EEVblog 1479 - Is Your Calculator WRONG? 

The parens are most likely free in an interpreting implementation, as once it's parsed the parens don't matter; provided that your implementation isn't an olde BASIC that re-parses everything on a line during execution.

Get an rpn calculator

@@edwarepl there’s always an RPN guy somewhere out there when these topics come up 😉

Good point re: modern interpreter implementations - I'm old enough to have cut my teeth on those 8-bit BASIC interpreters - you were lucky to get keyword tokenization, and that was likely more motivated by RAM conservation than execution speed. Yes, it's a moot point on RPN, and also on the Sharp EL-5100, which has no concept of implied multiplication, and yields an error instead of a result.

Always, always use parentheses. There is no way to get it wrong that way, period

same lol

They wouldn't let us use calculators until college and even then only certain ones. All this time i thought it was because the university got a kickback from the manufacturers for sales. This makes a lot more sense.

Same here, always put ((((( and )))))

Use RPN ;)

Absolutely. And if you write a/bc you _never_ mean (a/b)*c. There's no good reason why arithmetic expressions should be evaluated differently from algebraic ones.

Ugh, I'd hate if a calculator did this though tbh, I just want it to do what I tell it, not creatively try to figure out what I could have meant. With the clarifying parentheses added after the fact it's alright I guess, but it still feels iffy to me. I've always had 2x mentally modeled as literally identical to 2*x, so I wouldn't want my calculator to throw me a curveball and interpret it differently.

@@ska042 Yep. When I think about it, while programming, I *always* put in the parenthesis when I want to prioritize. I've done this naturally because you must be explicit... That way, on the backend a/bc *always* means a/b*c, in *that* order. If you don't, you will have the problem that a/2c is going to give you a different result than a/bc even if b = 2. Seems to me that we should go to the standard that makes sense for programming. Though, in programming a/bc isn't valid form *anyways*, as it shouldn't be. You would have to write a/b*c It seems to me to be at best laziness to omit operators between two variables. 2x makes sense, zx does not, after all, how am I to know that you don't have z, x and zx? Rather silly to limit yourself to 26 possible variables in a single equation...

@@ska042 "I just want it to do what I tell it" But it does - it is not the fault of the calculator when you are typing something different from what you mean. That is like the many old programmer jokes - carton of milk and eggs ....

2x is implied and must be multiplied before anything else. look at like yx.

its unclear because you wouldn't write it that way..... if you intend the fraction 6/2 then you'd write 6/2 in the fraction form ⁶⁄₂ and the whole fraction is the implied multiplier, if you intend 6 / the rest of the equation, you have 6 over the top of the rest of the equation.

This is pretty much what the, "show your work" methodology is for. Helps clear up inconsistencies in culture as well as a few other things teacher wind up doing like debugging your math to point out where you went wrong.

@@eideticex yeah when i first got my calculator in middle school i started typing in whole calculations (mind you i was able to type them in fully correct with all brackets and operators and such) but my teacher wanted me to write out the whole order of operations with intermediate results because as you say if all you messed up was a sign somewhere you could still get the majority of the points if the result was consistent with this single error. also having you write out the whole equation makes cheating way more difficult as quickly peeking at your neighbors end result wouldnt be enough

A bad student blames his/her calculator!

It's up to students to recognize what's the priority so I don't think that should be a problem. Students can always solve it step by step instead of putting the whole equation in the calculator

@@johnmiller0000 RTFM-Matters.

Replacement of 2( by sin( demonstrates exactly why the =9 answer is wrong. The notation tells us that we are calculating a numerical function in exactly the same way as we would calculate a geometric function. They are equivalent and should be treated in exactly the same way.

@@chrishartley1210 Not disagreeing with you, just saying if I were working the problem today, I'd still come out with a 1, not a 9... even though my kids would get a 9.

The goal clearly was not to make it more difficult. I suspect the goal was to make it simpler since there are fewer "cases" to follow (no distinction between implied and explicit multiplication - it's all the same). This is a simpler parsing algorithm, even if it doesn't match our natural way of reading the expression.

It's perfectly clear to me. I don't know anyone who would write it like the Casio's guess. But I am from the US, so...

actually it's because of juxtaposition

Funny thing with that is, on a TI you have a difference between the minus operator and the minus unary :D

Couldn't agree more to everything you said. If one has ever done any programming, one learns to state intentions clearly, at least after you've spent some time debugging the first version that didn't do as you intended.

