He must be a real joy during parent-teacher meetings hahaha

Man I feel bad for the 6th grade math teacher who doesn’t realize Sally Greene’s dad is Dr Brian Greene. Yikes.

This is how I learned - PEMDAS. I am an elderly, arm-chair physicist now, and what I have learned from physicists, such as yourself, is to apply thought; to look, and see numbers differently; to be playful with equations, etc. I have found 'the beauty of wonder' in mathematics and physics that I never realized I had the capacity for. You, Dr. Greene, sparked my beginning interests. Thanks!

Feynman was asked to review secondary school text books with a similar madness. If professional mathematicians and scientists had a platform to crowd source real working methods, we would have more students excited about STEM topics. I had to supplement and correct textbook material for my kids.

I've been a software engineer my whole adult life and when I started every byte you could save in your source code listing was important so back then the order of operations did matter, however with today's computers and even phone and tablets it's not that important anymore, memory is cheap. Code is written once (well, not counting the numerous refactoring) but read many, many times. So source code shouldn't be ambiguous, it should be easy to read the code. Therefor using parentheses is very important, especially since besides the regular arithmetic operators we also have a bunch of logical operators, and comparison operators, and the order of precedence of these can differ from one language (or compiler) to the next.

As a software engineer, I prefer making the order explicit via parens. Relying on rules that might be different from one language/compiler to the next is asking for trouble.

I'm a computer programmer. I always use parenthesis in my math code for the same reason Brian stated, it has to be readable and clear. If your code can't be formatted in such a way that it becomes clear to another programmer of what your intention was when writing those lines, then your code might work but it could become a pain in the ass to deal with on large projects.

It goes a fair bit further in computer programming. For example, multiplication (*) and division (/) and modulus (%) are higher precedence than addition (+) and subtraction (-), which are higher precedence than bit-wise "and" (&) operations which are higher precedence than bit-wise "or" (|) operations, which are higher precedence than logical "and" (&&) operations, which are higher precedence than logical "or" (||) operations, which are higher precedence than assignment (=), etc. A programmer uses a significantly expanded variation of "PEMDAS" when interpreting (and fixing with parentheses) ambiguous expressions that contain bitmasks or conditions. That's not even all, I left out several other operators that are probably not familiar to everyone.

Oh Brian, I LOVED this episode, you made me laugh and cry at the same time... I learned math using parenthesis, and that was when I got it. I had a love hate relationship with mathematics at school, and it always depended of the teacher!! I HATE when they are are arbitrary and they love to make you suffer... sometimes they seem to make up things that do not match nor make sense to any purpose. I just got an insight to apply math to my life, if math shows up patterns that happen and we can create formulas to solve and resolve and discover these patterns and create order and get results. Using a simple mathematical equation to reduce overwhelm in my life sounds like the magical solution LOL It makes so much sense to look at life in this way, from a mathematical perspective! You've no idea the train of thoughts this episode has started... I follow you every morning, I have breakfast with your episodes while being quarantined in Italy. I hope this series will go on and on!! Thank you Brian for your time and dedication. BTW, I'm an artist who loves to make sense of things introducing concepts from different areas :)

Couldn't agree more, Prof. Greene ! I graduated decades ago , I have been a teacher and a researcher in Mathematics, never used PEMDOS or anything like it. How could I?? On the contrary, always insisted in very very clear when writing equations or inequalities . Also, as Julia said in the chat,using the dividing line between the = sign is a simple but very useful practice!

Great episode. I loved math. Public school knocked that notion out of my head really quick. “Show your work!” “Solve it how the book says to solve it!” Don’t you dare be creative!!

a programmer should make the code readable... so adding parentheses is good style

What we were taught in elementary (in Canada) was to follow BEDMAS-brackets, exponents, division, multiplication, addition, and subtraction. It was always clear that being precise was important and very straightforward. However, it was not made clear the fluidity of mathematics. Thank you, Prof. Greene.

Omg! Thank you, Dr. Greene! I don’t have children of my own, but I have been tasked with tutoring many nieces, nephews, or pseudo-nieces and nephews, and have encountered this very concept!! It’s frustrating from my point of view, because as a language, mathematics can be much more elegant and easier to express. And I always stress that it is the NECESSARY language of science, engineering, and technology, so however one gains fluency is penultimate to success. Your endorsement of my alternative helps my arguments immensely!

