Why Calculators Lie: Can You Solve This Simple Math Problem?
Vložit
- čas přidán 9. 06. 2024
- Dave takes you on a journey beginning with an incredibly simple math problem that many people - and even calculators - get wrong. For info on my book on Asperger's, Autism, and ASD, visit: amzn.to/3DNfQao
- Věda a technologie
The problem with things like this is: you write the equation, so don't write ambiguous equations. Write the expression the way that reflects what you want to happen.
This right here. Math on its own as just numbers doesn't mean much. If you know the units you are solving, the numbers are just a quantity - and you know what you want operations you want to perform on them.
It feels like Dave has worked on everything
What didn't he work on?
Windows 11 🙂
maybe that's why it feels terrible
@@ProtekNickzThe basic code base is the essentially the same. Fix some bugs, add some more. Make everything look a little different to make people believe it's new.
Flat Earthers (aka Flatheads) still think it equals 1. lol
Evaluate 1÷2π. Does it equal 1.5708 or 0.1592? In a textbook formula it would be 1/2π. I was taught (in the '50's) that multiplication by juxtaposition took precedence over explicit operators. This included x(value) without an intervening operator.
Here's my problem with Dave's video and explanation, and PEMDAS in general. Using PEMDAS, and Dave's rules and logic:
1 / 2(pi) = 0.1592 but...
1 / 2(1 + 2.14) = 1.5708 therefore...
1 / 2(3.14) = 1.5708
The answer can't change because you replace a variable with literal numbers. That's the whole point of a variable. It represents a literal number that you may or may not know. Using PEJMDAS, the answer to all of the above is 0.1592, which is the correct answer if you were to use that equation in the real world. Therefore PEMDAS is wrong, or at least incomplete, and therefore Dave is wrong on this one.
Formula such as 6/2x(2+1) are exactly why when writing code, or elsewhere, I always included brackets to make the calculation order very clear. It solved a lot of potential issues when the not-so mathematically literate came across them and made them easier to tweak later too.
On the other hand, I once programmed in a computer language where the statement "a = a + 1" produced a different result to "a = 1 + a". That took some effort to get to the bottom of why, someone else's code, just was not working as expected. If the starting value of a was 10, the first completed with a having a value of 11. However, the second statement left a completed with the value of 2.
What language was that? That's a super interesting thing - was it some sort of weird typecasting issue?
When I write code, I use a lot of parenthesis to make both math and casts clear!
What? 1+10 != 10+1?
It looks like it is treating the + operator as an "increment by 1" function that doesn't accept any parameters after the operator.
@@DavesGarage
I used to write in LISP...
(Lost In Stupid Parentheses)
For AutoCad applications.
i giggled furiously at the 80085 on the calculator.
your videos are always worth the time to watch, great job
I thought there would be more ppl commenting on that.
Then does that mean that the 8085 microprocessor is BOBS?
@@rasszo8729 It is easier to see on a seven segment display. Gotta be old school!
Me too xDDDDD That was a great detail
I didn't notice that.
When I did maths at uni many years ago, we hardly ever used the divide by (÷) symbol. It was long enough ago that everything was hand written. Instead we would draw a long horizontal line, with all the other bits either above or below the line. The convention then was you first evaluated the bits above and below the line separately, then performed the division. I'm guessing part of the problem has been in moving to on-screen text, some have assumed the ÷ symbol did the same job, with everything on the left assumed to be above the line and everything on the right assumed to be below the line.
I *always* put that stuff in a bunch of sets of parentheses.
Yes, that's the right way to do it. PEMDAS or BODAS or whatever is just a short hand for (English-speaking) school kids that ends up doing more harm than good.
The actual rule is "don't be ambiguous".
And that's why you test calculators to check how they process ambiguous situations. They are neither right or wrong if they show 1 or 9. They just interpret in different ways
Those two and a third are what we learn here in Denmark. We learn how to write it in three different ways in primaery school, and told to use what we personally find the easiest to use. And nobody learn PEMDAS. It is way too complicated. We learn reduction. Meaning 6÷2(2+1) becomes 3x3. Way more simple.
THIS
Not a problem. Real issue is dave never learned math past grade school and is adding a operation that doesnt exist in between the number of a unknown unit and the unit itself.
Aka he doesnt know how to solve 6:2n where n=3 and instead writes himself some unrelated problem of 6/2*3
Wiki:In some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n.[1] For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division,[26] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.
And if they're specifying it's because they're not following the standard convention that everyone would assume they were using if they didn't.
And if they're doing something non-standard Without specifying, then they are doing their readers/students a major disservice.
@@laurencefraser It says that standards and agreements are different. PEMDAS is not absolute
PEMDAS is for little kids... and Americans.
They are specifying it because some people are from the US, and need it explicitly spelled out for them@@laurencefraser
And if they are simply clarifying, or simply doing it that way without comment, you can't conclude anything. @@laurencefraser
The way I was thought about it is to use a rule of implied multiplication (juxtaposition). So if there is no symbol between the number and perentheses, it means you need to multiply what is inside of it by that number first before doing any other calculations.
I was first taught this too, but it becomes even more important for higher maths. To me it's simple to see this as it's written. If they wanted it the other way they would have written the multiplication sign. I've always seen it as distributive multiplication, and that will never change; that's what it is.
Yeah PEJMDAS is the more accurate mnemonic but the US for some reason doesn't seem to teach juxtapose first... so for most of the world the 1 answer is correct.
Agreed. Otherwise, x/2x would not be one half (for non-zero x), but x/2*x = half of x squared instead!
@@leesaudan they have never come to this point they are solving elementary school math. to confuse more than half the population that did not take higher math classes so than they can argue. If the teacher had taught well we should have seen such video.
and that's clerly wrong. here is a nice expression for you (2+1)3^3
what is it's value?
you won't find a single calculator that solves it as (2+1)3^3=(3)3^3=9^3=729. either you can't imput the expression or they solve it as (2+1)3^3=(3)3^3=(3)27=81.
that breaks the rule you were taught. you didn't multiply what is inside of the parenthesis by the 3 before doing any other calculations.
that rule works fine when you are using fractions and superscript to represent powers because the size and position of the numbers removes any possible confusion.
when you are forced to use a single line without superscript or subscript, the rule breaks.
multiplication is commutative. having a parenthesis (implicit multiplication) before or after gives the same result. which means (2+1)3^3 and 3^3(2+1) are equal.using superscript makes it easier to see since it becomes (2+1)3³ or 3³(2+1)
Actually Dave, the Windows 1.0 calculator was entirely correct. It looks like you hit add instead of multiply, so you entered 6/2+1 which does in fact come out to 4 :)
(you can rewatch the footage and you'll see that the + was pressed but * never was)
You are right
What's the time
Clearly Dave should've typed at 75%, then his answer might've been 100%
He should rewatch it at 75% speed 😎
@@toolbaggersFor some reason, adding the word "speed" makes the joke less funny.
I'm so glad I sent you that Mach 10 board... it's so nice to see it actually running again after sitting in a box for 20+ years.
I still have the TI SR-52 I went off to the Naval Academy with.
There were somewhat similar products on the market from other companies. I remember I went into SoftWarehouse to buy one and they told me they had stopped selling their particular model "because it was a crappy product".
Implicit multiplication normally involves multiplying the value outside the bracket to each term inside the brackets
6/2(2+1) => 6/(4+2)
Then the brackets
6/(4+2) => 6/6
Then finally the divide
6/6 = 1
The problem is that PEMDAS does not truly handle implicit multiplication/
You didn't use PEMDAS. Implicit multiplication has no special meaning using PEMDAS. Why did you evaluate the M before the D when PEMDAS says you should have gone from left to right and so you should have done the D before the M?
PEMDAS does handle implicit multiplication, it quite intentionally treats it the same as explicit multiplication. That's because it is trying to be logically consistent.
No it doesn't. It means you resolve what is inside the parenthesis down to a single value first before proceeding outside the parenthesis. So hierarchically.
6/2(2+1) becomes 6/(2(2+1)) for process order to 6/(2x3) to 6/6 ending in the same result but different order of operations. At least how it was taught to me.
I intuitively expect the first 2 to be grouped with the parenthesis group
If we ever see this, it is a syntax error, it is ambiguous because we don't know what the author was intending.
I remember BODMAS.
Dave....you going through Windows 1, 2, 3.1 were a pure nostalgia trip for me. I'm only 36, but 3.1 was the first OS I used. It's pure emotion to see these old operating systems in use!
So you were like 4?
@@ahabsbane It would have been between Windows 3.1's release and Windows 95, probably closer to 6 or 7.
@@oisiaa I figured, I'm not much older than you, just having a laugh at our age. I remember installing the "turbo" upgrade chip for the 386 with my Gramps. If I'm being honest he's probably the reason I'm in the field I am today. He was always trying out the newest tech before the rest of the family, computers, GPS, cell phones, he was quite the techie old man!✌️
honestly same, i will be 40 soon thats why i fallow Dave, the man coded a large part of my childhood.