@@tengelgeer tfw 5-4 comes out as -20

Yup, as is shown in the video, the actual interpretation is pretty arbitrary, so being explicit solves all of the problems. Hell, it also makes programming easier too. I remember having issues when I first started C++ when my assumption on how a long string of boolean operations would be evaluated was wrong. The solution was to add all the parentheses, ever. Make that line of code wrap 3x if you need to.

anyone who has taken advanced algibra or calculus does not need such markngs what does XY mean? Do we really have to spell it out for you as (X x Y)?

Because I didn't learn RPN in school I had to install RealCalc and practice a bit. So for whoever understands parse trees but not RPN here you go: 6÷2(2+1) Casio (implicit multiplication at higher priority) Parse tree ÷ 6 × 2 + 2 1 RPN: 6 2 2 1 + × ÷ Result: 1 TI (explicit and implicit multiplication at same priority) Parse tree: × ÷ + 6 2 2 1 RPN: 6 2 ÷ 2 1 + × Result: 9

@@mielole I have an old HP12C RPN financial calculator that operates the same as your TI example.

@@mielole Casio is wrong.

​@@MrGreensweightHist​@CRalph3122 I'm simply repeating the examples from the video. Let me resubmit my comment as a top level one.

@@mielole "'m simply repeating the examples from the video. " Understood. And the Casio is still wrong because this problem has no implicit multiplication. The answer is 9

More famously, crashes autopilots. (The F16's original autopilot software crashed if the aircraft passed through sea level. The Israelis found that out, because they have an air base below sea level near the Dead Sea.)

@@evensgrey that's because Americans live on an island where nobody outside the island exists or matters to them ... Except AFTER they've sold you a "pup" or mixed up Imperial with metric with decimal Imperial DUMB or what THREE different measurement systems when the WHOLE of the rest of the world uses just ONE! 🤔😦😲🙄🙄🙄

@@boblewis5558 No, it was a divide-by-zero error. There was, deep in the guts of the system, a place where the software was unwisely written to divide by the altitude above sea level with no boundary check for a zero altitude because there was nowhere that could happen in that autopilot mode in the US. It would have happened regardless of how the altitude was measured.

This comment. Yes. 👍🏻

@@Yuxim I see it as 6/(2x3. Just be clear and this debate will end.

Yep. Tossing the implied multiplication in there is the error in the first place… Mixing line expression and algebraic. The 2(2+1) would appear as a whole below the line if written on paper so that’s why the calculator does that and it makes sense. Wanna use left-right order? Use the x symbol to keep it consistent

You can also illustrate this with multiplied constants, as Dave sorta did with square root of 2. I would certainly expect my calculator to give the same answer for 6/2π and 6/τ

Ambiguous it is!

That’s totally wrong, i got used to some stuff in math but when i understood how it actually works it was so fascinating

it has never meant that.

@@ChickenPermissionOG It has always meant that

Never heard about this rule.

same, i think that kind of reading comes more naturally especially once you start working with polynomials and that sort of stuff. though honestly, any decent mathematician out there would just tell you to not care about PEMDAS and just make your expressions clearer with parentheses.

@@ChickenPermissionOG 2(2+1) can be written as 2(a+b) where a=2, b=1. Expand the brackets gives 2a+2b = 2x2 + 2x1 = 6 (not 1 whoops - I had the first problem on my mind)

Amen brother.

Mr. Bill. You are a smooth operator. As well as ALL calculators should be. Implied should have died (ok, not a poet). Cheers

If you follow the RPN concept, you will get 9 6 ENTER 2 / 2 ENTER 1 + X

@@marko.692 No you won't. PEMDAS, with RPN you start at the parenthesis and work back. You don't start at the beginning of the equation. You're not following the hierarchy, parenthesis is done first, division is done last in this equation. PMD. Parenthesis first; (2enter1+) answer = 3; multiplication next; answer, enter 2 * = 6: division last; answer enter 6 / = 1. Actual keystrokes: 2 enter 1 +, input 2, *, input 6 / = 1. You're doing RPN wrong.

I was taught BODMAS. B is Brackets, same as Parenthesis. O is Orders which is Exponents + Roots. Multiply and Divide are reversed compared to your's.

