EEVblog 1479 - Is Your Calculator WRONG?

Is your calculator WRONG? It could be!
Looking at the issue of implied multiplication and how it can affect your calculations. Why do some Casio and TI calculators give a different result to others? And why do they differ from your phone calculator, google calculator, or Wolfram Alpha?
How Order of Operations matters.
TI explains the differences in models and why: education.ti.c...
The How and Why of Mathematics video on the issue: • The Problem with PEMDA...
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#ElectronicsCreators #Calculator #maths

[@KarlAdamsAudio]

The Casio adding extra parentheses for clarity, it's also a programming practice I highly recommend - you may know your chosen language's operator precedence rules like the back of your hand, but parentheses are free (in a compiled language, at least) and you don't want your code's intent to be apparent only to a select few.

[@no-defun-allowed]

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.

[@edwarepl]

Get an rpn calculator

[@DrBovdin]

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

[@KarlAdamsAudio]

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.

[@MichaelOfRohan]

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

[@mattb9664]

I recall making many mistakes with a calculator due to having trust in the parenthesis logic..that was some 25 years ago. Same with excel. I basically trained myself to use lots of parenthesis to avoid order of operations errors.

[@TorutheRedFox]

same lol

[@sumduma55]

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.

[@demoncloud6147]

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

[@conodigrom]

Use RPN ;)

[@Gameboygenius]

My impression has always been that the reason for giving implicit multiplication a different priority is not for use with parentheses, but for use with variables. If you type 1/2x it's more likely that you meant 1/(2x) than (1/2)x.

[@RexxSchneider]

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.

[@ska042]

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.

[@SirSpence99]

@@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...

[@ABaumstumpf]

@@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 ....

[@samjones1954]

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

[@koharaisevo3666]

This is why I always use exra parentheses to make it absolutely clear how the order of operation goes.

[@markc2643]

If you look at it as 6 / 2(x+y) , you would not do the 6 / 2 before the 2(x+y) . The same rules apply whether you are using numbers or variables.

[@keithnicholas]

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.

[@best591]

Funnily enough in Germany there are both variants: my casio fx-991ES, which I used in the exams for my high-school diploma gives out 9 as a result (as shown in the video) and the fx-991ES PLUS, which my girlfriend used, gives out 1. Both are allowed to use in the exams for the high-school diploma and while studying computerscience

[@eideticex]

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.

[@TheScarvig]

@@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

[@johnmiller0000]

A bad student blames his/her calculator!

[@sujanaryal833]

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

[@picobyte]

@@johnmiller0000 RTFM-Matters.

[@hjups]

Even including the explicit multiplication could be ambiguous, so I would opt to include parenthesis around the division as well. And if it's really that critical of a calculation, you want to be as precise as possible, where the result shouldn't depend on how one calculator might work over another. The other thing you could do to be explicit while entering into a calculator, is treat division like multiplication, where you explicitly do 2^-1 (via the inverse function).

[@tengelgeer]

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

[@SweDennis]

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.

[@baileyharrison1030]

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

[@edgeman83]

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.

[@samjones1954]

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)?

[@Pongo8844]

The ambiguity is from the inline typesetting of the division sign. It is unclear whether the intention is 6/2 x (2+1) or 6/ ( 2(2+1) ). The Casio resolves this with putting in extra ()'s to clarify the operation. I think that it is a great feature.

[@flybackrs]

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...

[@seethruhead7119]

actually it's because of juxtaposition

[@0LoneTech]

As usual, my favourite calculator Qalculate! does a little helpful step: like Wolfram Alpha, it displays its interpretation, and when you hit Enter/= it actually explains the ambiguity and asks which variant you want (which is then set under Mode - Parsing Mode). It's not an issue in RPN mode, of course (6 2 2 1 + * /). You can even get the chain syntax (1+2*3=(1+2)*3, strict left to right) if you want.

[@mielole]

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

[@CRalph3122]

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

[@MrGreensweightHist]

@@mielole Casio is wrong.

[@mielole]

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

[@MrGreensweightHist]

@@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

[@Grantherum]

When I was a kid, in the USA, we were taught the 6/2(2+1) would have been 1. Sometime in the 90's or so, they changed over to the =9 version. They wanted to make things difficult for some reason. I was always taught that you do parenthesis first, including the implied multiplication... then do the rest of the multiplication in order of precidence.

[@chrishartley1210]

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.

[@Grantherum]

@@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.

[@MuffinsAPlenty]

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.

[@metatechnologist]

This is the kind of thing that crashes a space probe.

[@evensgrey]

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.)

