level. I must have been unaware of this great debate my whole life. I've always liked that math is clear with one correct answer, so this just blows my mind that no one has formed a standard to get everyone on the same page.

[@robertcherman]

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

[@robertcherman]

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

[@mscottreynolds]

NO! NO! NO! NO! NO!!! I have never heard of PEJMDAS before. I am, yet another, programmer and I have ALWAYS used PEMDAS religiously. I grew up with HP RPN calculators (The HP42S being my favorite and the subsequent Free42s emulator) and preferred them because I too had noticed the inconsistencies with the way "regular" calculators interpreted PEMDAS vs. PEJMDAS. It *NEVER* once occurred to me that this was actually a thing--I had just always assumed that half of the world was crazy when using PEJMDAS! x-D I have also, as a software developer, written my own compilers and interpreters and they always interpreted formulas using PEMDAS. This is madness!!! I am going to have to study this phenomena more. Thank you for pointing this out; don't know why this never occurred to me before! I'm subscribing to your channel.

[@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....

[@CyanPhoenix_]

interesting how wolfram is seemingly inconsistent in how it interprets inputs, but at least it also will redisplay the question in proper notation to show you what it's actually doing.

[@judelarkin2883]

Great video. If it adds too much difficulty or not, it seems like something students should know. It’s really difficult when you originally learn that one way is this universal rule and then find out there is a whole world of people doing it another way and have to sort of reverse engineer how they are doing it because they take for granted that their way is the universal rule.

[@emilwandel]

One would think school prepares for life not for tests. But yet here we are. I only learned point before bars and we write the symbols like that + - : • and later learned use para theses for complicated terms. 2 1/3 doesn't actually mean 2*1/3 but 2+1/3 when introducing fractions and when introducing variables we learned juxtaposition implicity as they said. We are lazy now write 2*a as 2a and mean this. Remember the weird fraction rules now better just use it at the end of a calculation otherwise you will get confused.

[@geonalugala]

I am grateful to you for this video. Quite insightful and well resourced. An additional consideration for pupils, is the idea that the answer, or result, is always part of its own proof. 6 ÷ 3 = 2 is an implied proof that 6 ÷ 2 = 3 and 6 = 3 * 2. The calculators giving 6 ÷ 2(2+1) = 9 are wrong; because, 6 ÷ 9 ≠ 2(2+1), irrespective of whatever the priorities of multiplication are. [Note: We have neither introduced new parentheses, nor interpreted the controversial factor, 2 (2 + 1), either way. We've only tested their answer against conventional division: x ÷ y = z, implies that x ÷ z = y. ] And, their wrong answer, 9, is eliminated by common sense and basic arithmetic. On the other hand, using 1 as the answer: 6 ÷ 1= 2 (2 + 1) affirms the equality. 6 = 2(3) = (4 + 2) = 6.

[@Snommelp]

Not to pick a fight, but I think you accidentally mistook the implied proof in your argument against the answer 9. 6 ÷ 9 ≠ 2(2+1) but 2(2+1) is not part of the equation that gives an answer of 9; you could express that equation as (6÷2)(2+1) = 9 So the implied proofs would be that 9/(6÷2) = 2+1 and 9/(2+1) = 6÷2 both of which are correct

[@geonalugala]

@@Snommelp I hear you. The simple mistake you make, is to introduce parentheses that were not in the question. You were asked 6÷2(2+1) You answered (6÷2)(2+1). Now, if you are keen on math, and cannot tell that these two are different, then you need remedial coaching. Don't tell us that you think the person who asked that question, and inserted just one set of parentheses, somehow, was ignorant of meanings. When someone writes Mom. And another writes Mom? Do you just suppose that they are stupid, and intended to shout Mom! Meaning is always about careful reading. And specific answering. By creating your own question, you can always answer it however you want. Once again, the question was 6÷2(2+1) It was not (6÷2)(2+1). Obviously, the two are different things, with different results.

[@Snommelp]

@@geonalugala I'm not saying that's how I read the equation, I'm saying that's how the people who get the answer of 9 read the equation And I'll give you the benefit of the doubt and assume that you're using a general "you" instead of meaning me specifically when you use the word "you," otherwise I'd have to rebut your comment about remedial coaching with my own comment about remedial reading -- accompanying your comment about not inserting parentheses with my own about not inserting words

[@geonalugala]

@@Snommelp The point, and indeed the whole argument here is that a simple statement of arithmetic like this one should not be read in different ways by different people. The only purpose of math is precision. Precision should not, and cannot have multiple meanings. 6÷2(2+1) is not the same thing as (6÷2)(2+1).

[@casadelosperrosstudio200]

​@@geonalugalaso you are implying an extra set of parentheses after the division symbol instead of before it...

[@technerd9655]

When you use clear notation implicit multiplication (which is how I was taught here in Canada) or multiplication by juxtaposition is basically irrelevant. If you use clear notation BEDMAS/PEMDAS=PEJMDAS. Here in Canada, we're taught to be explicit in our notation, and our text books and teachers follow this. Programmable or graphic calculators were also not allowed in class or during exams, we had to show we not only could do the calculations, but also understood the concepts. Additionally, don't rely on your calculator to do the whole expression, do each individual calculation separately, and it was/is well known even in high school that different calculators could give different results, especially if you put the complete expression in.