I'm the same age and the same first OS. Or would it be DOS? I mean the two were kind of inseparable lol
The ambiguity is due to reducing a "chalk board" style problem that involves fractions (with numerator / denominator) down to a single line. The "slash" symbol could indicate that the question is 6 over 2(2+1), in which case the answer is 1. Or it could be interpreted as 6 divided by 2, times (2+1), answer 9. The question is deliberately imprecise.
It's not about the slash symbol. Some people intuitively treat multiplication by juxtaposition as having a higher precedence. Some people even claim that's how they were taught.
6 over all would be (6)/(all).
@JonathanGray89 if it were 2x instead of 2(1+2) then that is absolutely how most people do juxtaposed multiplication. 2x always means double x, no matter what comes before or after!
@@JonathanGray89
> Some people even claim that's how they were taught.
Because we were. Incidentally, it's also how mathematicians publishing mathematical papers in mathematical journals CONSISTENTLY interpret the order of operations.
@@cassiee.3969 I hate that CZcams automatically censors my comments and I can only find out with in-cognito mode. I have to rewrite them and paraphrase just to get my point across. Anyway, you're not only wrong, but you're also dishonest. I can link a video right now of an actual mathematician saying the answer is 9. The fact of the matter is mathematicians are overwhelmingly aware of this ambiguity and simply avoid it. At the very least that's what some mathematicians are claiming. Don't make assumptions about how mathematicians interpret the order of operations just because you already think you're right.
It should be noted that google is “inserting” the operator because its walking the expression tree that it builds and spitting out what the tree holds. The insertion is happening at the parse and tokenize time, probably when it sees an opening bracket following a number.
From Wikipedia: "Multiplication denoted by juxtaposition (also known as implied multiplication) creates a visual unit and has higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n."
What on earth are they teaching those kids in North American schools?
(And how on earth has this come to poison the software that the rest of us have to use?)
The calculator isn't necessarily wrong. It just uses a slightly different version of PEMDAS. The 2 multiplying the parenthesis is treated as what can be called an "implied multiplication" (because it has no multiplication symbol) which is supposed to be resolved before other multiplication and divisions.
Part of why I prefer to abuse parenthesis to avoid this kind of funky stuff.
Correct. The reason that people get 1 for an answer is 2(2+1) would be using implicit multiplication (or multiplication by juxtaposition) which has a higher precedence than multiplication/division.
@@mattgaia Which in turn is based on basic axioms of sets and substitution which anyone, especially dave, should know.
You can substitute any part of the equation with unknown element (6:2n, n:6, 6:2(n+1), 6:2(2+n) and any way you can resolve the problem without outright just doing something you arent allowed to results in 1.
Brackets and operation symbols exist for a reason.
I think that Dave is just magicking away the parentheses in this example. Ask yourself, does 2(2+1) = (4+2)? If it does, then what is the solution to 6÷(4+2)?
Yeah it's not wrong, both are valid. Almost every academic paper that I've seen will utilize implied multiplication as higher precedence. 1/2x for example resolves to 1/(2x). How many times I've seen stuff like pV/RT = n in school, or 1/2pi. It's obvious that they are linked. Just because the calculators shown are being forced into evaluating it strictly by a convention developed by educational institutions doesn't mean that the academic world uses it.
@@wwusirius How many academic papers are you reading that use inline division? Showing division unambiguously above or below the dividing bar is overwhelmingly the preferred presentation.
"I'd cut him some slack, because (A) He'd be a 108 years old, and (B) he passed away a long time ago." Logical as always. I Loved this episode, which for me does clarify using brackets in equations. Unfortunately my old Sharpe scientific calculator no longer works, nor does my original TI with plasma display.
Ooooh! I love plasma displays.
My Sharpe always uses parentheses.
@@DavesGarageThere is not a universally recognized convention for evaluating this expression. It is technically ambiguous as to what the answer is in the video.
6/2(2 + 1) = 6/[2(2 + 1)] = 1 is juxtaposed [and implicit].
6/2 (2 + 1) = (6/2) (2 + 1) = 9 is implicit but not juxtaposed.
6/2 × (2 + 1) = (6/2) (2 + 1) = 9 is explicit multiplication.
These questions are always written to be ambiguous to make people have long and pointless arguments about it.
@@John-McAfeeSorry, but you’re wrong. There is only one situation where you have implied brackets, and that is when you have a numerator and a denominator expressed as a fraction. You evaluate them both separately, then divide.
However, in this case, the fact that a 2 is written next to the parentheses without a multiplication sign between them is accepted shorthand for multiplying. The correct answer according to the mathematical rules we invented, and globally follow, is 9. End of.
"and (C), what is he going to do?".
I own three Casio calculators (fx82, fx991ES, fx991DEX), (40, 10, 5 years old respectively)
the two older ones outputs 9 for both "6÷2x(2+1)" and "6÷2(2+1)", the newer one outputs 9 for the first expression as well, but for the second one its 1, and my input got forcibly changed to "6÷(2(2+1))".
Thank you for mentioning the change in math rules as even I was under the impression that a implecit multiplication also implicates brackets around it. It never got directly shown or even mentioned during my algebra classes in school.
I agree with your calculator, the answer is one. Any number attached to parentheses should be completely solved and used as a single number
off line, it also said the earth was flat....
@@Neolantis So your response to someone following global scientific convention is to accuse them of being a flat earther? You do realize that globally most of the world uses the convention used by that calculator and only the USA seems to interpret it the other way. Which is why Casio makes US versions of the same calculators.
I've always seen this problem as a "notation" problem coming as a consequence of "lazy" writing. Is 6÷2(2+1) defining 6 ÷ [2(2+1)] ("missing the brackets [ ] and specify that the parenthesis operation is in the divisor" to say it somehow ) or 6÷2 x (2+1) (missing a multiplier sign)? Because of ambiguous writing, this problem can be misinterpreted.
Its very easy. ÷ is not a sign of division. It means ratio of 2 things. Stuff left and right of this sign.
But its easy to show where dave makes his mistake.
6÷2(2+1) , allright. Lets get rid of those brackets.
2(2+1) =( 2×2 ) + (2×1) = 6
Unless you want to absolutely break mathematics, the correct answer is 1...
Implicit operations was a feature in calculators that got removed. So his old calculator is actualy the one that have the correct result.
It seems like it may be a misinterpretation of PEMDAS. Instead of "Parenthesis" it should be "Products"(denoting groups) which aren't complete so long as the "Parenthesis" remains. The trickery is revealed when you write the implied product of your last expression: "6÷2 x 1(2+1)". Products don't exist by themselves. They are a factorized grouping of terms. The factor must be distributed in order to complete the first operation in PEMDAS. Your first expression could be rewritten as 6÷1(4+2) because "one group of 4 and 2" is the same thing as "two groups of 2 and 1"
To me this is the failure in reasoning/notation.
@@nagyandras8857 Everyone actually in mathematics tells you this has two answers. You over simplify. This equation is missing information. That missing information creates two DIFFERENT equations. Thats it. Its that simple. There is two correct answers depending on what you decide that missing information is. PEDMAS does NOT contain any rules for assuming missing information either. According to pedmas this is an invalid equation.
The problem is how you view 6÷2(2+1). It actually has been solved by mathematicians both ways, and both are correct depending on the paradigm. If you view it as 6÷2 x n then you are correct, but if you view it as 6÷2n then you are wrong. In most math classes it is expected that you see it as the latter. If we use the standard way of writing it on a computer then it becomes more clear, 6/2n implies it is a fraction 6 over 2n and you would never expect it to be 6÷2×n.
@@Wylie288 everyone with any meaningfull math education will Tell, that first order expressions can only have 1 or 0 solutions.
There are no 2 solutions.
Dude, I love your random stuff… I have ADHD really bad, which makes reading nearly impossible… People like you, doing things like this, is sooo helpful… Adult education is key to building our world… Educated people can be very intimidating… it’s a special talent, to bridge that…
No, you don't "have ADHD really bad". You have something that's screwing your brain up which you need to be solving, instead of taking on a meme diagnosis as an identity.
@@Acetyl53 you mean well… Do you wanna learn about what you have wrong?
It’s more than “getting distracted”…
I don’t think “in words”… for me to “read,” I have to “visualize” it… I can do it. It just takes forever. Now, live in a world, where everyone else reads something, and then waits for you, to take 3 times longer, to read… That stress causes a “fight or flight” response, in people… When I’m trying to read, my subconscious is looking for excuses to avoid continuing reading… After a while, you naturally avoid those situations.
@@DairyAir I understand this, however you have to realize I more than mean well, I'm right. You have to run around in this labyrinth of compensation because your brain is being disrupted. Get off whatever drugs they have you on in whatever course is encessary, get on a multivitamin, and try cutting out rgains for a while. Look for evidence of mold in your environment. Lastly avoid wireless devices.
You can do it. Don't accept the FALSE soothing of submitting to the disabed identity.