[@boblewis5558]

@@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! 🤔😦😲🙄🙄🙄

[@evensgrey]

@@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.

[@billcrowell5096]

It is concerning that the results are region dependent. Math should be universal. My takeaway is that one should always use the operator as this appears to be an implied operator quirk. 6/2x(2+1). Of course, some of us use RPN and our calculations are always correct.

[@ovalwingnut]

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

[@myselfremade]

It's 5:31 in the morning. I have been awake for 21 and a half hours, and I'm listening to an Aussie tell us about how calculators can be even more pedantic than you would otherwise think. Obviously the correct solution is a new button on the calculator ( I propose it be labeled the "1/9" button ) to make it even more confusing for casual calculator users.

[@adriansdigitalbasement]

HP 48G: as you type it in, it adds a period between the 2 and ( basically explicitly saying there is a multiplication there. The result is of course 9 on this calculator.

[@jenselstner5527]

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

[@Kidna]

Ti89 here. Same thing

[@13231wmw]

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

[@KG5IF]

My trusty HP42 comes up with 9 also

[@granitepenguin]

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

[@lerssilarsson6414]

"Young man, in mathematics you don't understand things. You just get used to them." -- John von Neumann

[@FadiFadi00]

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

[@jimwolsiffer4397]

I am in the US and I was taught in School and College math to use the system that results in the answer of 1 for this equstion.

[@tommihommi1]

If you replaced the 2(2+1) with 2x, with x equals 3, anyone would expect the result to be 1. Yes, pemdas is a thing, but anyone who stops to think about it should realize that this case is ambiguous.

[@fredfred2363]

This comment. Yes. 👍🏻

[@timgooding2448]

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

[@not_your_raccoon]

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

[@substantiaalba]

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

[@stephanc7192]

Ambiguous it is!

[@skunkmyrddyn]

These kinds of errors with algebraic input is why I prefer rpn input on my hp. It is much more difficult to get order of operation incorrect when dealing with a single operation at a time with values on a stack.

[@bertblankenstein3738]

Amen brother.

[@arminth]

RPN on an old HP 48SX and the calculator does what you want!

[@mosfet500]

Thanks Dave. I use the HP's but in RPN mode which, as an engineer, I prefer. So I get to decide the order, not the calculator. My answer is "1". I live in the USA, by the way. I think the system I learned in school is PEMDAS (Parenthesis, exponents, multiply, divide, add, subtract.) The system doesn't seem to differentiate between implied or not implied, however, when I see equations like this it is implied to me.

[@marko.692]

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

[@mosfet500]

@@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.

[@katbryce]

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.

[@marko.692]

@@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…

[@mosfet500]

@@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

[@darkforce2803]

One more reason I prefer RPN style calculators :). RealCalc that you used on the phone at the end in RPN mode has mostly replaced my hardware HP calculators though.

[@russellhltn1396]

RPN for the win

[@zitt]

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.

[@filipbataz6684]

This is one of milliard of reasons to use RPN

[@ErikN1]

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.

[@robertwatsonbath]

@@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.

[@KamiraXIV]

Bridgekeeper: What is 6/2(2+1) King Arthur: What do you mean? (6/2)(2+1) or 6/(2(2+1))? Bridgekeeper: I don't know that? *Bridgekeeper gets tossed into the chasm*

[@radry100]

It's not a yankee thing. I bought a FX-991ES in europe and it gives 9 as a result. I learned in school to never put in such ambiguous formulas but always add parenthesis to make it unambiguous and that calculators generally use the left to right priority when you don't add parenthesis.

[@AxMi-24]

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.

[@usagi_hase]

The fx-991DEX sold in the german market also has a higher priority for the implied multiplication.

[@M4RC90]

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

[@Yotanido]

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

[@11mitsaras11]

I always put all the parentheses by my self. That way I'm always sure about the result. Also it helps when i have to read my code again after a few months. I know it looks stupid sometimes and it takes more key strokes but it gives me peace of mind!

[@mal2ksc]

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.

[@rpocc]

I also should mention that after all there are two approaches to input methods. When a mathematician or an engineer deals with algebraic formulas, the members are usually just variables or constants with some added coefficients (with all multiplications implied), all ratios are usually straightly expressed with horizontal bars, and all pure-number ratios and extra series of divs and mults are usually already simplified. This way formulas are always very clear. The one-line method is nice for school problems or programming and needs extra parenthesis to eliminate any disambiguations, so those 8:2(2*4) or 6/2(2+1) are intended traps that shouldn’t exist in the first place and should be replaced either with a straight simplified constant coefficient like 1, 9 or 16, or by a formula like a over (b*(c+d)) or (a over b)(c+d), so there will be no multiplication signs at all, I mean all multiplications will be implied. In my experience, when I deal with real calculations, I take a mathematician’s formula and convert it into an explicit series of assignments, operations and returns, so there will be no any nonsense like the subjected example.