@@mosfet500 I’m sorry, but I disagree. Why are you trying to proceed this from right to the left? PEMDAS is from left to the right. If you don’t want to take advantage from the registers in RPN calculator, then calculate what is between the parenthesis, rewrite the equation without the parenthesis and follow from left to the right…

@@marko.692 PEMDAS doesn't have a direction, it has a priority. I followed that priority. The parenthesis are integral to the equation, rewriting the equation with different priorities changes the equation. solve for x: 6/2(2+1) = x; 6=x(2(2+1)); 6 = x(2(3)) ; 6=6x ; x=1

RPN for the win

Yeah; my college HP48SX doesn't even seem to allow me to create a function without the extra parens. Maybe I've just forgotten how to force edit a text equation.

This is one of milliard of reasons to use RPN

I'm a big PCalc fan myself. And I use a HP 35s. Can't stand anything other than RPN once I discovered RPN. I wish HP still made great RPN calculators.

@@ErikN1 I'm not on commission I swear, but I recently got a SwissMicros DM42 (which runs Free42) to replace my old hp 42s. Very pleased with it.

That's only, when you use the formula editor. When you use the stack/RPN logic there will be no surprises.

Ti89 here. Same thing

HP48G's successor, HP50G, also has the same behaviour on the "baby" ALG mode.

My trusty HP42 comes up with 9 also

@@jenselstner5527 yup. This is another reason I'll never use a non-RPN calculator ever again. Still rockin' the HP48sx :-)

Yes that was the way we used 991ES during the Uni, otherwise with brackets

You're brave. I've never dared to put mine in that forbidden mode

@@ErikN1 I learned my lesson.

I had to try it on mine, didn't I? Now I feel the need to take a shower...

When it is written as a fraction with 6 as the numerator and 2(2+1) as the denominator, the result is 1. But that is because parenthesis are implied, not because implied multiplication has a higher priority.

Rule in Europe ( and everywhere else where maths rules are followed) that in case of same priority operators they are applied from left to right. Correct answer is thus 9.

RAM also isn't a thousand dollars a kilobyte anymore either. Just like the convention of two-digit years in dates, some efficiency measures only make sense up to a certain point but don't get abandoned when they really should.

I'm from Germany and my TI-84 Plus doesn't.

Yup, tested it on my fx-991DE plus too. I'd never actually encounter this, though - I always use fractions instead of division.

It's the calculators that are wrong NOT the priority order. The ones that allow use of brackets but give 9 are unfit for purpose OR cannot cater for more than one set of brackets (still unfit for purpose!) Also since WHEN does precedence of Muliplication over Division, or Addition over Subtraction matter? When has it EVER? 6*2 is the same as 2*6 6/2 is the same as 0.5*6 5-4 is the same as -4+5 However 6*2-4 is NOT the same as -4 * 6*2 but it IS the same as -4+6*2 If order of operations is done correctly Does NOBODY comprehend this?

The issue is that implicit and explicit multiplication are the same operation expressed using two different operators - one of which happens to be also the function call operator. In programming languages, and really, any time computer parsing is involved, it's operators, not operations, that get assigned precedence levels, and the function call operator has one of the highest precedences, far above that of M/D (otherwise 1/f(x) would be interpreted (1/f)(x), which will probably fail type checking).

You say you rhink giving multiplication and division equal precedence is misguided and confusing. So which would you give the higher precedence to - and why? After all... (6*2)/4 = 6*(2/4) = 6*2/4 ... no matter what the priority order is.

@@RottnRobbie There is no problem with the * and / operators having the same precedence. However, the juxtaposition operator is primarily a function application operator, only serving double duty as multiplication, and even that is better analyzed using the rule "when a number is used as a function, it evaluates to a function multiplying its argument with the number", making juxtaposition always a function call. Giving the function call operator equal precedence to the M/D operators makes no sense - it needs to be higher so expressions like 1/f(x) can work. The key takeaway is that while in pure mathematics, it's convenient to think of operatIONs having precedence, efficient parsing requires precedence to be assigned to operatORs instead.

@@NetRolller3D the whole point of algorithms is to follow the maths rules NOT the other way around! Provided your algorithm follows the rules and provides the correct result it matters not. However Maths rules have been in place for centuries not 70 years ergo forget trying to define those rules in terms of programming! Any programming MUST adhere to the rules not vice versa, or there is NO prediction of accurate results! Programmers, please note and STOP confusing the issue. After all, computers only ADD, shift bits to multiply and divide and complement when trying to subtract!

I learnt (in Europe) to solve divisions before multiplications. But as for multiplications, no difference between regular and implied ones. It would be like differentiating between x, * and •.