@@DairyAir Sounds more like dyslexia than ADHD but go get a proper diagnosis from a professional. Can help a lot.
@@DairyAirAbsolutely, Tons and tons of people dont "believe" in mental illness etc, typically because they cant "see" it. If they saw a broken leg, Yeah they would believe it was broken. But they cant see the difference in pathways that certain brains take or lack of etc And they also wont spend the time to understand. So they just stay stupid and arrogant. Thats ok though.i have been diagnosed add and bipolar type 2, my brain is very different and im a natural with music and music production, therefore its what i have always done. It just makes sense to me. I wish i could program, and i have dabbled a good bit, But holy hell its a mouthful of learning. It needs to be practiced like trying to be advanced on an instrument. Blah blah blah ✌️
My calculator I’ve had since high school, the HP-33S is an RPN calculator and I still love it even as a computer scientist! My chemistry teacher recommended it back in high school and said once you got used to it, you wouldn’t want to use any other calculator and he was kinda right
It annoys me that the "RPN" calculator on MacOS handles the stack wrong. I should write a correct one that takes key inputs.
I was using an HP25C in 1978. Then went on to use an HP67 (with the mag stripe reader) then the HP41C.
RPN makes these equations trivial.
I began using an HP RPN calculator about 45 years ago, and still do. I find it much more intuitive. I still have a couple actual calculators which I love, but also have an HP-41 app on my iPad which I use all the time. I 'm so used to RPN that I have trouble using a conventional calculator for anything with more than a couple of numbers.
1. It can mean whatever you want it to mean it's just notation and notation is not math
2. They teach PEMDAS and such to children so people think it's axiomatic or something, it's nothing of the sort.
3. In advanced math, probably the most common convention, and it's only that, is that adjacency is higher than infix. Hence 1/3x is not (1/3)x but rather 1/(3x).
=== from wikipedia (and this jives with my experience)
Mixed division and multiplication
In some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n.[1] For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division,[21] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[b]
This ambiguity is often exploited in internet memes such as "8÷2(2+2)", for which there are two conflicting interpretations: 8÷[2(2+2)] = 1 and [8÷2](2+2) = 16.[22] The expression "6÷2(1+2)" also gained notoriety in the exact same manner, with the two interpretations resulting in the answers 1 and 9.[23]
Ambiguity can also be caused by the use of the slash symbol, '/', for division. The Physical Review submission instructions suggest to avoid expressions of the form a/b/c; ambiguity can be avoided by instead writing (a/b)/c or a/(b/c).[21] [24]
===
It turns out to be very convenient that ab/cd is not the same as abd/c
But a wise person will just avoid this stuff.
The only "real" answer is that it's ambiguous. The proof of which is that people, even clever people, disagree. Even those that wrote calculators.
But it wouldn't be hard for me to find a page in one of my calculus books that shows the more common way to interpret this after grade school is to boost adjacency. Still it is whatever you want it to be. It's not axiomatic as some people like to say. None of the axioms deal with notation. RPN would work just as well. Another way to avoid this problem is to never use infix division but always make fractions. But that doesn't work so well for writing single lines of text.
Anyway, it's whatever.
I guess another interesting thought is this. The purpose of order of operations is to make it so that you can write more expressions with less parens. Hence multiplication goes first because that's more convenient and for no other reason. Adjacency being stronger than infix is also more convenient because it gives you a few more ways to avoid parens. But none of this is law and indeed even advanced math publications do not universally agree.
Exactly. PEMDAS works for elementary school math before algebra and more advanced formula notations are taught, but falls short in real world applied mathematics where juxtaposition and many other mathematical notations are just not covered by PEMDAS. (We also learn about seven colors in the rainbow and the Bohr model of the atom - oversimplifications that we later learn to set aside. )
Exactly this. I recall having a bunch of lessons in school class of Algebra where we were exercising in interpretation of division notations. The solution is straight-forward: default to Landau-Lifshitz-Feynman, and if the expression in the video has to be interpreted to yield 9 not 1, simply rewrite it, move the part in parentheses to the dividend, to avoid misinterpretation. In this (Landau-Lifshitz-Feynman notation) case, reading one-liners like ab/cd is unambiguous
@@ricomariani There is another reason why adjacency is generally treated as stronger: consistency.
In mathematical notation, we often use expressions such as e.g. "2x² + 3y + z" or some such; so, either nx is supposed to be treated the same as (n*x) at all times, or it is inconsistent. Therefore, mathematicians (rather than math afficionados) tend to prefer this analytical approach.
Still, the reality is that it's deemed as ambiguous and therefore preferably avoided altogether (the way e.g. linguists avoid using archaic words which are typically misused by the general public, unlike journalists who insist on butchering them).
Great video, but counterargument:
Multiplication by juxtaposition or implied multiplication may be interpreted as having higher precedence as division.
So your old calculator is also right (as well as my expensive Casio calculator I bought a year ago)
The real takeaway is: Just use parentheses
I say get rid of the quotient symbol and only use fractions, even for kids grade math
This, as shown by the Sharp calculator, is the traditional version, known as PEJMDAS in response to the misdescription of PEMDAS. The background is documented in some detail in a video titled "The Problem with PEMDAS: Why Calculators Disagree" on the channel "The How and Why of Mathematics". The short of it is, some American teachers (whose experience came from recent and incomplete textbooks, not reading applied maths) decided to claim their unusual interpretation was "the correct way", and convinced Casio to release some calculators with their version, causing severe confusion ever since. Most textbooks espousing that strict PEMDAS without regard for juxtaposition don't even follow it themselves.
It's called PEJMDAS and is the only way in science and nearly everythink outside USA.
@joweraDE, yes and also knowing what's the purpose of the calculation, because (6/2)(3*1) will be different than 6/(2(3*1)). The math is not a problem, but what do you want to do with it and how do you express it.
yup, but in the end, as said by some other it's more the way the math problem is written that is the problem, it's ambiguous ...
Just as a point of information, my Casio fx-85gt gives the answer 1 if the 'x' is omitted, 9 if included. The user guide explicitly says it will do this, it has a precedence table and 'multiplication where multiplication sign is omitted' is listed as higher priority than other multiplication and division, so the design is deliberate, not a software bug.
Yes, you're right.
Depends on the scientific calculator but here are some that give one or the other:
These give 1:
Casio FX 83GTX, Casio FX 85GT Plus,
Casio 991ES Plus, Casio 991MS,
Casio FX 570MS, Casio 9860GII, Sharp EL-546X, Sharp EL-520X,
TI 82, TI 85
These give 9:
Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES,
Casio 570ES, TI 86, TI 83 Plus,
TI 84 Plus, TI 30X, TI 89.
Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation which implies grouping (1). Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation (1). TI later changed to the programming/literal interpretation (9) but when I asked them were unable to find the reason why.
Some commenters have said it was pressure form American teachers but I've no confirmation of that.
So, yes, features not bugs.
9:05 “for operators of equivalent precedence you proceed from left to right”
Except exponentiation, which is right-to-left.
I feel the problem with all these kinds of problems stems from the division operator being written on a single line, as opposed to how it’s written in advanced math classes, because then the divisor is cleanly what is below the line.
@@ThomasVWormalso it depends what's your background, in some places the implicit multiplication is considered as having implicit parentheses.
Like if you go 6/2x it's implied that it's 6/(2*x) and if you actually want (6/2)x you'd just write 6x/2
@@ThomasVWorm There is a bit of an issue in that / also is a single-line fractional notation, the less ambiguous division operator being ÷ (which low and behold represents a fraction too, but at least it wouldn't make for any fractional notation). Therefore, the math problem as given in the beginning of this video is correct, but with how it's noted usually and even later in the video, as 6/2(2+1), it's deemed truly ambiguous.
Besides that different priority systems still in use internationally could still make for different solutions.
@@ThomasVWorm The world has never worked off of one clear set of rules.
Different industries have different standards because the context is different.
Why try to shove everyone in one box?
@@ThomasVWorm The spaces would probably indicate it's not a fraction, but in a multi-line notation there is nothing wrong having multiple levels. The kind of line could make a difference though. As a single line there's no difference between 1/2/3 or 1÷2÷3 or 1/2÷3, but it could also be 1÷2/3 in which case one would write it like that or as 1/(2/3).
The great thing about standards is that there are so many of them.
Try using:
6/2A
You say it is: (6/2)*A
But in formulas it is usually accepted as 6/(2A)
Or do you say a () is handled different from a letter?
In my HP Prime calculator it is not a issue. It do not use RPN, but shows divide as true horizontal line with factors above and below.
If you're using variables with inline division, you're basically a criminal.
But then you'd have to stick with the rules: your monomial there is 3a, not 3a^-1
Yep, some people will still say it's (6/2)*A, declining any other opinion. May I ask where are you from, which country?
@@evgen5647 From Denmark, Europe.