[@donepearce]

1 is the answer for me. When I see a 2 followed by an expression in parentheses, that means two of whatever is in the parentheses. If there is an explicit multiplication sign the normal priority applies.

[@ChickenPermissionOG]

it has never meant that.

[@donepearce]

@@ChickenPermissionOG It has always meant that

[@M4RC90]

Never heard about this rule.

[@quackers969]

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.

[@shanejimmink]

@@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)

[@Scouarn]

Please just use fractions, this calculator has a fraction key. The meaning of the division symbols is not always the same. But the fact it adds brackets is really cool.

[@travelthetropics6190]

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

[@gcewing]

Interesting. I've long thought that the convention used in most programming languages, where multiplication and division have the same precedence, is a misguided and confusing thing to do. But giving different precedence to implicit and explicit multiplication seems even more confusing, especially if different calculators do it differently.

[@boblewis5558]

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?

[@NetRolller3D]

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).

[@RottnRobbie]

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.

[@NetRolller3D]

@@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.

[@boblewis5558]

@@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!

[@burritocodes]

Okay, hold on. 6/2(2+1) = 9 is an American standard? I wonder when that became a thing. I'm a Yank, born and bred, and the second I saw that calculation, I came to the answer of 1. I learned that you always calculate anything in parens, then multiply that by whatever preceeds it (assuming implied) and then divide. None of this divide and then do the implied multiplication rubbish. Just for context: I graduated in 1998.

[@IlBiggo]

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 •.

[@rwbishop]

The 'newer' HP35S in algebraic mode came up with 9. Had never had it in that 'mode' till now. It was stressful un-nerving & bizarre... need to go lay down a while...

[@ErikN1]

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

[@rwbishop]

@@ErikN1 I learned my lesson.

[@KarlAdamsAudio]

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

[@vampiremuffinman7183]

My take would be that 6÷2(2+1) is the same as 6 ÷ (2x2 + 2x1), where 6÷2x(2+1) cannot be thought of the same way. Ideally, it should be written as (6÷2)(2+1) if the 2 can't be distributed into (2+1), but an explicit 'x' included should indicate the same. I could be very wrong in this thinking, as I haven't cracked open a math textbook since the 80s... Now I wonder what my Casio fx-451 would think?

[@okaro6595]

No,it is not related to the parentheses.

[@squidkid2]

This video does a great job demonstrating the pitfalls of letting computer programmers get involved in math!! BTW try this calculation on your Casio. 11^6 /13. It will give you a topic for a whole other video.

[@skonaslp]

First of all, you should always use fractions (and such) for clarity, but it's not always possible. Juxtaposition precedence rules are more a scientific thing, for example when formulas get in-lined (meaning only one line is there for utilization), they can't use stuff like fractions. Extra brackets could cause confusion when reading, as it's a bit messy, and many are too lazy to add them everywhere. That's where juxtaposition comes into play. 1/2x=1/(2x) and examples shown in the video would normally be interpreted like the Casio did, but for clarity it should always be made clear (say at the start that some paper uses juxtaposition rules). The Casio is doing a super job at this,even adding brackets for clarity, but it's not something you'd likely encounter in school. Not even schools in Europe (in my experience) teach kids about implied multiplication and its precedence rules, as mentioned before it's just a "style"-thing and it's not likely to be encountered anywhere for the average student.

[@tookitogo]

This is something I love on my HP Prime: in the default input mode, it parses the input directly into fractions as you enter them. So the instant you type the division sign, it creates a visual fraction with the 6 in the numerator, and the insertion point in the denominator. So regardless of what my intent was, I get instant visual feedback of what the calculator is going to do when I press enter. (The HP Prime is not perfect, but I think it’s a very underappreciated calculator that deserves a lot more popularity than it has.)

[@brlisong]

This is why I love my HP 11C which uses RPN. With RPN there are no implied operations.