So, whether using 6/ or 6÷ you're assuming that everything that follows is the denominator? If that's how we treated math expressions it would be even worse than the implied multiplication "rule". No, order of operations is left to right with division and multiplication having equal priority. If you want numerators and denominators then you can draw it that way on paper or write it that way using parenthesis like the newer calculator did. There's no "reason" to distribute first. The Law of Distribution is like the Law of Gravity; it's not a rule that must be chosen to be followed. It just is. You can't change it. It's a property of math just like gravity is a property of physics. It doesn't matter if it's a 2 or a (6/2) next to (2+1); the Law of Distribution will still be proven.

@@joshuaewalker No, I am assuming that what follows is a denominator because it's a parenthetical expression. The parentheses interrupt the left to right that you are talking about and the 2 outside of them is part of the expression they identify so it is calculated at the same time. To express what you are describing the proper way is to include the x for the multiplication of the 3 and 2 in the parenthesis. It seems odd that you admit you are omitting a symbol and expecting the result to be the same as if it was included. But that's the way I was taught and never had a professor of a higher math class complain but clearly we've been taught differently. And somehow science continues, despite these differences.

@@schwamforfreedom How would you say 2(1+2)? Two multiplied by the sum of one plus two. That's what it means. We all know that's what it means. We all know there's a multiplication happening. What's odd is your choice to add extra parentheses around the whole thing when those parentheses were never there to begin with, but the multiplication was always there.

Order of ops not being standard outside of the 4 standard symbols really does suck. I see it with programming when mixing binary and decimal math within the same statement. Some languages the binary ops are higher than arithmetic, others lower, while others treat 'or' as addition and 'and' as multiplication with no regard for binary or decimal. Start to throw in powers and logs, gets even hairier. Good luck fitting vector ops into those categories beyond the basic operations. Just by changing an syntax in the definition (as in literally just a word is changed like explicit to implicit), I can pick in some languages to put things like dot-product and cross-product higher or lower than ordinary multiplication. Even though logically a dot product is multiplication and cross product closer to 'xor' (which has it's own unique rules). My C# vector structures are full of odd tricks to make the operators obey the order of operations I was taught. Most of the time it doesn't matter but sometimes it can be the difference between garbage results and useful results.

As an engineer who’s taken all the maths from pre-algebra to way past DiffeQ, I’ve had a totally different experience. Same goes in industry, I’ve never come across a case where another engineer has written an equation similar to this example with implied multiplication and intended it to be calculated as you’re saying so I’m quite surprised to read this. Funny how math is supposed to be the universal language but it’s a little concerning when things like this come up and the possible ramifications of an incorrect interpretation. I’ll def keep this other viewpoint in mind going forward.

The problem here is whomever wrote the question is an idiot that doesn't know how to math their way out of a wet paper bag - aka a person that writes mathematics curriculum for the American school system.

It really is an issue of whether you give implied multiplication precedence over division. If you don't put parentheses around 2(2+1) then the ambiguity in 6/2(2+1) remains and you are making a choice when you decide to resolve that as (6/2)*(2+1), rather than 6/(2(2+1)). Try to resolve a/2b where a=6 and b=(2+1). Do you get 9 or 1? You get 1. So why should 6/(2(2+1)) automatically be a "wrong" resolution of 6/2(2+1)?

@@RexxSchneider Agreed. You can't accurately represent the original problem with a single row of text without parentheses. When the numerals lie below the horizontal line, the parentheses are implied. At least in America, coefficients appear to the left of fractions rather than the right.The right side is reserved for terms in the denominator. Had the equation been entered 6(2+1)/2 or even (2+1)6/2, all ambiguity would've been eliminated. I don't know if this convention is unique to the US, but to my eyes I would've said 1 is actually correct and 9 was incorrect as the problem was entered.

Quick question. Math is a defined language that should be able to validate itself. 2*3=6 6÷3=2. So I guess the question is, does this equation validate itself when done both ways? It seems to me there should only be one correct answer unless I'm screwing something up.

@@sumduma55 I guess my answer would be that math isn't one language, or at least there's dialectal variation in the way it's written. The problem as written uses elements that are from two different dialects and you're left to guess which dialect you should use for a third element that exists in both dialects but means two different things. It would be like saying you need to put some gasoline (American word) in your lorry (British term) and then saying you feel like slapping a bum. Do you want to slap a poor person or do you want to slap someone's butt? (It's a ridiculous example, but hopefully my point is received.)