I have seen this problem before, it has basically existed as long as calculators with implied multiplication. In books 3/2A is usually taken as 3/(2*A), but this is not much of an issue anymore because modern typesetting can use a true horizontal line. Today some people want a strict PEMDAS rule, other accept a modified rule where implied multiplication take precedent.
@@henrikjensen3278yep, Europe. You see, USA are freakingly fanatic about PEMDAS because they were tought this way. India as well I believe (which is a bit strange taking into account that India was British colony for some time).
@@evgen5647even in American they don't use PEMDAS in scientific papers. Google the America Physical Society style guide. on page 21 it shows research papers should have multiplication before division
arithmetic in English does not have precise precedence rules ... so use parentheses to clarify
(6/2)*(2+1) = 3*3 = 9 OR 6/(2*(2+1)) = 6/(2*3) = 6/6 = 1
or we all could adopt the precedence rules of APL (a programming language, Kenneth E Iverson, 1962) and process all operations (strictly) right to left. 6÷2*2+1 = 6÷2*3 = 6÷6 = 1 ☺
To be honest, my first expectation was the floating point problem, wherein 1/3 * 3 1.
I started with a solution of 1 in your example, but your logic changed my mind. Left to right.
A long, long time ago in a Pascal programming class the assignment was to write a program that could evaluate expressions, including exponents & parentheses, as well as input errors such as entering a letter instead of a number. The assignment was designed to teach us about the use of stacks. Basically I pushed operands on one stack, and operators on the other until I reached the end of the expression, at which point I would pop two operands and one operator, then evaluate. Of course an open parenthesis branched into a separate routine that evaluated whatever was in the parentheses, before returning the result.
I was really quite proud of it at the time. Not sure if I even have the source code, now.
this is why i am in the habit of surrounding everything in parentheses to make sure it calculates the order im expecting
This is my philosophy in C, whether the expression is arithmetic or logical. What's the strictly defined order of operations? Who cares. Don't leave it ambiguous and it doesn't matter.
@@nickwallette6201 Depends. An expression like x + 2 * y is perfectly clear on it's own. Though I'd use whitespace there: x + 2*y. I've heard people argue for adding parentheses because others might not know the order, but I think you should be expected to know that as a programmer. It is never ambiguous, regardless of how you write it. Parentheses can help readability by visually grouping things together though.
@@jbird4478 That last bit is my entire point. :-) I know you _should_ know the order, and there _should_ be a single correct answer. And in the case of a simple arithmetic expression like that, sure. But... I still use parentheses out of habit, intentionally formed habit, to ensure that there's never a time (particularly more with logic expressions) that would be implementation-defined, or difficult for a reader (like myself, 5 years later) to parse. If nothing else, it just relieves your brain of having to sort through it. It's spelled out for you. Mistakes happen, and anything I can do to minimize the chance or severity of those -- it's a good thing. :-)
@@nickwallette6201 My point is that it is _never_ implementation-defined or ambiguous. The order of operations is fully defined by the standard. x || y && z will always parse as x || (y && z). That should not cause any confusion. The risk in always using parentheses is that it will cause you confusion if you encounter code from people (like me) who assume this is understood. I do use extra parentheses sometimes, but that's more in the same way as one uses whitespace; to give some visual structure that makes it easier to skim through the code.
@@jbird4478 I'm not going to argue about this. I recall at some point reading a text book on C that mentioned some combination of pointer dereferencing or something like that, that was evaluated differently on two different compilers. I can't back that up, because I don't remember what text it was, so feel free to take that with a grain of salt.
The point the author was making, and the point that _I'm_ making, is that _usually_ there is one single correct answer. And sometimes evaluating what that answer is, is enough to make somebody give up and move on. I don't care whether it is, is not, or "shouldn't be" ambiguous. If it's even one millisecond faster and thus easier for a human being to parse (x + y) * (a + c), then I'll write it that way, because at least on my keyboard, I have unlimited parens, and they're free. For trivial examples, it hardly matters. For more complicated examples, it matters more. So I do it everywhere, and never have to think about whether it would be helpful to add hints. They're just there. Feel free to ignore them if they're superfluous.
Not only did my dad indeed swear to an old HP calculator using postfix notation, he managed to get me hooked on the nicely unambiguous notation which also removes the need for parentheses. I even have an RPN calculator app on my phone today!
I have a RPN calculator app on my phone and I still have my HP 11C calculator too. I just wish I had bought a HP 16C at the time.
Try the Free42 or the updated Plus42 calculator - available on most platforms - which implements the HP-42S calculator.
Same here, HP41C emulator on Android. I wish I had never gotten rid of my actual HP41C. I still have an original HP-35. Slower than molasses on a winter day but when it was first marketed, it was the most amazing thing since sliced bread.
I liked the 'get off my soapbox' text on screen. I did recognize the language and did understand.
Thank you for including your humor! (gotta love the first step in solving something that is apparently not working: reboot, and if that doesn't work, second: turn off the power, wait and restart, ... and if that doesn't work how about reinstalling the component. Never bother just looking around for simple sources of the problem.)
i love this problem. they way it was explained to me by the professor that showed it to me is that the question is really asking what exactly the in line division symbol means and how it works with order of operations. and to my surprise, my professor told me there is NO CORRECT ANSWER! Instead he explained this is why we never use the inline division symbol anymore. it is effectively a non-standard symbol with some debate on how exactly it works (similar to the debate on if 0 is a natural number or not). and you can find text books that support both answers here. some say to always multiply before dividing when using inline division symbol, and some say to follow left to right always with parenthesis, then exponents, then multiply OR divide as read left to right, then add or subtract as read left to right.
and i've even found examples in a text book that show to convert the inline division symbol into a standard fraction notation before doing anything at all which is a weird third option.
all that said, i maintain that the correct answer to 6÷2(2+1) is false or invalid because ÷ is not a valid math symbol. if instead you write 6/2(2+1), then it is clearly just a matter of which calculator engine you use and if it does strict PEMDAS or left to right when you leave out parens. For what its worth, wolfram alpha will take 6/2 as an irrational number and do an implied multiplication which is the standard method. i was taught long ago that a lack of parens after the / symbol means that the / symbol applies to exactly the next term only (in this case a 2). or more specifically, that / only affects the next single term and never includes implied multiplications
That doesn't apply to implied multiplication, which takes precedence over multiplecation and devision. Someone else explained why in another comment. Basically 2(2+1) is the same as 2y and 2y or 2x, or whatever, which is always treated as one entity.
But the discrepancy arises from the brackets, not the division.
You see 2 * (2 + 1) ≠ 2(2 + 1)
2 * (2 + 1) = 2 * 3
2(2 + 1) = (4 + 2)
@@GordieGiino????
Brackets come first
2(2+1) = 2(3) = 2x3 = 6
@@GordieGiithere would only ever be a 4 if it was 2(2) + 1 = 4 + 1 = 5
@@Xnoob545 in Maths it's okay to go from 2(1+2) to (2+4) because you multiplied the individual parts by the outer part.
6
_________ = 1
2(2+1)
Long horizontal fraction bars do not respond well to PEMDAS. They're commonly used in scientific and mathematics journals and papers.
Applying PEMDAS mindlessly is not recommended.
Best to avoid ambiguous notation.
Long horizontal fraction bars leave no room for misinterpretation, and yet do not follow PEMDAS.
Yes, ...I can anticipate that some will mention invisible brackets, but why did I have to mention this point first?
That's the point, PEMDAS is silent on this important counter example. PEMDAS, by itself, is clearly defective.
Do not worship it.
You are correct and you beat me by 8 minutes. Funny we are the only two here to be the outliers.
Long bars work like this 6/(2(2+1)) they are essentially just hiding ( ) exacly like 2(4+1) hides 2x(4+1) so it still applies
@@Aderaen You must be pretty naïve if you actually believe that anyone would accept your flimsy explanation.
If you divide 6 and multiply 2 and 3 all at the exact same time, you get a prismatic mobius strip.
Sorry, Dave, but, after expanding the parentheses, I see "n / a(b + c)" as "n / (ab + ac)". It should be no different just because the terms have been substituted with their values. "n / a(b + c)" is not the same as "n / a * (b + c)".
Left to right isn't a law of maths. Implied multiplication is just as valid and is practically the only version used in academia in physics and maths. Treating the left of the ÷ as the numerator and the right of it as the denominator is completely valid and more intuitive.
The equation is deliberately imprecise to provoke discussion. It's why even well-educated mathematicians are disagreeing, why different calculators and tools produce different results and why there's still no clear answer even though the puzzle has been floating around for years. If you're asked to perform this calculation for anything more important than a Facebook survey, ask where the equation came from and clarify exactly what was intended. Either add parentheses, rearrange the terms, or format it such that all fractions are unambiguous numerator-over-denominator fractions.
9 or 1 are both valid answers based on interpretation.
"1" is also correct since implied multiplication has higher priority than regular (explicit) multiplication and division. So:
6/2(2+1)=1
6/2*(2+1)=9
That’s a rule in older TI calculators - not a mathematical rule.