[@ricardomarques1769]

The issue is how the equation is written. 6÷2(2+1) is not the same as 6÷(2(2+1). According to math rules, when you have an equation like that, with multiple operators and parenthesis, you start by solving first what's inside the parenthesis, in the first case it would be (2+1) = 3 on the second case it would be (2(2+1) = 6, furthermore it is implied that if you have a number and a parenthesis next to it, without an operator, that number is multiplying what ever is inside the parenthesis. And then there is the operator precedence or priority, multiplication and division have precedence over sum and sub. And equations without parenthesis you should start by solving them from the left to right. So after solving everything wich is inside parenthesis, and then solving from the left, for the first case you will get 6÷2(3) = 9 And for the second case 6÷(6) = 1 I've encountered this problem while programming some equations in C, if you don't use the appropriate method, include all parenthesis you will encounter similar errors. The best way of how to solve those kind of equation, for me would be knowing the basic math rules.

[@AndrewSopov]

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

[@RexxSchneider]

@@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?

[@AndrewSopov]

@@RexxSchneider repeat school course of algebra.

[@RexxSchneider]

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

[@chrisose]

The removal of priority of implied multipliers in the US is a "new math" thing. When I went through algebra and higher maths in the 70's, implied/distributed properties were handled before base multiplication/division. Granted, this was a time when we wrote everything out and the only calculation tools we had access to were the tables in the textbook and a slide rule. Something I noticed on the Wolfram page you showed. It reconfigured the equation to completely remove the distributed property by making it a discrete fraction times the result of the parenthetical expression. Yet the inputted equation was clearly written as a fraction with "6" as the numerator and "2(2+1)" the denominator.

[@surferdude4487]

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.

[@Vibe77Guy]

Coefficients should always be calculated as part of the parenthetical, just like every other factored value. Take the example of a 24" dia wheel rolling along a 120" track. 120"÷2(π)(12)= Anything other than approximately 1.5 cannot be physically duplicated.

[@cmuller1441]

This has been discussed 1000* on the internet. It's the problem of implicit multiplication or no * before the "(". Some consider it a higher priority operation. Some don't... The problem is that the calculator writes operations on one line as a sequence of operations. Mathematics on paper doesn't have this problem because you put an horizontal line for division that removes any ambiguity. Like that: www.hpmuseum.org/img/48s/48eqnwr.jpg

[@okaro6595]

Well you can do that also on modern calculators.

[@alch3myau]

Yeah some quick ad rev.

[@cmuller1441]

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

[@chitlitlah]

To me with my American education, it's mixing two different notations in a way that was never taught in class. As we transitioned from basic arithmetic classes to algebra classes, the division sign was completely replaced with fractional notation and implicit multiplication was first introduced. To me, it's like mixing an English noun with a French adjective and then asking if it would be grammatically correct to put the noun or the adjective first. Just stick to one language or the other and you won't have that problem. I almost exclusively use RPN/RPL, so the order of operations is absolutely clear anyway.

[@sumduma55]

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.

[@chitlitlah]

@@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.)

[@sumduma55]

@@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.

[@schwamforfreedom]

I was taught in the US from pre-algebra through differential equations that the implied multiplication is a priority because the 2(1+2) is an expression that needs to be resolved before it can be divided into the 6. The actual calculation is just a fraction being reduced, so the expression in the denominator, 2(2+1) is resolved first and the remaining fraction is 6/6 or 1. And I would expect a calculator to give 1 as the answer. Glad to know order of operations isn't standardized and not all calculators see it the way I do though. Thanks for diving into this.

[@joshuaewalker]

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.

[@schwamforfreedom]

@@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.

[@joshuaewalker]

@@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.

[@eideticex]

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.

[@MagFlux]

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.

[@johnclawed]

I had a Casio programmable from 1990 and its manual said "right to left", which is what it did.

[@crissd8283]

I'm from the US but I agree the answer should be 1. Implied multiplication should take priority.

[@HAL_NOVEMILA]

In Europe (at least in Italy) we get the ES model and I can confirm that conventionally here implied moltiplications and regular moltiplications have the same level of priority, if we wanted to prioritize one over the other we would use brackets...

[@okaro6595]

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

[@HAL_NOVEMILA]

@@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...

[@Zerc00]

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

[@okaro6595]

@@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.

[@HAL_NOVEMILA]

@@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

[@Darxide23]

This is what happens when you write an ambiguous statement. In Algebra, if you do have a number to the left of a parenthetical without an operation symbol, then that number gets distributed within the parenthesis and this operation takes precedent over multiplication and division. So that distribution happens __first__ meaning you end up with 6 / 6. If you expect that 2(2+1) should actually mean 2 x (2+1) then you __MUST__ have that multiplication operator in there. tl;dr: Write a bad form or ambiguous mathematical statement, you're going to have a bad time. EDIT: Actually watched the video. Am a Yank and I learned that the distributive property as described is a different function that multiplication. If I answered 9 to this statement then I'd be marked wrong. The expected answer was 1. This applied to not only high school algebra (AP courses), but when I got to college level mathematics classes as well. I don't know who learned it the way Dave says the "Yanks" were supposedly taught, but I've never seen it that way in a classroom environment.