@@chitlitlah I think I was wrong. All math should be able to prove itself by inverting the equation. When I originally tried both answers, I only had a validation of the 9 answer. But I think I was doing it wrong for the 1 answer and eventually did end up with it by suspending all that is good and proper. In other words, I thought there was only one correct answer because I was failing to fully account for the difference in operations while checking. I'm pretty sure I was wrong now. Thank you for entertaining an idiot.

Correct. And if anyone disagree with that, he need to learn mathematics.

@@AndrewSopov I disagree with it. There is no reason whatsoever that "6÷2(2+1) is not the same as 6÷(2(2+1)". That's a perfectly reasonable interpretation of the ambiguous expression. It is also untrue that for "equations without parenthesis you should start by solving them from the left to right" because that forces implicit multiplication to have the same priority as division, and there's no reason for making that the only possible choice. As an example, would you evaluate a/bc where a=6, b=2, c=3 as 1 or 9? The entire world evaluates that as 1, but your "left-to-right" scheme gives division priority over implicit multiplication and would give 9 as the answer. Perhaps I'm not the one needing to learn mathematics?

@@RexxSchneider repeat school course of algebra.

@@AndrewSopov You repeat the course. Even better, I'll teach you.

ES was replaced by ES Plus in 2008 and it has the higher priority.

@@okaro6595 weird, do you know why they did that? We're taught in school that implied moltiplications and regular moltiplications have the same priority, it seems counterintuitive to sell calculators that use a different convention...

Just curious, what would be the answer to 6÷2x when x is 3? In this case there is an implied multiplication between the 2 and the x so if implied and regular multiplications have the same priority the answer would be 6÷2x3 or 9 but my understanding of algebra is that the answer is 1

@@HAL_NOVEMILA I truly doubt they told so or they told because they thought so. There a difference of what you know and what you think you know. Everyone doing math knows that implicit multiplication has higher priority but they might not know they know it. When you write sin 2x you do not think it is same as sin 2 * x.

@@Zerc00 If you wrote 6÷2x I would consider it as 6/(2•x) so for x=3 the answer would be 1 BUT if you wrote instead 6/2x I would intend it as (6/2)•x so the answer would be 9... Still the writing is a bit ambiguous

No,it is not related to the parentheses.

I'm American and we were taught PEMDAS or Parenthesis, exponents, multiplication, division, addition, subtraction. I've always just done whatevers in the parenthesis first.

Yes! Is this not taught in schools anymore? I was scanning the comments to see if anyone else would say it and surprised so few did.

Finally someone else uses bomdas or BODMAS where the "O" is OF, dot notation & indices. I had someone become very insulting when I referred to ”of”. I just know that when I use my education, I do get correct answers that match measurements made in the real world, unlike what would happen if the example was a load bearing capacity on a building floor. My method would indicate 1 ton where the other would indicate 9 ton load bearing. I guess that’s why there are buildings in this world that pancake!

PEMDAS does not include Implied Multiplication or the Distributive Property which is the issue here.

I learned it as PEDMAS...

No, multiplication and division are at the exact same priority, in which case it is strictly from left to right. Same for addition and subtraction, which are also at the same priority.

@@M4RC90 Yes, exactly. I hate that the Casio makes up stuff by itself adding those parentheses.

It was taught as BEDMAS when I was growing up (and the dinosaurs were still roaming the earth). "B" stood for Brackets. It's interesting that some of the various mnemonics swap D and M. Either way, the implied multiplication seems like it ought to be taken into consideration and I'd call this a bug.

Another ambiguous example is the expression -2^2 which could mean either (-2)^2 or -(2^2). Due to the same symbol being used to indicate both (negative) sign and unary minus operator.

Well you can do that also on modern calculators.

Yeah some quick ad rev.

@@okaro6595 "modern"... My 30 years old hp48s does that too... Check the linked jpg...

Not at all - there should be no difference in result whether the multiplication before the parethesis is implicit or explicit. It should be 9 in both cases. Casio simply got it wrong.

@@Rosscoff2000 No. The answer is 1.

@@louf7178 brackets take highest precedence, then multiplication and division, then addition and subtraction. Within that the order is left to right. So first you add the 1 and 2, giving 6/2×3, then it's 6 divide by 2, leaving 3x3, which is nine. To get the answer 1 instead the final stage would have to be 6/(2×3), but that second set of parentheses is not there in the given sum - Casio chose to insert them unasked, making its answer wrong. It doesn't matter in the slightest whether the multiplication is implicit or explicit - the rules of precedence remain, including left to right order in cases of equal precedence which is critical here, but Casio chose to separate the two multiplication types and put implicit multiplication way down the list of priorities, effectively inserting parentheses that weren't specified, and changing the calculation entered to something different.