I was about to say this too, not that I argue that the answer isn’t 9. But I learned at school that the 2 belongs to the brackets part because there is no mulitplier sign. So you complete the entire brackets part including the 2 first and then do the division. Like when the the entire 2(2+1) would be below the 6, under the division line.
@@stephanszarafinski9001 that would be true… if there were a division line. This is the division sign, so it can’t be made clear what is “under” it. And as Dave alludes to in the video, the division sign hasn’t been equivalent to the line since around 1915.
@@thatsunpossible312 The rule of multiplication by juxtaposition going before regular multiplication or division is used in most mathematics papers and university level math. Its mostly just there because as humans we want to be lazy and be able to write something like 2x/5y without adding the brackets around 5y. But yes, there isn't really one global way of doing mathematical notation, so this can be different between schools.
The videos from "The How and Why of Mathematics" on the problem with PEDMAS is a really good explanation of both the disagreement as well as how we got ourselves into this mess, I would highly recommend that video.
@@HroiG yes, this problem is deliberately misleading. Imagine if students were presented with the distance formula as d = vt + 1/2at^2
Academic papers don’t generally go for ambiguity 😁
I think the use of the expression "2(2+1)" (without an explicit operator after the first 2) could be taken to make the larger expression mean 6÷(2*(2+1)), which really is 1. Of course, you can't enter it into a calculator that way (with no explicit operator).
Yes the implied multiplication is the main issue. And that was surreptitiously avoided through the inability to enter that in to many of the examples
Suprisingly on non-US made calculators you generally CAN assume they'll handle Juxtapositions correctly as far as I know only TI and HP calculators don't work this way and even then some of theirs do actually understand juxtapositions you literally have to check the manual to be sure if they do or not each time you get one >.
Order of Operations = correct result
@@bluedark7724 only if you know the actual order and don't rely on PEDMAS as your order
I agree 6/2(2+1)=1 which is not the same as to 6/2*(2+1).
Implicit multiplication, has priority over division, the 2(2+1) is evaluated first. This has been standard practice in mathematics and engineering for decades.
Adding an explicit multiplication symbol, changes the expression, the priority and the result to 9.
I feel like an endangered species being able to see this for the first time and confidently have the right answer in a few seconds. And I don't do anything mathy for a living. Just went to school back when schools taught valid curriculum.
My youngest is in grade 4 and immediately knew how to do it correctly.
PEMDAS says nothing about omitted but "implied" operators.
The way the problem is written determines the result. Omitting the * between the 2 and the parenthesis is simple sophistry that confuses an otherwise simple operation. Decades ago it was taught that an expression like 2(2+1) has to be solved first, as it is part of the parenthesis group, then that result is divided into 6, equalling 1.
The expression: 6/2*(2+1) is altogether different, and the PEMDAS rules can then be applied.
My scientific calculator does exactly that... omitting the * results in the answer 1. Using the * gets 9.
It's confusing, and that's why RPN is better.
Indeed. For PEDMAS to be "right" it ignores basic arithmetic foundational corner stones such as the Distributive Law. Unfortunately things have to be dumbed down today because most everybody seems to have become dumber since the 1970s.
I'm not entirely sure, but in my mind the implict multiplication (the one where you can leave out "x"..) binds stonger than any operand. So my mind will make 6/(2*(1+2)) out of it. Simply because I'm used to working with formulas and have been taught to keep the variables in until the expression has been simplified as much as possible, before adding any number into it. And I'm used to doing that on paper so I always know what the dividend and what the divisor is.
Totally agree, 2(1+2) is not the same as 2*(1+2) the first notation means give me content of parenthesis twice the second regular PEMDAS multiplication
What's you're talking about is called PEJMDAS and it's a real thing (J for juxtaposition). Some modern calculators do use PEJMDAS and many allow you to switch between PEMDAS and PEJMDAS, so there is really no "right" answer here - it just depends on which order of operations you're using.
Meanwhile I put anything ambiguous into parentheses, so there is no room for an error.
Most code language linters will force you to do so anyway, because consistency is important.
@@ksarnelliYou don't have to add the J. This works with PEMDAS. The parenthesis takes precedence and cannot be eliminated the way he did in the video. He performed the addition inside the parenthesis and eliminated it without first performing the implied operation of the parenthesis, you cannot do that.
Why did he not do this years ago? Well, because back then he would have not gotten any traction from all the idiots in the Internet.
You're describing the distributive property
a(b+c) = ab + AC
Solving left to right as described in the video would break this law of mathematics
The expression is ambiguous, as implied multiplication by juxtaposition is often taught as having a higher precedence, and you will often find this rule followed in physics textbooks, for example. The multiplication by juxtaposition is implicitly showing grouping. That is, 6/2*(1+2) and 6/2(1+2) are communicating two different ideas.
This rule actually works better in the real world. For example, take this system of equations:
y = 6/2x
x = 2 + 1
I expect that most people attempting to solve this system of equations would give the value of y as 1.
Assuming that you insist PEMDAS should still be followed in this particular case, I would point out that:
6*x/2 and 6x/2 and 6/2x and 6/2*x would all be identical, but the only way to notate my intended meaning would be 6/(2*x) or 6/(2x). This is both confusing and inefficient.
It's worth mentioning that, while I am an idiot, there are numerous papers, statements from mathematical societies, and real world examples that would agree with me.
Part of the problem with the parentheses having a single number inside and the single number outside is that modern textbooks show this to be proper multiplication notation. I taught algebra and pre-algebra all of my career but that was post-millennium. I can’t exactly remember what textbook in the late 20th century taught in regard to this notational instance.
Appreciate your comments on the RPN calculator. Once I started using one, I never looked back. Most of the younger engineers don't use one anymore so when they grab mine, they can't figure out how it works! 😂😂😂
My calculator in college was the HP-29C, at the time the most popular non-RPN calculator was the TI-51. I don't remember if we ever checked what the TI did with the above expression, but I remember being able to enter complex expressions much faster than those who used the TI model.
b.t.w., both of the above calculators were programmable. The one advantage the TI offered is those programs could be saved (or purchased) on small magnetic cards. Of course my programs would not be lost when I turned off the power so I didn't need those cards, although I was limited to 99 steps.
I still have an old RPN calculator. I was surprised at how quickly I picked up on how to use a calculator without an equals key, and how easy it is to use once you know how. It also has a "program memory" which is just automatic button pushing, with each button push occupying one memory space (I think it has 64 total). It even has a "halt" feature which pauses a program so you can enter another number before continuing the program.
I love RPN.
@@melkiorwiseman5234 - Did a lot of programming on an HP-28S (which I still have). It's retired since I replaced it with an HP-50g. That's my day-to-day work horse now.
I always enjoyed the puzzled look on the face of the person I would lend my calculator to, right around the time they started to realize they couldn't find the "=" key!
In 1982 I saved up and bought a trs-80 pocket computer. That thing was awesome at school. Computers were still new enough that teachers had no clue that a pocket computer even existed, let alone the power of it. They just figured it was a fancy calculator. Which it was. However, you could also program it to do all kinds of things. Like not only calculate complex problems but to also show you each step in getting to the final answer... in case you needed to "show your work"...lol. I personally seen no problem with using it in this way since I had to have a complex understanding of how to do the problem in order to program the computer to do it for me. All the program did was speed up the process significantly. Which was especially helpful for homework since I held 2 part time jobs during HS.
That’s awesome! Knowing how to do the problem is one thing. Knowing how to program a computer to do the problem and output steps should just be an automatic A.
I did something similar with a programmable TI calculator during tests in my calculus class in the early '80s. I could enter an integral into the thing and it would approximate the solution using Simpson's rule. While it spent minutes calculating, I would solve the integral, and then compare my precise answer against the approximation returned by the calculator. If they were very close, I knew I had solved it correctly.
In college I bought me a HP48 calculator, I was a Physics Major with a math minor. The nice thing about the HP48 is you could program it. But since I didn't do any repetitive calculations, it was easier to do it by hand rather than program the calculator.
I did program one thing on it. I programmed a stopwatch on it. I was a Cub Scout leader and needed a stopwatch for an activity, didn't have a stopwatch so I programmed a $400 calculator to be the stopwatch.
One day I stumbled upon a Dave video. I never left since. Keep it up sir!
PEMDAS is merely a convention. A mathematical notation's purpose is to simplify comprehension. Ken Iverson recognized that PEMDAS fails at this because of the non-intuitive special cases required to make it work as demonstrated in this video. With Iverson's computational notation described in the book 'A Programming Language' published in 1962, he used simple right-to-left evaluation with optional parentheses (fewer than required by PEMDAS) when needed to improve clarity. In APL:
6÷2×(2+1)
1
So, the correct answer is 1 unless you are using PEMDAS.
If you wanted the expression to yield 9, in APL it would be:
(6÷2)×2+1
9
Common grammar school symbols for multiplication and division also improve comprehension.