[@soctnights]

TI-36X Pro: Answer 9. USA. My favorite calculator at this time. I worked for a popular pharmaceutical company 80s-90s in their Engineering/Calibration group. The QA department had to approve the calculators we used. Some preferred HPs and most of us preferred TIs.

[@GoSlash27]

I think the problem here isn't one of implied multiplication, but rather one of implied division. The coefficient (2+1)... Does it reside in the numerator of 6/2 or the denominator? If you put parentheses around 2(2+1) then that term firmly resides in the denominator. If you don't, then (2+1) lives next to the fraction, which is in the numerator. I was raised on single line calculators where it was imperative to solve your problems in heierarchial order to avoid such ambiguities and many tech exams still insist on using these old- school calculators. This gives the younger techs and engineers fits. I mentor technicians on the exam to qualify as lead technicians, and I spend a lot of time teaching them how to use the older style calculators properly.

[@gorak9000]

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.

[@RexxSchneider]

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)?

[@GoSlash27]

@@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.

[@juntendo6104]

Casio fx-100AU says 1, and does order signifying bracket indication as well. Never really paid much attention until today. God I love this calculator

[@Cherijo78]

American here, and this stuff drives me nuts. When I was taking computer programming courses 20 years ago in college, we were taught that we should always avoid allowing any kind of implicit math like this in our code precisely because we may not be able to predict how a compiler will interpret it down the road. I was taught to always be explicit and use parentheses liberally even if a language allows implicit multiplication.

[@jonathanxhu]

RPN supporter here. Another reason why RPN calculators are better. :D

[@AJMansfield1]

Putting implied multiplication at a higher operator precedence makes a lot of sense to me as a convenient way to let the user express denominator quantities succinctly. When writing out mathematical notation on paper, we generally only use implied multiplication within a single conceptual "term" of the equation, and having explicit multiply/divide peel off the first symbol from that rather than taking the whole thing violates that conceptual model. Having 6x³y² ÷ 4x²y³ evaluate as 6 × x³ × y² ÷ 4 × x² × y³ feels nearly as counterintuitive to me as having 6x³y² + 4x²y³ evaluate as 6 × x³ × (y² + 4) × x² × y³

[@jcxtra]

At school they taught us BIDMAS (or BODMAS) Brackets, Indices (or 'Of'), Division, Multiplication Addition, Subtraction. That's usually how I try and remember the order of maths, but fascinating look at the way calculators do the operations :) Edit: But yeah, probably best to put things explicitly for how you want it processed, to assume makes an ass out of 'u' and 'me' :)

[@ericg7044]

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.

[@ThePopolou]

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.

[@peterbrown172]

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!

[@johng.1703]

the problem is, each device is making an assumption for the problem because the problem is ambiguous is it; 6 ____(1+2) = (6/2)(1+2) 2 or 6 ______ = 6/(2(1+2)) 2(1+2) computers really struggle with multiplication by juxtaposition, plus, in PEMDAS and those calculators follow the US standard, have multiplication, division, AND multiplication by juxtaposition at the same level. this can be demonstrated by doing 8/4(3) which should equal 2/3, but in the US / PEMDAS that would be 6. as it treats it as (8/4)*3 not 8/(4(3)) using brackets forces the calculator to do the operation as you require it to be done regardless of which market the calculator was designed for.

[@LundBrandon]

I'm a software engineer. I always double, even triple check my calculator for this simple reason.

[@iainwalker8701]

I was taught all Casio calculators operate on the basis of BODMAS. bracket's (inside), (bracket's) Outside, division, multiplication, addition, subtraction. i would say the Casio calculator is correct as operator has not specified a multiplication after 2. Looking Outside the brackets there is only one symbol which is the Division. So the calculator sees the first 2 as Outside so does that before it calculates the Division afterwards.

[@AintBigAintClever]

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

[@iainwalker8701]

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

[@okaro6595]

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

[@AintBigAintClever]

@@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.

[@tda2806]

This is to do with RPN (Reverse Polish Notation) the expression 6÷2(2+1) converts to RPN (postfix) 6 2/ 2 1 +* (the * being the implied multiplication) answer 9 which is what an HP calculator (designed in the USA and were all RPN in the day) would give you. RPN or posfix is a neat way to conserve memory and the storage of rules when using computers to carry out calculations, there are no bracketed operations in RPN.