[@GanonTEK]

Exactly. To modern standards the question is ambiguous and badly written. Documents like ISO-80000-1 mention about fractions on one line and how brackets are needed to remove ambiguity now. So, something like 6/2(3) should be 6/(2(3)) = 1 or (6/2)(3) = 9 instead to remove the ambiguity caused by the multiplication by juxtaposition having higher priority or the everything after / is in the denominator conventions used in academic papers that arent used in programming. Better yet, don't write fractions on one line anymore 🙂.

[@mathewcherrystone9479]

I'm not sure what you mean by using it for individual calculations, but this only works in a neat "class room" environment. At university we had to do the maths exams without a calculator too, but they always had to make the tasks calculatable by hand. For every other class this was not possible since we used real world values. Addtionally you wouldn't have time to do all calculations individually, because it takes far to long, not to mention the rounding errors you would accumulate over time. I had a long discussion in an exam review, because I had a different value than the sample solution (almost an order of magnitude), since I haven't rounded my calculation and derived a formular to calculate the value directly, while the sample solution did it in 3 steps every time rounding to 3 digits (not significant digits!).

[@mokooh3280]

i like pajamas too

[@BBC600]

Yes, breaking the problem into bite size chunks is a good idea where possible. As a Canadian (although I'm not very mathy) so it probably doesn't count but I had never heard of the concept of multiplication by juxtaposition. How would you even know when to do that? What am I supposed to do read the problem writer's mind? I'd rather what I call BIDMAS (the "I" stands for indices which is what they call exponents in Britain).

[@BBC600]

@@mathewcherrystone9479 what a pain in the rear end that would be. Typically, I would leave all rounding until the very very end as a non-math person. It feels to me that then I get the most precise answer possible.

[@tanna_k]

In Russia, we were taught that ab=a*b in all cases so I'm a bit confused why those can ever be treated differently in the order. As far as I can remember, the "/" symbol was never used again after we were introduced to the concepts of variables and fractions. In fact, I think we were specifically told not to use it. If I ever had to in calculators, I made sure to explicitly mark everything down with paranthesis.

[@Capybarrrraaaa]

PEMDAS/BIDMAS/etc. work perfectly fine when you understand what terms actually mean. Most scientific sources make perfect sense from a PEMDAS perspective when you see the symbols and constants as individual items. When you have a calculation for a speed, it's fine to write it as 6/2(1+2) where '6' is 'distance' and '2(1+2)' is time. Knowing that speed=distance/time is the 'format' is enough to understand it correctly. While formatting it as 6/(2(1+2)) would be much clearer, knowing what the terms are is enough to identify how it should be treated.

[@RS-fg5mf]

Knowing what the terms are speed= distance/time then you see 6/2(1+2) let's you know the person did not write the expression correctly... If time = 2(1+2) then it should be correctly written as 6/(2(1+2)) putting all the operations within the denominator... written as 6/2(1+2) only the 2 is in the denominator... With context you can tell if the person wrote the expression correctly or not. Doesn't change the fact that as written 6/2(1+2)=9 not 1

[@jiminverness]

​@@RS-fg5mf If you want only the 2 in the denominator, your need to put in parentheses so that (6/2)(1+2). Written as 6/2(1+2) everything after the slash is the denominator...

[@TheBrainDunne]

I just consider / divide to have a bottom of the the rest of the expression on the right. And 2(3) is a single term with implied multiplication so 1+2(3) is the same as 1+(2x3) and not 1+2x3 .

[@Milesco]

Actually, 1+2×3 would be the same as 1+2(3) and 1+(2×3) because multiplication always takes precedence over addition. But I know what you're _trying_ to say. 2(3) is a single term in the same way that 2(a) [or simply 2a] is a single term. So that 1÷2(3) isn't the same as 1÷2×3. 🙂 __________________ Postscript: Or maybe you were confusing the addition sign (+) with the divison sign (÷)? Because if you substitute the division sign for the addition sign in your example, it's correct.

[@TheBrainDunne]

@@Milesco thanks 👍 we both have the right idea. I see now my example was not the best. Yes, I meant your 2(a) example.

[@Milesco]

@@TheBrainDunne 😊✌

[@DEMERN]

here's how i rationalise it: i don't see 2x as two separate numbers, i see it as a single term. division is an operation that uses two terms. in 8 / 2(2+2), 8 is the first term and 2(4) is the second term. therefore, the answer is 1. i don't think pemdas is being broken, i think it's all working as intended. maybe that's why the old textbooks didn't see any contradictions, because honestly i don't see any contradictions either. if you do it the other way, you're just splitting a term apart for no reason

[@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

[@William3DP]

It is so refreshing to see a such clear explanation of why PEMDAS is wrong that includes good, solid references. Thank you so much for this video.

[@RS-fg5mf]

Another individual who suffers from Cognitive Dissonance.... 🙄🙄🙄

[@Robert_McGarry_Poems]

As an older person now, I kind of knew about this problem. But, I never took the time to connect the dots between why I struggled so much with back of book answers... Until hearing you say it out loud. And I can agree completely, it took hours out of my life without being particularly useful. That does seem to be a problem for students (who do not know better), but what if this is purposefully done as a means of separating... Classism. Why separate professional use from common use conventionality unless obfuscation was a part of the conversation...

[@matterofrights2344]

The problem is not in the math or the calculators, it is in your equation.