And of course, not only engineers loved RPN calculators. I had a spreadsheet application running on an HP3000 that used RPN instead of infix for formulas.
I remember being exposed to RPN (reverse Polish notation) in the 70s when I first picked up an HP calculator. It took about 30 seconds before I heard angels singing and I realized that I had the only rational system for performing calculations on the fly. You characterized RPN’s stack-based entry system as forcing the user to “ translate” computations. For me it was simply the way I visualized them. Now ~50 years later with a PhD in applied math and using Mathematica as my primacy programming environment, I continue to use RPN calculators and view those using so-called CAS (computer algebra systems) calculators with a combination of pity and horror.
Ok, we got it. You are so much better than us.
yeah.....God I hate TI calculators. My favorite was the HP55.....but I loved my 25C also and it was a lot cheaper....on my laptop and desktop I have a 55 emulator...my phone has the 25C
I was first given an RPN calculator for a competition in high school. I didn't do so well in the competition because I couldn't thoroughly learn RPN in time, but I came back to it later and never went back to algebraic notation. Algebraic notation is obviously more like how we'd write a problem, but I think RPN is more like how we'd do it in our heads.
(4+5)/(1+2)
"Take 4. Add 5. Take 1. Add 2. Divide the last two results."
If you put the verb after the noun like you're speaking Japanese, you get almost exactly what you'd type on an RPN calculator.
4 [enter] 5 + 1 [enter] 2 + /
This can easily turn into a Linux vs. Windows debate, but really RPN is objectively superior, like Betamax. Shame it did not become the standard.
@@rkadowns Yup. Sony really screwed up by selling VHS to JVC before Betamax was ready to launch. (to recoup their R&D money) It's amazing what 6 months can do when the public and content publishers are chomping at the bit.
I still do use an RPN calculator!
But I was a late developer, I made the switch about five years ago. I now find it more difficult to use an infix calculator, because my brain's switched to entering expression as postfix.
Two most used are a Swiss Micros DM42 and a WP-34S, but I also regularly use an HP15.
Now do a video on the HP16 programmer's calculator, which at one time was a must have especially for the well-healed assembly language programmer.
I'm 76 and have been using RPN on HP calculators since the 1970s. I too have trouble evaluating complex expressions without using RPN.
First thing I do with each new Android phone is install the RPN calculator.
Me too. Been on the HP48G since ~1995. It also has a clone on the android store. @ianjlilly which android RPN do you use?
I still have and use my HP 16C.
After years using RPN every single day, every time I have to use an algebraic mode calculator I trip and stumble like I'm drunk for a few tries. Then I start cursing.
Some would say that you are doing math "wrong" because it is not PEMDOS.
6/2(1+2)=1. An implied multiplication *always* goes before any explicit operator. 1/2π = 1/(2π), for instance. That's the standard in mathematics, science and engineering since centuries back.
Agreed. A simple google of "Multiplication by Juxtaposition" will find thousands of references explaining that implied multiplication is treated as the same precedence as brackets
The problem is that that is up to interpretation due to the "standard" used. Even PEMDAS is being interpreted differently depending on the calculator you use.
@@forkless How? The syntactical conventions has been settled since the 1800s... just look at older scientific texts!
I studied mathematics and science in the 1980s and 90s. Never heard of "pemdas", "pejmdas" or similar... until CZcams...
The fact that some calculators give the wrong result must be due to these strange american attempts to change the standard. Perhaps sloppy programming in some cases.
RPN is what I prefer. I still use my HP48 GX just about every day. Great video!
Last example is basically how i always think about these, because here you know that dividing by 2 is the same as multiplying by 1/2 (and there is nothing that tells you the parenthesis is inside the division) so you end up having three multiplications in a row, which makes it easier to understand when solving left to right.
In single line notation, if multiplication is used without a sign, then it is the relation of a coefficient and not just simply multiplication. Coefficient relationships cannot be separated without equivalence.
f(x) is what those parenthesies are, so you HAVE to clear the f(x) first.
So 1/2x is exactly the same as x/2, right? So why do bother ever writing 1/2x? Why should we have to add unnecessary parentheses to write the reciprocal of 2x?
6 /(4x+2) where x = 1 is 1, if you do the maths that equation is (funnily enough) is equal to 6/2(2x+1), which you're saying is now equal to 9
I know people have said this a lot of times, but there is something called juxtaposition. If two things are written together without a multiplication sign, I read them as one thing, such as in 6/2x, and I also apply that to when the factor is written next to parentheses, such as in 6/2(2+1). The thing is that the way math is done is invented by humans, so humans can have preferences on how it's done, and unfortunately, juxtaposition is a subject where people are very divided (no pun intended) on the use of it. It's okay if you think my way of reading it is wrong, but when writing questions, you need to be aware that people can interpret it in different ways, and that's why you shouldn't write the expression like 6/2(2+1). Instead, write it as either 6/(2(2+1)) or (6/2)(2+1), or simply write it as a fraction if doing it on paper. If I want something to be read as being multiplied with a fraction, I usually write it in the numerator, not after the denominator, so I can avoid confusion. So that would be 6(2+1)/2. So basically, when writing expressions like that, you need to be aware that PEMDAS isn't the only way to interpret expressions, and there are ways to write them so everyone gets the same answer. And of course, if you don't like juxtaposition being read first, just write out a multiplication sign. It doesn't take that much effort, but it prevents a lot of confusion.
Dave's comment about maths not being a good place to have your own rules is very true - but there is enough evidence of juxtaposition taking precedence of normal multiplication that you can't be sure. After 35 years writing various types of code, I always go for the pedantic, but unambiguous method of using parentheses rather than rely on the opinion of the programmer who wrote the parser.... but I'm a H/W engineer so I never rely on the programmers anyway 😜
@@flyball1788 Prior to 1900 the rule was as follows. If multiplication sign is ommited, then the multiplier is unconditionally associated with adjacent parenthesis. Please hear me out, they used this rule in academic papers and scientific work in XIX and XX centuries.
Moreover, this rule is still applied in some contries e.g. Russia, ex-USSR, and I believe Great Britain as well.
The PEMDAS rule is a simplification in a nut shell. It is a mnemonic to help people to remember the "correct" order of operations. Except it actually doesn't tell you what to do if multiplication sign is ommited. In case your math teacher tought you "If you don't see multiply sign just imagine it is there and then apply PEMDAS rule", he is just wrong.
Regarding calculators, despite the fact lots of modern software calculators give 9 as an answer, there are scientific calculators which still will give you 1 as an answer.
Modern mathematitians, knowing that the default assumption would be different in different countries, recommend to use explicit multiplication sign in those cases where it will be ambiguity otherwise.
@@flyball1788 Writing code and writing math are different things.
@@evgen5647 The issue has nothing per se to do with parenthesis. Parenthesis is just the thing that allows implied multiplication. You just cannot use it between two numbers. More often it is used with variables.
@@okaro6595imagine we replaced all numbers with variables like this: a÷b(c+b)=? Does it remove the ambiguity? No it doesn't!
For some reason you think that the rules for arithmetics and the rules for algebra differs. They don't!
Actually, if you ask a Maths PhD they might say 1 is actually the right answer. If you were to put the explicit * between 2 and ( then it's really 9. Because 2(2+1) actually implies implicit parentheses, so: (2*(2+1))
I love the basic (scientific) calculator you showed at the beginning. I snapped up 3 of them when my last one suffered from a fall which rendered it inoperable.
As a mathematics major, the correct answer is 1. Implied multiplication binds more strongly than division, so algebraically the 2(2+1) is treated as a single unit with respect to the division. It’s easier to understand why implied multiplication has a higher precedence with variables, so let’s imagine the same problem except replace the (2+1) with a variable y.
6÷2y is unambiguously 6/(2y) not (6/2)y. And if you think otherwise then you never made it far enough into math education to where this becomes the normal way to represent multiplication. The 2 in 2y describes the y, that’s why they are concatenated together. It’s “six divided by two y” (two y’s) not “six divided by two times y”. Clearly there’s no “times” to read from the equation. However ambiguous math, such as this question, is itself wrong. Math questions like these don’t really have correct answers because they are written to intentionally be ambiguous. Math expressions should never be vaguely stated like this, and where there exists ambiguity (even when there isn’t actually any from the mathematicians point of view as I started this response with) the author of the question has the responsibility to make the question more precise to avoid situations like this. That’s why parenthesis exists, to make clear which operations should be completed first.
Side note: I think the main reason why a lot of calculators do implied multiplication wrong (if they support it, because implied multiply is not very common in calculators) is that they translate #() to # * () before parsing the equation, so the parser doesn’t know that it was implied multiplication as opposed to explicit multiplication in the first place.
EXTRA SIDE NOTE: math on paper existed before math in computers, and the paper math rules have been clear for a very long time, so the calculators are the things that “do the math wrong” in this situation. They were supposed to implement algebraic math, and anything else is a bug.