@@Rosscoff2000 I'm well aware. I'm futher aware of shorthand habits.

Or it converts into 6 2 2 1 +*/ It is all how YOU convert it.

It has to do with where the math came from in the first place and what the intended order actually was to fit the problem. RPN is superior because the control of operations order is never given over to the calculator. Also mixing implied multiplication and explicit division is just bad form because it’s mixing paper algebraic format and line expression format

​@@okaro6595 No it doesn't, you have evaluated the equation 6÷(2x(2+1)), not 6÷2x(2+1). The rules for RPN are published and fixed, there is no dependance on a individuals calculator manufacturer's interpretation of presumed brackets.

@@tda2806 But the original expression is 6÷2(2+1). *You* made the decision to treat that as (6÷2)x(2+1) rather than 6÷(2x(2+1)), and it's nothing to do with the rules for RPN.

@@RexxSchneider No I didn't, the only decision I made was to add in the implied multiplication sign, giving 6÷2x(2+1) I then evaluated the expression left to right following the operation precedence, I added no additional brackets. I followed the rules for Polish Notation, postfix.

how does your rpn calculator help here? You have to guess what was meant when it was written as 6/2(2+1) in the first place to know how to type it into a regular calculator OR a rpn calculator. The problem here comes in the notation used in the question, which is ambiguous - hence the person writing the question is the one that's the idiot - place all the blame on them where it belongs and forget about the rest

@@gorak9000 You are assuming that I am solving an expression found somewhere in a textbook. My usecase is much different. I have some real world problems to solve and I come up with my own expressions to solve, so I know what I need. I'm not in school anymore solving excercises :)

@@AntonioBarba_TheKaneB Who said anything about a textbook? Just because I said question doesn't mean it's a textbook question. Have you never had someone else give you an equation to solve in a non-academic setting? Sweet naive summer child, clear communication is always the utmost importance, ESPECIALLY when it's real-world math that has physical implications.

@@gorak9000 I'm sorry but I am not a native English speaker, any communication mistake is probably my fault. Anyway I don't think I am a naive summer child lol, I'm a 36 year old software engineer 😅

More importantly.... It took me 5 mins! to read your name. OMG. My eyes hurt. You can't do that to people :) Cheers [enter name here] you RoCk

I know I was taught this stuff but I went to a parochial school so not exactly national curriculum there.

Yeah, I'm fairly sure I remember in US schools that the implied or explicit didn't matter and you should just add the multiply symbol so you don't forget it, then do PEMDAS.

Which calc?

@@EEVblog HiPER Calc Pro. But it's an android app, not hardware, sorry for misleading.

How does an RPN calculator fix the issue when the issue is the notation of 6/2(2+1) itself is ambiguous?? You have to know what was intended to type this correctly into either a standard or RPN calculator, and with it written that way, there's no way to know for sure what the person that wrote it meant (all you know is they suck at math - aka probably someone writing math curriculum for the US education system)

@@gorak9000 I never claimed it did. Indeed, an RPN calculator doesn't in itself fix anything here. It does however require you to think a little more about how you enter calculations at which point ambiguities or anomalies generally become more obvious. If you just blithely bash the keyboard without thinking its generally asking for trouble.

@@gorak9000 The calculator doesn't resolve the ambiguity at all. You do when you decide how to enter the numbers. Do you enter 6, 2, divide, 2, 1, plus, multiply? Or do you enter 6, 2, 2, 1, plus, multiply, divide? Either way, you're not relying on the calculator to figure it out. It's up to you.

I was taught that the second part of BODMAS was Order, as in powers.

AintBigAintClever you are correct. Has been about 20 years since I learned it though.

You should really forget those PODMAS etc. They are first grade stuff.

@@okaro6595 I wasn't aware there was a different order of operations to be followed when you grow up. Do enlighten us, o wise one.