As a Bachelors Degree Mechanical Engineer who spent 10 years teaching Intermediary and college algebra and various other subjects at a community college to students just trying to get through the class and finished with the semester, I most certainly agree with you.

As a computer programmer for over 40 years I agree with you completely. If a programmer brought a program to me that contained an equation like the one without the parenthesis, I would give it back to him and tell him to write it properly. If a teacher gave me a problem like that, or a textbook, I'd tell them to take back their test (or book) and rewrite it properly.

Thank you Professor Greene for your much-appreciated positive efforts in doing The 'Daily' Equation which I am loving ! I'm catching up on the back catalogue of YDE and 'Live' discussions- brilliant stuff. re : Daily Equation #22. I agree absolutely with Professor Greene. I believe order of execution within an expression such as [ 8 - 2 ÷ 2 x 3 + 4 = ? ] is not ambiguous when following PEMDAS ( in UK, we sometimes learn the equivalent BODMAS-- Brackets, Order [ i.e. power ] , Division= Multiplication, Addition=Subtraction ) and means consistency when entering large equations on a calculator if, and only if, operations of equal priority ( add=subtract, multiply=division) are carried out strictly left-to-right. This teaches us to be able to predict the output of a calculator and be assured of consistency. e.g. If I use BODMAS, then the division of 2/2 is left of the multiplication, so divide first, then multiply, then add/subtract left-to-right gives 9. When entering the expression [ 8-2/2*3+4 ] on my calculator, it gives 9, as expected. 'As expected' is the key point here. It's important that if you buy a calcuculator, then it's output will always be as expected. Note 1 : many modern calculators display 2/2 as a fraction automatically, which is a visual clue as to the fact that it is 'doing PEMDAS/BODMAS. left to right. Note 2 : Polish Notation calculators ( I believe Sharp make them still ) is a completely different kettle of fish-- good videos on KZfaq by various people already there if you're interested in following *that* up ! Please keep up your magnificent efforts ! How about a YDE on conversion of units by including the units as algebraic symbols and using multiplication of the start value by unity expressed as a a ratio of the units to be converted to and from... e.g. to convert 1km to m : ( 1 * km ) * ( (1000 * m ) / ( 1 * km ) ) = 1000 * m = 1000m ? I find including units always is desirable to avoid ambiguity or mistakes, and think you could teach it in a clear way. Thank you.

Simpler and clearer. It’s the way. 👍 That PEMDAS is like a secret code for emotionally unbalanced mathematicians. 😃 I like this so much. This is about philosophy of thinking and of life! Great stuff. 🌷

I agree with this so much. Math in school is about memorizing silly rules and not thinking in the language of mathematics. Thanks Dr. Green.

This may come up when using an electronic calculator. It's why "reverse Polish notation" was standard on a lot of them, because entering parentheses wasn't possible on some of the simpler ones, and even if it's available, you end up having to write out the expression so you get them in the right places. Since most don't display the formula as a whole, but calculate on the fly, it's easy to make errors.

I'm with you Brian. Trained as a physicist and professional engineer for the past 40 years, and I've never seen PEMDAS or whatever they call it used professionally. Or course the operator hierarchy as used in polynomials is used all the time (exponention->multiplication->addition/subtraction), but the ambiguity in the order of applying division is always made explicit with parentheses or other layout cues.

Thank you Dr. Greene. You have given me some insight on how to teach my 5th grade nephew about order of operations. Maybe you could show us some more basic arithmetic ideas, for young math learner's. Possibly how to introduce the idea of variables to the uninitiated.

Excellent video. What are your thoughts on teaching prefix/Polish notation or other unambiguous notations to avoid the problem altogether?