This should be stickied. The whole problem of inline math and the ambiguity it brings is due to trying to input expressions into computers/calculators. It's also why MATLAB straight up requires the user to resolve the ambiguity themselves to ensure the result it provides is for the actual problem the user wants to describe.
Fellow mathematics graduate here. I agree with everything you wrote. Thank you!
Thanks for explaining this… I really couldn’t put my finger on it, but I knew that the answer was one. That is because when I was learning these presidents rules in algebra class, calculators were still pretty rare, and most of the computing languages that we have today didn’t exist.
Thank you for the detailed explanation. Yes, this is exactly why the answer should be 1. If all you do is resolve what’s in the brackets without the implied multiplication, then you haven’t resolved that value fully.
I was about to write something very similar. I'd have said the answer is ambiguous because it depends on interpretation but the implicit multiplication makes most sense if you are writing complex equations
In over 73 years in the real world, I've never had a problem like this in actual practice. In Elementary / Middle / High School, tests, etc., the problem might be presented as 6/2(2+1) and I'd just have to do the right thing. But in solving word problems (remember those!) and real-world problems, you (the operator) have inside knowledge of the meaning of the problem you're trying to solve, and you'd just do it right.
If i wrote a problem like 6/2(2+1) i'd get scolded for not property writing a non ambiguous equation
Agree. Never remember this being an issue through university calculus and physics BUT that was just before the computer explosion. Is VERY important now.
@@johndorian4078 As I was taught it... we learned order of operations before algibra. Before algibra, you'd probably get told to right the multiplication sign in, and may or may not get an brief aside about how this was short hand that would be learned later on in algibra, but wasn't useful for what was currently being taught, so please don't do it. After algibra was a thing, people would might look at you a little funny, because while 6/x(2+1), or (x/2)(2+1) were generally considered pefectly normal, 6/2(2+1) was something that would generally only come up part way through your working for something that started out more complicated, not as a start or end point. (whether it was an acceptable answer or not would depend on the question... but I'd be hard pressed to think of a situation where a question with that as an Answer would come up (they'd usually want you to resolve the multiplication and division and produce a final result, by the time you got down to something that only had numbers in it like that.)
When you write it by hand, you'll move text baseline to either ⁶/₃(2-1) = 9 or ⁶/₃₍₂₋₁₎. Nobody writes 6/3(2-1).
@@QwDragon 6÷3(2-1) is exactly the way the problem was presented in the video. Except Dave had the luxury of using a real division sign (the line with dots above and below) which I didn't bother to look up at the time. So I used the '/'. Same problem, same "ambiguity". Nobody writes it as 6÷3(2-1), either, in the real world. Which was my point.
There is a mathematical expiation for for the answer 1 it is monomial numbers. When a number is written without the infix sign such as 2y it is a monomial and so should be interpreted as (2*y) not as 2*y. You can verify this by the monomial and polynomial therms.
6/2(2+1). Consider this: how do I factor (4+2)? I can take a 2 out of each term, which I would rewrite like this: 2(2+1). In this case 2(2+1) is a term that must ALL be factored together, in which case 6 would be divided by 6, and the answer would be 1. It is a “not enough information” assumption to assume that the 2(2+1) is NOT a single term, which MUST be assumed for the answer to be 9. The reason for not enough information is a lack of a standardizing rule that makes one or the other solution no longer possible. I believe that the example I have given demonstrates why the rule should be that considering a lack of an explicit multiply symbol considers all conjoined characters as a single term should be the rule of the day OR that parenthesis should always be enforced such as 6/(2(2+1)), though the latter still leaves some logical problems to be solved. Logically, given the limitations introduced by the ambiguity of the rules at the moment, the answer is either 1 or 9.
Thank you, thats exactly my thought.
6/2(2+1) can also be written as 6*½*(2+1) so your factoring is wrong.
It would be factored as (1 + ½) which is ³/². So that would be 6 * ³/² which is also 9.
«It is a “not enough information” assumption to assume that the 2(2+1) is NOT a single term»
If you want it to be a single term, then you must "spell" the expression differently. If you think there's "not enough information" then you missed out on some fundamentals during your education. So the real problem when encountering expressions like this one is that you have to wonder if the person who wrote it is someone who thinks as you do.
There is no special rule for multiplication by juxtaposition. It's multiplication like any other, and has the same priority as division, so you have to evaluate left to right. Just because the multiplication symbol isn't written out explicitly doesn't change anything about the order of operations.
Don't use the division sign. It's eeeevil :D
And now I remember why I always hated math class growing up. Most the time my cynical teachers spent more effort trying to trick us than making sure we understood the subject.
Try learning PEMDAS, then moving into engineering calculations where the parentheses bind more tightly to an adjacent number, therefore are not immediately replaced with a multiplication, effectively treating the entire expression as if it were in parentheses. His example of 6÷2(1+2) =9 would instead effectively be calculated as 6÷(2(1+2))=1
Then again, perhaps this is just a logic trap for aspiring engineering students to deal with.
@@davidrush4908poor formatting that's designed to spark a war in the comments every time. You get people who are outright wrong, and generally two groups that understand OoO in different ways.
for me, the answer is 1
not because thats the "correct" answer
there isnt one
its because that's the answer according to the writing convensions id use
id interpret 6 / 2(2+1) as
a/bc
and id consider bc to be one single whole complete product
i make a distinction between ab and a * b
Most of the modern calculators assume "bc" are seperate products, which is probably for the best, cuz how else would you clarify that?
You can simply use another pair of brackets to make the calculators read them as one product:
6 / 2(2+1) = 9
6 / (2(2+1)) = 1
Try it out =)
@@SilverSpade92 adding brackets suck
I prefer writing as efficiently as possible
It's just easier to keep ab as one product by default and make a * b an operation than it is to have ab default to a * b and having to add 2 brackets to two sides to determine the values are together.
So I'll stick with the old calculators.
@@SilverSpade92 No way, that first expression should have parenthesis around the fraction, especially given that it's going to be multiplied by something after it. You shouldn't have more numerator after the denominator unless it's clearly marked.
(6/2)*(2+1) is the only correct way to write that first expression, that or reordering it to (6*(2+1))/2.
@@chad_bro_chill You start by doing a misstake and then the rest is wrong.
since there is no operator writen out it is 6/(2(2+1))
You are simply showing that you have not done enough math to grasp it.
Traveling the tree in a third way yields pre-fix notation, which has the same “never needs parentheses” feature as post-fix but can a bit more intuitive for some. AB+ versus +AB. Both allow easy stack-based processing. Function calls are a form of pre-fix notation: add(A, B).
As someone that used a calculator with postfix notation in university, the comment to ask your dad, made me feel old. My father always talked about his slide rule. My first ever programming project was a "keep busy & learn the ropes" kind of project and it was to write a postfix notation calculator for Windows NT in C++.
It gives a different answer because it has (as stated in it's manual) a more complex/complete order of operations than just PEMDAS. It has a level between E and M for "implied multiplication". This is also often how scientists write in their papers. Calculator devices still come in both this configuration and the strict PEMDAS configuration today. It's a case of sloppy input from the me problems and inconsistent interpretations of the "right" way to math.
Part of the problem with the viral problems is that there are multiple conventions in mathematics. In some systems the number next to parenthesis has higher precedence than a multiplication or division symbol.
That was my thought. I remember being taught in math that when you have an expression like 2(3+1), the "2*" part is actually part of the equation in the parentheses.
Anything that is outside a parentheses without any operator means it is factored out of the parentheses and the terms inside of it.
So yes. Constants, values and variables outside the parentheses, regardless if it's in front of or after, belongs to the parentheses and is not dependant on PEMDAS or any other silly rule. It is ALWAYS multiplied into the parentheses first, or atleast at uni.
I think big problem is that there is no corporal punishment for writing 2(2+1)... That notation should not ever be allowed with real numbers. If you are going to follow some elementary rules write every symbol there. Or fail the test automatically.
y = mx + c has an implicit multiplication between the m and x
2ab ÷ 2a ≠ 2ab ÷ 2 x a
as b = a²b only if b is zero or a = ±1 for real values of a and b
Replacing an implicit multiplication with an explicit multiplication to reason about the precedence does not demonstrate anything about the precedence of implicit multiplication vs division.
Gawd damn, I picked up degrees in maths and engineering 40 years ago and I still think the answers 1. Looks like I might have to learn something new today ;-/ thanks Dave
It is one. Because every person who does these videos fails to do the distributive property. The parenthesis has to be evaluated WITH the distribution of the thing attached to it. If you write the expression as a fraction it becomes obvious that everyone is doing it wrong.
Don't worry , 1 is the correct answer. Implicit multiplication is not something calculators usually understand.
@@nixboox No, it's not. Maybe if you're from USA the lack of good education didn't teach you that one correctly but you're forgetting how brackets work and when left to right notation is applied.
You want to learn about order-of-operations, really the best resource in CZcams is two videos by the channel "The How and Why of Mathematics". I forget the exact titles, but you'll recognize them when you see them.