The thing is, this isn't really the "new math" or anything like that. This issue has always existed, it was just some silly internet thingy that brought it to the attention of the masses who all thought they knew why it was one or the other. The issue is due to contexts and how the expression is mixing them. The world of binary operations is where the division ÷ operator is used, but this is also where BODMAS/BIDMAS/PEMDAS is also taught. However, when you get to algebra, where the implied multiplication is taught and used, it is expected that you would use the division line to show division and the expected understanding of the order of operations is BOMA/BIMA/PEMA. The reason why the order of operations is simpler is because you are expected to understand inverse operations, negative numbers and rational numbers which makes the division and subtraction special cases redundant. An example of this is 1 + 2 - 3, this would obviously be 0. But let's say that you really want to use the - 3 first. If you understand that n - m = n + (-m) then you can rewrite this as 1 + 2 + (-3), then rearrange to (-3) + 1 + 2 and still get 0. The reason why the entire situation is ambiguous is the mixing of contexts. Algebra tends to imply extra parenthesis on implicit multiplication after a division line (1/2x implies 1/(2x)), but arithmetic doesn't imply the same after a division operator. Arithmetic also doesn't use implicit multiplication, so it is unknown whether you should expect it.

"New Math" is just code for "today's teachers are dumber than a bag of hammers". Now, that said, as "professional" mathematician, if I gave you the expression ax / by and gave you values for a b x and y, would you follow PEMDAS blindly and go a * x / b * y, or would you naturally evaluate it as (a*x) / (b*y)? And what if x and y where just variables and didn't have values? Would you treat a and b like weights of x and y or separate terms? When you get into more advanced math, it's rather implied that "juxtaposition" multiplication has a higher precident than explicit multiplication. Now anyone worth their salt would never write it that way, and usually terms with juxtaposition are in an equation joined by a lower class operator (like add and subtract) so there isn't any ambiguity. This whole issue comes down to sloppy notation of the original problem presented.

But most of the time we don't use the division symbol anyways we write it as a fraction and then all these problems vanish anyways

Yep, schooled in the UK in 50's/60's without a calculator (brain computation only) the answer was 1.

Which country?

Same

I doubt. The point is that precedence is taught years before implied multiplication. In fact implied multiplication has no clear precedence: sin 2x = 2 sin x cos x. Math is human readable.

except when people type math out on a regular keyboard - doesn't really matter if it's the division symbol or / writing it that way makes it ambiguous

@@gorak9000 Well, that’s what I’m saying. Because you’re forced to used a single symbol and everything is on the same lame, there’s inherent ambiguity unless you use parentheses. A horizontal line with the numerator above and denominator below clearly indicates how everything gets treated. If every text box everywhere allowed us to use LateX, then we wouldn’t have this problem.

I'm curious where are you from? When I went to elementary scool (grade 1-9) in the 80's here in Sweden, we never used division symbols. All divisions were written as fractions from day one. Calculators were not allowed in elementary school here.

@@ncmaothvez I’m from Canada, AFAIK they also use the division symbol in the US up until at least Pre-calculus (grade 11) as well. It’s bizarre because I think representing fractions with the horizontal line is more intuitive and easier to explain than using an abstract symbol to represent division, which is how they teach elementary school children here. (I think it’s especially weird considering the division symbol “÷” literally just represents what a fraction is actually supposed to look like; the dots being placeholders for the numerator and denominator of course) Yet again North American education falls short haha

@@confucheese I'm also from Canada - division was written as fractions as far as I can remember, or the "square root-ish" division symbol if actually doing long division. That was in the late 80's early 90's though before computers were super prevalent. I'm sure now there's a lot more "/" used and writing things all on one line with lazy teachers that don't know how to typeset math properly. I've also heard that they don't teach "long division" anymore at all - I guess it's just too stressful for today's weakling kids /s

I disagree. You still have to decide which order to evaluate the expression? Is 6 / 2(2+1) to be evaluated as 6 2 / 2 1 + * or as 6 2 2 1 + * / ? RPN doesn't resolve the ambiguity for you any more than the other methods of entering arithmetic expressions.

@@RexxSchneider I think his point is that if you avoid algebraic expressions entirely you don't get this ambiguity. It's not about translating it into RPN, which obviously requires interpreting which order of operations to follow.

Agree completely. When I was studying maths at uni it was an unspoken rule never to use the division symbol. In fact I'm pretty sure it was during A levels. I think it changed when people stopped writing equations by hand and instead typing them into a computer. Then it became more tricky to do the whole multi line equations of the form a over b.

on paper you just... don't use this kind of notation at all. If you have to use a slash or obelus for space reasons, you use brackets.