Prof. Greene, I am older than you, and studied math in the 60’s and 70’s. We were never taught PEMDAS, but were taught the order of operations along with other principles like commutative and associative. We didn’t spend a lot of time up on them, however. It’s true that they could’ve made the bigger picture clearer to us, but at least we weren’t being trapped with tricky rules. Girls were not expected to do well in math; as a senior in high school, I got the highest SAT score in math our high school had ever had, and my math teacher told me he was really surprised because he didn’t realize I was that good in math. Well, he had not paid much attention to me. The bias against girls in math always annoyed me, especially because I saw it continued in the 90’s with my own daughters!

I totally agree although, from a pedagogical perspective, I've always wondered whether order of operations discussions couldn't be a starting point for a discussion of associative, commutative properties that would be more mathematically engaging. I'd be interested to hear from teachers whether that's a reasonable approach.

Another great episode. As a student I was NOT taught PEMDAS. 20 YEARS later as a teacher I was required to teach it. I had never heard any reasoning behind the PEMDAS pedagogy, so when kids asked 'WHY' I finally gave them my only reason. "Some old DEAD guys wrote the rules a thousand years ago and now WE are stuck with these rules. I agree .. more parenthesis would be a much better solution. The person making up the problem should make the problem understandable by all.

Finally! An equation I can understand! And even without the brackets! :D

As a software developer with a maths degree, I absolutely agree - no mathematician or programmer ought to be writing expressions like that, and kids ought to be taught not to. A mathematician would have the possibility of using layout to make their intention unambiguous, of course. A programmer should use parentheses - or even well-named variables for the intermediate results. But PEMDAS isn't even right - doing multiplication before division and addition before subtraction is just wrong. Maybe it should be written PE(MD)(AS).

Very enjoyable video, Brian has passion. Great to watch.

Thank you Dr. Brian Greene! You are saying all of the things I wish education programs would realize.

I completely agree with the general sentiment, 100%, and I also hate when problems like the one you broke down go viral online. However, PEMDAS is very cleanly applied in situations like polynomials, which would be very cumbersome with parentheses/brackets around every operation. It is useful to know that 5x^2 - 3x + 4 is not the same as [(5x)^2-3]x + 4, which it would be just going left to right without the standard implied order of operations, [5(x^2)] - (3x) + 4. At the elementary level though, I agree that clarity and simplicity should be taught, which would mean removing ambiguity by using parentheses, and then later students can learn to drop parentheses in commonly-understood formats, for example exponentiation before multiplication before addition as in a polynomial. Thank you for the video! I'll save it to post next time somebody shares one of these annoying problems.

I thoroughly concur with your sentiments about the beauty of math to encapsulate wondrous patterns found in the natural world and think it applies to chemistry as well. On a technical note, what software do you use to record your video with handwritten notes.

Reminds me of Richard Feynman‘s comments on textbooks in California. Love it!

I never heard of pemdas. We always learned to use Parentheses in 1970s Ireland :)

Thanks, love the videos! Yes precision is also quite important in written communication, as evidenced by the emails we sometimes get! I’ve never become comfortable with math, I plan to go back and watch this entire series again! At university my symbolic logic prof took pity on me and gave me a ‘D’, as I tried- just didn’t get it. Calculus was a bust too on paper, but in the real world I regularly ‘calculated’ the speed of the semi in front of me , the speed of the oncoming car, and whether or not my motorcycle has enough acceleration to pass the semi and tell the story afterwards. I was actually going to ask you to do operation orders - so you’re psychic too! Nice!

I am a similar age to you and was taught the equivalent of PEMDAS at school in the UK. I have always used parenthesis to remove ambiguity though.

So true, finally, thanks for that Professor Greene!

Conputer programmer here, unfortunately ambiguous operator order is legal in math and in many languages. You can use code analysis to enforce a rule of no ambiguity, using parentheses, which is certainly recommended. I've encountered bugs in the wild more than once that came down to this ambiguity. I think the real fix would be to totally disallow ambiguity both in languages and in math. So that your example wouldn't even be well-formed syntactically to begin with.

Hi Prof. Greene, When I went to school (a long time ago), we were taught about BODMAS. Bearing in mind that in Britain / England we tend to call them "brackets" rather than "parentheses", which in the UK, usually mean square or curly "brackets". However, my Maths teacher at the time emphasised the same point that you made in the video - the expression should always be clear & consistent in its ordering. Then, if the expression was clear, BODMAS showed the order in which to perform the various operations. To give you some idea of when I did elementary maths at school, we were still taught to use Log Tables and Slide Rules !! Honestly. All the best, Paul C.