[@okaro6595]

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

[@not_your_raccoon]

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

[@tda2806]

​@@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.

[@RexxSchneider]

@@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.

[@tda2806]

@@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.

[@JarOfRats]

PEMDAS- Please Excuse My Dear Aunt Sally Parenthesis then exponents, multiplication, division, addition, subtraction. Parenthesis- 2+1=3, no exponents, multiplication- 2x3=6, division- 6/6=1. End.

[@tannerbass7146]

Dude that PEMDAS error in the thumbnail is killing me Good work 👍

[@marcus_w0]

So much for maths as a universal language.

[@louf7178]

I'm from the US, and my answer is 1. I understand the "9" answer, but that is completely ignoring the shorthand meaning of the number adjacent to the parenthesis.

[@Rosscoff2000]

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.

[@louf7178]

@@Rosscoff2000 No. The answer is 1.

[@Rosscoff2000]

@@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.

[@louf7178]

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

[@AntonioBarba_TheKaneB]

yeah, I don't like the division in expressions, to avoid any confusion I usually use brackets, or flip around the division to get a multiplication, such as A * (1/x) instead of A / x. I'm glad that RPN calculators exist which don't suffer from ambiguities like this.

[@gorak9000]

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

[@AntonioBarba_TheKaneB]

@@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 :)

[@gorak9000]

@@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.

[@AntonioBarba_TheKaneB]

@@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 😅

[@ekaa.3189]

Make it explicit. This is why I always make it explicit and add "(" and ")" to all equations to get rid of implied operations.

[@idimitrov]

The scientific notation gives a higher priority to the implied multiplication. It's not just Casio calculator. Wolfram Alpha will give you answer 1 as well when the problem is given as 6:2(2+1). There is an explanation of this in the following Wikipedia page: en.wikipedia.org/wiki/Order_of_operations section "Mixed division and multiplication"

[@bottenbotten91]

"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.

[@alexanders6474]

​@@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.

[@rubemjr9623]

My fx-82MS (second edition) bought in Brazil also prioritizes implicit multiplication. It's interesting because as a child in school, this priority difference was never mentioned, but as an engineering student, it is quite intuitive to me.

[@surisuri8993]

You can also enter (6/2)(2+1), then you get 9 on an fx-991EX. Or enter 6/2 as a vulgar fraction using the fraction key, then (2+1). In both cases it will calculate "correctly" =9, even with the implied multiplication.

[@gylkag]

I didn't have one of those "smart" symbolic calc, they were too expensive for us in mid-90's in Ukraine and I already felt blessed having the generic engineer Casio calculator. So I don't know what VPAM calcs would evaluate, but the only intuitive thing I would expect is 9.

[@RexxSchneider]

Thanks for taking the time to investigate this issue, which has been doing the rounds on the internet for some time. The simplest summary, of course, is that an expression like 6/2(2+1) is *ambiguous*. If you don't take the time to resolve the ambiguity by entering 6/(2(2+1)) or 6/2*(2+1), you can't complain if the calculator decides the priority in the original expression differently from what you expect. Neither 1 nor 9 is the "wrong" answer.

[@SpitfireMkIIFan]

When you are about to write exams and then your exact calculator model shows up on EEVblog with the titel "Is Your Calculator WRONG?" :P

[@Thirsty_Fox]

It isn't a matter of what is the answer, but what is the question. It's always given how it's typed into the calculator, rather than how it's written (ie. what the actual math problem is). Of course, if it were written then the user could just enter it properly, such as by using brackets for (6/2). In any case, I much prefer having implicit multiplication take priority ahead of explicit because it seems more like a bracket operation than a multiplication... ie. the value is factored from the brackets and must be distributed back. But then I also just use so many brackets as to be sure as to what's being calculated (and the 991 EX fractions help clarify).

[@TrueThanny]

I've always seen implied multiplication as taking precedence. So 2(2+1) is identical to (2*(2+1)). Converting that to 2*(2+1) instead strikes me as lazy parsing logic. There's a _reason_ people use implied multiplication in certain places. And it's not to simply save space on a character.

[@TrueThanny]

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

[@MrPnew1]

Wouldn't order of operations dictate PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). So parenthesis 2 + 1 = 3. The multiplication is next so 2 x 3 = 6 then division, 6 divided by 6 = 1

[@EEVblog]

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

[@Philip8888888]

I learned it as PEDMAS...

[@M4RC90]

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.

[@tkermi]

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

[@gunderd]

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.