@@tommihommi1 My comment relates to multiplication, not division in the video (hope this makes it clear). Anyway, multiplication and division are (usually) treated as having equal precedence and the order of evaluation should be well-defined (left-to-right) regardless of whether multiplication is implicit or explicit. BTW, if you need to use brackets, you are expecting the reader not to follow/know the usual rules and the brackets are just a safety net, not a necessity.

The issue is that, unlike a lot of science, there is no complete international standard for mathematical operation precedence. PEMDAS/BIDMAS/BODMAS is a widely taught convention but the nuances of juxtaposition and other operations are left for people to define themselves and then tend to become regional conventions lead by exam bodies. In the UK, the exam bodies expect juxtaposed multiplication to be evaluated after parentheses are so that parentheses are all resolved away before continuing with explicit operations.

I should add I'm in the US. So your theory about why the one model is different doesn't ring true to me.

"The scientific notation gives a higher priority to the implied multiplication." No, it doesn't. Write 6/2(2+1) in Wolfram Alpha. It gives you 9. When you write 6:2(2+1) it treats it as a fraction with 6 on top and 2(2+1) on bottom. That's why if you write (in Wolfram Alpha) 6/6*2 it will give you an answer of 2 and if you write 6:6*2 it will give you an answer of a fraction with 1 on top and 2 on bottom. You are simply incorrectly inputting the problem, that's why you get a wrong solution and reach the wrong conclusion that implied multiplication has priority there.

​@@bottenbotten91 And that's a known Wolfram Alpha limitaition. Even if you try _a=2+1; b=6/2a;_ it will evaluate b as (six over two)a = 9, whereas almost every mathematician, physicist or a good engineer would give you an answer b = 1. Except maybe some North Americans.

The problem isn't so much the calculator, or if you use RPN, but when it's written on paper it's rather ambiguous what was actually meant in the first place. RPN or not, with the way it's written, it's really a toss up - do the calculations from left to right and treat explicit division and implied multiplication the same, or is the division higher priority, so you have to evaluate the whole right side and treat that as the divisor or not? Just a crap way of notating that particular equation.

@@gorak9000 when it's written on paper it's rather ambiguous -> absolutly but RPN method avoid bad-reading of it in most cases :) with adding both protection PEMDAS & RPN syntax so you always have all intermediate calculation in right priority order (but human side errors)

@@francoisp3625 Exactly, with RPN you decide the order, there is can not be any ambiguity. Also having the intermediate calculations also help you to understand what is going on and if there is a problem. I would recommend Free42 if you are after a good HP42s on your phone or computer.

@@chrisdrew9767 But which order do you decide for RPN? Is 6 / 2(2+1) to be evaluated as 6 2 / 2 1 + * or as 6 2 2 1 + * / ? RPN doesn't resolve the ambiguity for you any more than the other methods of entering arithmetic expressions.

@@RexxSchneider You chose which one, if it is a fraction like here you want the first one if it is 6 divided by every thing you chose second one. This, why not solving the problem, dosen't confuse the user when entering like Casio here

I no longer use multiplication or division symbols - I use only implied multiplication and fractions. This goes back to learning algebra...

Exactly my thought. There's no reason at all for processing implied and specified multiplications differently.

When I was taught math, we had to clear out brackets first (BEDMAS), hence I get 1 when doing it in my head :D 6/2(2+1) 6/2(3) 6/6 1

@@flandrble Divisions and multiplications have the same priority, so they get solved left to right: 6:2*(3) 3*(3) 9 To make your solution right, it should be written 6/(2*3). The problem is, when you imply the multiplication it stops looking like arithmetics and starts resembling algebra. In this case, one might solve 2(2+1) as 2*2+2*1 o_O

@@IlBiggo with bedmas and bodmas (basically everywhere in the world except America) you clear out brackets first.

@@flandrble Exactly. So first is (2+1) => 6:2*3. Then you solve left to right 6:2=3, 3*3 = 9. I don't know if the "DM" in bedmas means "divisions before multiplications" (doesn't matter in this case), I think it should be read as "brackets, exponents, div and mul, add and sub". For additions and subtractions the order doesn't matter, but for divisions and multiplications I've been taught to solve left to right. If one really wanted "six divided by everything else", "everything else" should be contained in brackets: 6:(2*(2+1)).

I thought the same with the first couple examples in the video.

Hah. In fact those symbols meaning the opposite way.

@@gandalf94013 It's certainly possible that I misremembered that. ^_^'