Who's gonna tell these people that there's no such thing a 'division' in a Field

Thanks for explaining Dr. Brian

I absolutely agree. The order of operations can be arbitralily decided by us mortal humans. No big meaning behind it. It is a waste of time to spend more than a day on it. As you say, bracketing clearly shows you what you mean. So have that dealt with quickly and talk about something more interesting than made up rules.

I'm a maths graduate and computer programmer of many years standing and share your views and frustrations. I only became aware of these acronyms when all these "Try this - 99% get it wrong!" "PEMDAS" / "BODMAS" memes started appearing on (e.g.) FB. We were never taught that way (why would we have been?). Yes we were taught PRECEDENCE - but surely this took at most an afternoon. From what I've seen, an awful lot of those who have been taught these acronyms apply them blindly (why wouldn't they?) and often get the wrong answer as a result - insisting that multiplication ALWAYS comes before division (if "PEMDAS"-trained; conversely if "BODMAS"-trained) rather than applying operators of equal precedence in left-to-right order. They should at least have taught it as "PE(MD)(AS)" or "BO(DM)(AS)". This misguided teaching is now showing up in computer code - with who knows what consequences.

As a pogrammer with a university maths degree I always use parentheses in my code. The result after compilation would be the same with the compiler applying PEMDAS, but I prefer code that is easier to interpret and thus maintain. Just another way math is like language studies, you have elaborate constructs you don't use in everyday conversation either

I am so glad that Brian said that is this ambiguous. I taught myself both semesters of college algebra (to save money) and then went on to 2 semesters of calculus. Never heard of PEMDAS and found the above equation pretty crazy. I was so happy to see I was correct. No punctuation in that sentence.

ok I know i am coming into this conversation way late but i do agree whole heartedly with you Dr. Greene. i would also like to add my own perspective which has been driving me crazy and i been arguing with mathematicians for a few years. This Idea of 1=0.999... and simply that one divided by 3 equals 0.333...I wrote this response to a mathematician in regards to our argument 1/3. Hello Professor Ely, Robert It is Rolin Bankus again, I hope this finds you well, and not too busy to reply. I understand with School in session that your time is very important. I thought long and diligently about our last conversation we had in your office; I wasn't even sure if I was going to respond but then with as frustrated as I got, I knew if I didn't that it would drive me crazy. As always, I must apologize first. I am sorry for two reasons one this will be quite long, and two if I offend you in any way then I am sorry I mean no disrespect. When we were talking in your office about 0.3 + .1/3 and 0.33 + .01/3 etc. you had told me to forget about the remainder part of this (which I think mathematicians do too often) well now I am going to throw this back at you because I think you will see where I am coming from if I do. Ok in the proof that 0.999... =1 you set 0.999... = to x and multiply by ten etc. (which I am going to write out SRY.) x = 0.999... 10x = 9.999... 10x = 9.999... - x = 0.999... = 9x = 9, 9x/9 = 9 /9, = x = 1. well, you already know that I have a problem when you multiply by ten and like you told me in your office someone already wrote a book about how the values are not equal etc. So, like I told you in your office if I add 1/3 and 2/3 in their decimal form my way then I can come to a value 0.999 + .003/3 well now here is where I throw this back at you, forget the remainder value set it equal to x and prove it is equal to one. because the only difference in my value and your value of 0.999... is your ... and so, they must be both equal to one, correct? I know you're not going to but we both also know what happens if you did. it would look something like this. x = 0.999 then multiply by ten so 10x = 9.99 and when you subtract you would get 9x = 8.991 so, what I am saying is that 1/3 does not equal 0.333... because this ... is a poor example of what the value is. for example, if you take out your phone divide one by three then it is 0.3333333333 my phone only goes out ten decimal places but then if I erase this and put that same value in and multiply by three, I do not get one. when I am doing math, I was always told if you find the answer stop. well, 1/3 = 0.3 + .1/3 because like I said in your office you can only divide 1 by 3 once. If you continue you are only seeing how small you can make the remainder so that if you were then to multiply by three it still equals one. because 0.3+.1/3 is the same value as 0.33+.01/3 you cannot just simply forget the remainder and say that if you do the value is bigger. Because yes 0.33 is bigger than 0.3 but see neither of these two values are a product of 1/3. They are either 0.99/3 or 0.9/3. you know it was Einstein who said that the definition of crazy is doing the same thing over and over again and expecting a different result. I also do not think that I am arguing over semantics because with my value you never get an incorrect value and is simple to explain. See without your special little ... you cannot explain how your value is equal. Well as always, I appreciate all your help and look forward to your response. Sincerely Rolin Bankus.