[@akioweh]

fx-991ES **PLUS 2nd edition** also prioritizes implied multiplication, but does not change input/add parenthesis for clarification

[@fantesykikachu]

Its not just implied multiplication, there is a lot of variation in the interpretation of order of operations; as a result a lot of scientific journals (especially math focused ones) specify the way they want calculations done. In fact a lot of American journals do differentiate between implicit multiplication and explicit multiplication.

[@markevans2294]

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.

[@yourcurtainsareugly]

Coming through US schools, I don't think I was ever taught a difference between explicit and implied multiplication. Didn't matter to me in terms of calculators since I bought an HP RPN calculator in junior high.

[@ovalwingnut]

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

[@npiper]

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

[@matthewmiller6068]

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.

[@frogandspanner]

The obelus (÷) symbol has different meanings for those of different ages and cultures, and is outwith the normal BODMAS scheme. For this UK 70-year-old ÷ means means the same as a bracketed rhs, so 6 ÷ 2(2+1) ≡ 6 / (2 × (2 + 1) ), which would result in 1. I wish they would replace the obelus with a slash on calculators, then it would not be quite so ambiguous.

[@brucepickess8097]

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

[@poorlydrawnstickman1660]

Coming from US schools we have never discussed implied or explicit multiplication. However, my programming brain always adds the parenthesis in as the Casio explicitly writes them out… just makes so much more sense!! No clue why this nonsense has existed for so long, should be standardized by now! Funny enough my ti-89 titanium sometimes even removes my extraneous parenthesis after I explicitly type them in… I wonder if newer calculators do this?

[@mielole]

In case this video makes you want to learn RPN and you already understand parse trees: 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 So as you can see to get from a parse tree to the RPN you do a depth first traversal of your parse tree, left to right.

[@neplatnyudaj110]

Mine shows one, but gives me a warning and lets me select between strong and weak grouping.

[@EEVblog]

Which calc?

[@neplatnyudaj110]

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

[@robertwatsonbath]

Been a RPN calculator convert for over 35yrs (current daily is a SwissMicros DM42). I knew those banana shaped things you kids have on your fancy calculators would be trouble.

[@gorak9000]

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)

[@robertwatsonbath]

@@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.

[@TrueThanny]

@@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.

[@hakonsoreide]

Yup. Apparently, even the people who invented PEMDAS prioritised implied multiplication of parentheses in their examples, but seem to have implied it so much they forgot to mention it in the write-up. Arguably, an implied multiplication of a parenthesis is part of the parenthesis itself. I didn't actually know calculators messed this up before buying one recently, but I was happy to find it does it in what i consider the only mathematically correct way, prioritising implied multiplications before explicit multiplications and divisions. Mine is a Sharp EL-W531TL.

[@zokonjazokonja]

I don't have such problems, I'm used to use old style calc which do not use expressions, on phone I'm using realcalc. I'm deciding about priority of execution by how I type data inside, for more complex calculations I use memory and it works just fine for me.

[@durexyl]

Mathematicians must pull their hair out when equal operations' precedence/priority is treated differently. Implied or not, should be the same (on paper too).

[@tommihommi1]

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.

[@durexyl]

@@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.

[@roberthindle5146]

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.

[@aliveandwellinisrael2507]

Although I don't see why it would be anything other than 9 (order of operations exists for a reason), I do like that calculator at 3:49. Nice design, and if it's going to return 1, it's great that it removes ambiguity as to why, allowing you to rephrase your input if needed.

[@he1py]

In Algeria here, the Casio FX-991ES PLUS gives a 1. Weirdly enough the solar panel is not connected but that is another issue..

[@MrNside]

"Math is the Universal Language..." Except when it isn't.

[@oninbridders]

Windows Scientific removes the 2 as you demonstrated on the latter example. You have to put the X before the ( ) to get it to work.

[@SustainedFuture]

Had to check it out myself: Windows 10 removes the 2 and gives a final answer of 2. Windows 11 adds an explicit × before the parentheses and results in the answer of 9.

[@kinpatu]

Bring back RPN!

[@KrS14]

Just to be clear, I live in Canada, which is part of North America. If i had a calculator that have me an answer of 9 for that, it would go directly in the garbage. We actually do BEDMAS with distributive up here. At least we did in the 90s in highschool :) Just didn't want to be lumped in with the US education system ;)

[@FluorescentApe]

i'm glad i have the casio fx-991ex, like shown in the video. i have always been taught that there is an invisible multiplication sign before the parantheses if nothing is declared. Edit: bought it in Sweden.

[@aaroncurley2377]

About eight years ago I hit this problem w a cheap Casio calculator. I figured the calculator had a defect so I just went out and bought a slightly different Casio model. Imagine my surprise when that one was “defective” too. I raged and threw out every Casio calculator in the house in disgust. I got yelled at by my mom because I threw out calculators that I didn’t technically own (were my siblings’s). I was still super annoyed w Casio and replied that I would gladly buy replacements from a company that knew how to properly implement standard math rules. Ok. Probably a bit of an overreaction in hindsight on my part 🤪🤪

[@Kris_M]

I'm not sure what distributive property has to do with it, because that applies to both implicit and explicit multiplication, i.e. A*(B+C) and A(B+C). 13:44 I don't agree, I'm an engineer and I say 1 is the correct result, implicit multiplication takes precedence.

[@LutzSchafer]

Well, but once you leave the division 6/2 the next operation *(2+1) belongs to the numerator again not the denominator. I am surprised to learn that this notation does not seem in common understanding. At least from programming or Matlab/Mathematica this is fundamental.

[@bobvines00]

Before retirement, I worked in an Engineering office where a co-worker was using his "shoe phone" calculator for calculations. Fortunately, he remembered enough basic trig to catch an error. The ME sitting next to him, using a different calculator app, did the same exact calculation, entered the same way, and got the correct answer. Long story short, the ME found that the error in the other Engineer's calculator app was that the Programmer didn't properly convert between degrees & radians in the algorithm used. **ALWAYS** verify that your "shoe phone" calculator app works correctly before using it for relatively important calculations where you aren't able to easily catch errors before you get too far! I didn't realize that the different calculators were "tweaked" for sale in different markets like Dave showed here. At work I actually used a cheap scientific calculator that I bought in the early to mid-'80s, and it still works fine today. (I also only had to change its (non-rechargeable) battery 2-3 times since I bought it. ;) After seeing Dave demonstrate the "school-approved" & "Engineering/Scientific" calculators, I'll definitely remember to test/verify how they use precedence!

[@NOELQUEZON]

GEMDAS or PEMDAS does support All Casio Model. Not Divide Symbol Instead Fraction Symbol. Im using FX-CG50.

[@confucheese]

This problem always just comes back to the division symbol. After grade 10 you will almost never have to deal with division symbols, and instead just use horizontal lines to properly denote fractions and the order of operations becomes objective.

[@gorak9000]

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

[@confucheese]

@@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.

[@ncmaothvez]

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.

[@confucheese]

@@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

[@gorak9000]

@@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

[@JanCiger]

It is not only North America, I have also been taught the multiplication like that - i.e. there is no distinction between the implied and normal multiplication.

[@EEVblog]

Which country?

[@ErikN1]

Same

[@okaro6595]

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.

[@ct1igu401]

Hi, Great video. According to Wikipedia, "Different calculators follow different orders of operations. Many simple calculators without a stack implement chain input working left to right without any priority given to different operators, for example typing 1 + 2 × 3 yields 9, while more sophisticated calculators will use a more standard priority, for example typing 1 + 2 × 3 yields 7." I have 6 calculators and tried out both 6/2(2+1) and 1+2*3, here are the results: HP48g+ 9 and 7 respectively but in equation writer mode it gives 1 and 7 HP50g 9 and 7 equation writer mode it gives 1 and 7 HP35s 9 and 7 Sharp EL-W506 1 and 7 TI 36xPro 9 and 7 Android app. True Calc 9 and 7 Only the Sharp and 2 HPs in eqw mode were correct for the 6/2(2+1) calculation. Generally I use RPN on the HPs which avoids these problems. Or a paper and pencil😀

[@phillipsusi1791]

I don't understand why this is a thing these days... I was in highschool in the '90s in the US and I was taught that a dot product or implied multiply had the higher precedence. And in those days it was usually on paper; calculators that could do this were only just coming out towards the end of that decade.

[@thomasives7560]

That result may be the "new math", but I can verify that in the 1980s American schools taught PEMDAS [Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)]. Helping my kids with their elementary maths homework during the 90s-00s enlightened me on how far public education has fallen over the years. Anyhow, once one gets into 'professional' mathematics for engineering, they'll have to re-learn PEMDAS, or bridges and buildings will be falling down all over the place - maybe they already are? Cheers!

[@darranrowe174]

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.

[@gorak9000]

"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.

[@MLeoDaalder]

I initially thought it was going to be about the old typographic convention. The ➗ symbol used to denote that everything left of it was over everything right of it, and / was the normal divide with the same priority as multiplication.

[@f.f.s.d.o.a.7294]

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

[@gandalf94013]

Hah. In fact those symbols meaning the opposite way.

[@MLeoDaalder]

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