Genuinely, the rules of math given by instructors were aggrevating as a child. Understanding why said rules were implemented was more useful. I failed math because of teachers that didn't take enough time giving the reason behind why we do these things. They would always mention in the next class you'll understand. By the time I reached trigonometry (27 yrs old), retaking it a third time, I started reading on why calculus was created and it made learning square root trees easier. Mind you, I worked as a Network Engineer mentee and was studying Embedded Systems during this time too. My comment is more so a thank you for the validation. Equally it sparked the question, why PEMDAS? Why this order? Similar to the question I had while learning C++.

Thank you for saying this Dr. Greene. Part of the problem, maybe, is that non-mathematicians are often teaching mathematics, and a certain social belief that prescribing steps may make mathematics, physics, abstraction, or complex ideas accessible, and they mean usable and useful, to anyone even without understanding. I find this type of thinking even in community college "developmental" courses ... and is there any wonder people repel away from math!

Thank you, Brian!

9:42 Facts sir. I got goosebumps when i heard that from you. I felt the same way and now I'm sure .

Love it! 😎👍

7:00 listening your this experience and feeling it is Amazing

These are great, EVERY one of them! Thank you!!!!

I was so happy to hear this lecture, thanks.

I learned 50 years ago BOMDAS no left/right precedence (pre computer era) that meant M was before D, A was before M. So my 50 year old answer would have been 8 - ((2 ÷ (2 x 3)) + 4) = 3.667 My current answer would be to put it into google to get the wrong, but modern accepted answer of 9. Left -> right is a recent and western biased concept based on the English language left -> right system.

Looking through old maths books history, reasoning and recognizing principles was more important that teaching a lot of routines.

It was BODMAS when I was at school, it may be the uk version.

Love it!

If i was given this equation, i might be inclined the proceed to calculate from left to right (as in reading English). The problem can be simply entered into a calculator while proceeding from left to right, giving 9 as the solution. Agreed that parentheses should be included if another order was intended. Should a person whose language reads from right to left do similarly (-1)? ;-)

30 years ago in Poland I learned that multiplication and division is stronger than addition and subtraction. The rest must be explained by parentheses. So when I saw the title equation, I found it ambiguous and its value could not be determined without parentheses.

I have no issue with the basic idea and I am consistent in my use of parentheses. But what do you do when presented with a math problem that is ambiguous and you need an answer? How do you proceed? Do you stop and request clarification from who knows where? There is a reason for having a default method for these cases.

Spot on Prof!

Late to the game, but really enjoying the Daily Equations. As a retired engineer and volunteer high school math and physics tutor I couldn't agree more with your comment on order of operations and avoiding ambiguity. In Canada the in word is BEDMAS, but the different letters solve nothing.

I learned this too but no acronym for it, just that multiplication and division were associative as were addition and subtraction. But we learned this early in school, while we were doing arithmetic, long before algebra.

Dr Greene i think that since you start by requiring a set of parentheses to begin with, maybe you actually apply this PEMDAS rule, given that the rules apply in that order after P. Anyway i agree with you on the general context. I am Greek and remember back in school that time we were taught the sequence of operations, i was really frustrated.

I agree with you! If I'm a graduate student, and if I learned how to solve differential equations why should I be tested on PEMDAS? What we know should be tested and not if we make a accidental mistake!

Even if we use parentheses/brackets, we still are doing PEMDAS but putting the "MD" and sometimes "AS" into P. Which just shortens it to PEAS. Right?

This strikes a real pet peeve as well. I am a computer programmer and over the years I have programmed in many languages so I always use parenthesis to ensure my formulas are interpreted correctly. One of my first languages was APL that is strict right to left, MUMPS is strict left to right, FORTH was (is) RPN, lots of equations involve conditionals NOT, AND, OR and XOR etc that are not covered explicitly by PEMDAS and often even different implementations of the same language do it differently so why take the chance. Long live the parenthesis !

I am a teacher and can't wait for a fresh explanation!

Great episode!

Professor I am a chemistry prof and tutor math in high school. Your comments on ambiguity and no brackets is right on. I will use your idea next time i have to teach this concept. thanks!...

PEMDAS matters when you're using a calculator. You can do things a lot more quickly if you use PEMDAS. The ambiguity in your equation isn't with the plus and minus sign but with the division and multiplication signs. It's ambiguous whether it means 2/(2x3) or (2/2)x3. I think that division sign is what makes it unclear, which is why it's better to use a fraction to represent it so we know where the 3 goes.

I'm a Physics Student. Luckily in the first semester at university, they teach us about Group theory and also Field (Körperaxiome). After learning this, you will actually get what did prof. Greene means haha. You will feel that this ambiguous mathematical expression is like an insult when some random dude asks you to solve that problem lmao

Yes Professor, you are damn true. Teachers in my school focus on the rules and tells us the rules that guide the mathematics. But I am trying to get our of this rules and understand the mathematics as a whole. Thank you professor it is very kind of you for taking me further into the magical and wonderful universe of mathematics.

Preach! lol, well lucky for me I was taught not to apply that rule PEMDAS so extensively but to rather sharpen it up in a precise manner.

I believe this is more about learning pre requisite to designing a computer than math or physics. Prof. Brian the world revolves around other subjects too!

I just remember it like this: ×, ÷ just more important so I do them first. And parentheses- makes it easy to see. Of course when you have all black board written with formulas. You have to make it easy to see.

Dr .Greene ,please make a video on laws of motion.

that is a great method of encryption your order of operation!

OMG Because of this video i have learned how to do math operations ! It is unbelievable what they were teaching us in school...I knew how to them but never understood why do the math has to be so weird that it lets me do so many different operations with that single equation...what you did for me today....you just made math more interesting for me ! Thank you for that !

To be honest after years on passing different courses of math and physics in school and then university, I could never imagine there would be a day I start to look at math and physiscs with all my heart not only the brain. That is the point: all the beauties of math can be revealed deep down in physics and physics in math. If from the first step, instead of "just" math or "just" physiscs, the authors of the school books start to join the building blocks of both sciences together and describe the very complicated math subjects by the help of physiscs, maybe there start a revolution in students point of view regarding both scienses. For sure the subject will be more understandable and will not seem to be rigid and though for students any more.

Hello, Brian! I never heard of PEMDAS. In my country, I was taught that you do parentheses first, then multiply or divide, and only then add or subtract. So, the bit that confused me was the order of the operations in the middle. Adding or subtracting first was never an option, but what to do with 2/2x3 - no idea. To top it all, this expression does not exist anywhere else. I agree with you - why make maths complicated and abstract, rather than allow its students to enjoy its relentless logic? Thank you for what you do and who you are!

PEMDAS? When I was in school we learned BEDMAS: Brackets, Exponents, Division, Multiplication, Addition, Subtraction PEMDAS is interesting. Multiplying before dividing? that is interesting

I agree totally. In primary school, I would *always* put brackets (sorry not saying paren..ehhh) around everything that needed them to show the order of calculations. However, my teachers did not agree with this unambiguous representation of a mathematical equation, and would mark me down for using brackets when I "didn't need" them. The real issue here is some (not all of course) teachers are so rigid and don't think for themselves about what they are teaching.

I heard PEMDAS when I was in my elem year and most of my friends use it until now that we're already in college, i don't know why but i still never really paid attention to it and never use it even thought they sometimes encourage me. Thankyou professor now have more reason to continue ignoring it.

Absolutely! No ambiguity. Also learned as in engineer to ALWAYS include units. Not everyone does, but it often saved my bacon. If the units don't fall out properly, then something somewhere is wrong.

Depends on convention/ prescribed rules that play the role of punctuations, and tell me how we all are going to read this otherwise ambiguous sequence of symbols.

I taught high school AP physics, AP calculus and AP chemistry for most of my 35 years in a public high school. I’m 100% in agreement with you on this arbitrary use of “rules” such as PEMDAS. I don’t make many friends on social media when these silly math sentences are debated as to the correct answer. I insist parentheses and or square brackets are needed to completely understand the nature of the mathematical sense of the equation. Ban PEMDAS.

Every time I hear the phase "small business owner" I think of Danny Devito running a shop.

So !... The correct result for the given example would be 11/3 ? It seems to me that none of the alternatives presented was the right one, according to the rules of the operations priorities. Am I right or wrong ?

As a programmer you need to know the order of operations of the language you work in, in case you get handed some code that does not use parenthesis. That said, I have always parenthesized my code for readability (probably was my physics background -- how else are you going to punch the equation into your HP-11C!). I even did it back in the 80s, as the compiler threw them out when it was done (no, it was never a space issue for me, and I doubt for anyone else).

PEmdas is the way to prioritise operations. How it was taught and what problem statements have been given to the daughter may be confusing.

Greene, brilliant! As usual... Not only this is torture for our our students but I dare say that our whole Educational system is a torture for our students. And, even worse, a tremendous waste of money for the whole society Except for the owners of companies profiting from this system... Summarizing a long story. Why should kids wait more than 20 years until they get a profession? A lawyer for instance. More than 12 years learning a lot of encyclopedic knowledge (Math, Physics, blah, blah...) when after in his career, as a lawyer, she will not use most of this knowledge. Let’s teach them early in life basic concepts of all Sciences, including Economics (financial planning), Politics, Civil Rights, Philosophy, Scientific Method, etc, etc… and most importantly, help them decide which career track to engage on! In this way, in our early teenage years we could have already professionals ready to produce for the whole society! Remember in Greece. “Hey boy, what do you wanna learn? Math!!! Go there and talk to that group led by Phytagoras…

Here in the UK we were taught BODMAS, which amounts to the same thing but at least the BS is more obvious in our acronym. Your final point about coding is pertinent for me. As a former coder, I would never have left the opportunity for ambiguity in any code that I was prepared to expose to others.

Master's thesis in Computational Physics (Bose-Einstein Condensates). Always parentheses. I do not have time to learn the precedence rules of every programming language I use. Keeps you out of trouble and is just good style. As for the paper-part of my thesis (or the rest of my career for that matter): Never used PEMDAS again since elementary school. But to be fair, our teacher did not try to trick us. Just told us the rules and went on to associative commutative and distributive laws. Much more useful.

Point before line (Punkt vor Strich)... one of the first things we learn at school in Germany. At least in my time.

The P in PEMDAS stands for parentheses, but yeah they don't stress on the importance of brackets... Even in general life we never use this rule because the context of the problem makes it clear which operation we should do first.

Not from the US (Or any other country that use this method), did not know PEMDAS was a thing, that is super arbitrary, I always used (), [], {} If you don't use them like this on College you pretty much gonna have a 1 (where I live notes goes from 1 to 10). I love math, I'm not that super good with it, but it is so beautiful... specially when you start with Calculus. To think Sir Isaac Newton had to "invent" his own math's to explain things he observed is astonishing, he was, no doubt, a genius.

I only slightly disagree since we would write something like 2x + 5y - 3 rather than (2x) + (5y) - 3. PEMDAS is ultimately to eliminate parenthesis where the meaning is visually implied, but in more ambiguous circumstances like 8 - 2 / 2 × 3 + 4 I definitely think parenthesis should be added. Ultimately, It is more about how it appears then what is technically correct since a large purpose of math is for the purpose of communication, and as long as the person on the other side of the communication understands the intent, the method is acceptable.

I'm perplex. We did use pemdas at the university all the time. We never used parenthesis for equations like: x = 2y + 1. We ony used them when we were transgressing the so-called pemdas rule, like in: x = 2(y + 1)