@@mufaro_xyz Left to right is not a thing. The correct answer is 1 because juxtaposition, or implied multiplication. When you have 6/2(1+2) after solving the parenthesis (which are always done first) you go for the juxtaposition and get 6/6 which is 1. Left to right just happens to be a right sometimes as a coincidence, I never saw that in my school years and I'm from Brazil.
Pemdas was invented by someone to explain it as easily as possible, but nobody used it in reality. There's an extra step: multiplication by juxtaposition, which takes precedence over explicit multiplication and division.
If you look in math text books, you will see that something like √12 will often be simplified down to 2√3, and then if you do math with that, say 2÷2√3, it will come out as 1/√3. By your logic, we should be getting just √3. The true order of operations is PEJMDAS. It was just stickler teachers who insisted that what the book says in text (without looking at what the actual math evaluates out to) is right 100% of the time that pushed pemdas. Pemdas is inconsistently followed. If you go back to google and type in 2/2pi, you will se they follow pejmdas and convert it to 2/(2*pi). Ultimately the most important rule is to be consistent. Pejmdas is what most people actually use before "corrections" to pemdas, and pejmdas still sneaks through in cases like 2pi.
likewise, the reactance of a capacitor is 1 / 2πfc; where all of the implied multiplication happens prior to division.
Yeah I remember this formula. So if teachers followed PEMDAS, they should have also written this as 1/2/π/f/c?@@Boffin55
Exactly. And it was the same stickler teachers who would think they were clever when they tried to correct our use of contractions by saying "ain't ain't a word because ain't ain't in the dictionary" without considering that they might should check the dictionary before making a claim like that. They did *not* think it was funny when you then went and found ain't in the dictionary and then you showed them in front of the entire classroom.
Which is extra funny, because that sort of teacher usually at least claimed to be Christian. You know, that religion that says pride is a sin? I guess I know where Ms. Paguaga is going to be spending eternity 🙄
@@cassiee.3969 she'll be in good company with my Mormon stepfather, he had the same attitude about ain't but pride is far from his only sin.
To avoid ambiguity stop using × and ÷. If you use fractions exclusively for division and parentheses for multiplication then things are much clearer.
It’s 1. Brackets, implied multiplication through distribution (aka multiplicative juxtaposition), then division. Because it’s attached to the bracket, 2(3) is solved PRIOR to sign multiplication and division as any proper scientific and algebraic calculator is programmed to do (VPAM and CAS). Thanks for highlighting this for the kids in my college calc 2 course who still know their order of operations 😂
Proud owner of HP-41CV. It had linear algebra and circuit analysis pacs so it was pretty much mandatory. And once you learn RPN, there's no going back. Also, I remember a rule about proximity (?) , that would mean 1 is the correct answer.
Agree- I bought an original HP35 when I was in college and have used RPN since. That calculator didn't even have a model number on it. I assume when they made it they didn't know if it would be successful or if they would every make another model. Later when they came out with the HP45 they started putting a model number on the HP35. Search pictures and you should be able to find them with and without the model number. I also have a 41C.
Have an HP 35s and an HP 50. Apparently the last of HP's RPN calculators. Of course I use the calculator on my phone most now, which is why I use RealCalc, which supports RPN.
@@TevelDrinkwater Yeah, RealCalc is awesome. My every day is 32, but have the 12 and 35 in case my 32 dies one day! Between the three, hopefully I'm set for life LOL. Also, how many RPN users had the "hey, can I borrow your calculator?"
"Yes, but...ahhhhh it's RPN sooooo"
"Whatever, let me use it....HEY! where's the EQUAL sign???"
I have a 41CV too. My father passed his HP35 on to me when I was in High School. It got a lot of use.
Understanding RPN and how to translate algebraic notation into postfix helps one learn to avoid ambiguity.
The HP12c RPN calculator is still sold by HP.
And HP now has some graphing caculators that are RPN.
@@linusfu515 if the equation were:
N=1+2 ; 6/2n
Then you would be correct, but the implied association only applies for algebraic variable letters and symbolic constants like π. When the only thing being is a parenthesized expression containing numbers only, that does npt apply.
6/2(1x+2x) is 6/(2×(1x+2x)) by default because it is algebraic expression with a variable and requiring parentheses every time you have 1/2x complicates things.
But
6/2(1+2) is (6/2)×(1+2) by default because it is an arithmetic expression.
However, in both cases, there is ambiguity that should be resolved with proper parentheses.
If the problem just says to provide the answer to 6/2(1+2), the question is about grade school arithmetic (even if the problem is in a higher math class), and the answer is unambiguously 9, since there is no precedence level in PEMDAS/BODMAS for implied multiplication. If a grade school teacher is teaching that implied multiplication has a higher priority than regular multiplication/division, then that teacher is teaching incorrectly.
If there is more to the problem, and 6/2(1+2) is only part of the presentation, the context should indicate the official meaning, and a comment should be included in the answer that the formula as written is ambiguous.
If 6/2(1+2) is part of the answer you write, expect to have points taken off for not writing it in 2 dimensions and using a vinculum and/or not parenthesizing properly.
I had always used the term 'reverse polish notation' for the postfix representation. It was never the topic of discussion when I worked with other engineers, so hearing it hear was both informative, and a nice blast of nostalgia. I recall deriving the same tree structure to implement RPN for a class assignment.
I always heard "reverse polish logic" as the term used by engineers and scientists. This is the first time I heard "postfix".
Yes. 'RPN' (Reverse Polish Notation) is what HP calls it too.
Now, class, can anyone tell me why it's "reverse" and why it's "Polish"?
@@tedlassagne8785 Yup.
(And this is all info I already new, but I did have to look up to confirm the guy's name.)
It started with what we can call prefix notation, invented by Jan Łukasiewicz. It was called "Łukasiewicz notation", and then, because people were lazy and couldn't remember how to properly spell his name, "Polish notation"(since he was Polish).
Polish notation had the same advantages that postfix notation does - it's unambiguous. The only difference is that, when writing Polish notation, the operator comes before the operands, not after. So, for example, 5 * 3 + 2 could be written as + * 5 3 2.
Postfix notation is the complete reverse - reverse Polish notation. (The same formula above would be written 2 3 5 * + in RPN.)
@@Acorn_Anomaly interesting. I knew the Polish part, but never did understand where the Reverse came from. Thanks
You have such a satisfying way of explaining, you secured yourself a sub, sir! 😁
The other thing I cannot resist to ask: What's that cool gadget in the background swirling blue light dots around a mysterious globe of some sorts?
This looks like something I need for my tabletop repertoire 😄
When Dave said he'd be spending weeks arguing in the comments, I assumed that he was just being funny.
Then I looked at the comments.
I'm extremely pleased that on the HP Prime (in algebraic mode), this expression is a syntax error, and in RPN mode it's a constant function (6/2) applied to another constant (1+2) and the value is 3! (The calculator DOES warn you.)
The value is 3factorial?
I argue that the answer is indeed 1. The ambiguity of the equation comes from the use of one line division in conjunction with the use of implicit multiplication. Implicit multiplication just feels like it has an implied grouping to it as well. i.e. 2(2+1) should always be expanded to (2*(2+1)) when used in a larger expression rather than simply 2*(2+1).
Another way to think about it is n(x) is the n-times function. i.e. n(x){return n*x}. Would you still try to tell me that 6/n(x) should return (6/n)*x instead of 6/(n*x)?
So true. But "garbage in, garbage out", is the reason.
most people would have little problem recognizing implied over explicit priority in examples like "5 ÷ 2n"...
No, professional mathematicians and physicians constantly use implied multiplication in in-line expressions and it has a higher precedence. Who are you to tell professionals they do it wrong because of what your elementary school teacher said?
Another fascinating video, really enjoyed it!
I've used a postfix HP 15C that my dad gave me when I started college. Amazing thing, and while postfix might sound complicated when you get the hang of it it's actually much easier to use when you have complicated math expressions
I still use an HP-15C. It's a great calculator
I've been using an HP11 for general use since the early eighties (and other HPs, and even a SInclair Scientific at school in the seventies) - the nice thing about them is that very few people every ask to borrow RPN calculators a second time.
Thanks for this, Dave, nicely explained. But I expect people will still get it wrong...
p.s. the HP11 is recently on its *third* set of LR44 - does nothing _last_ these days?
"very few people every ask to borrow RPN calculators a second time"
@@tectopic how true (and I had both of those calculators too!)
It's been decades since I actually used that division sign that I was actually confused for a moment.
Most open-source distributions providing X window server are pretty rad for the compose function. IN the absence of a dead key to enable (⋄), most desktops provide an alternate button to press which overrides normal keyboard use while in effect. Short of that, Windows users can use WinCompose to emulate this with macro inputs.
⋄:- = ÷, ⋄xx = ×, ⋄ = ⋄ etc.
Wow! How did you even survive!? Are you just going through life winging it blindly like a bat?? ( o.o)
@@fsmoura No I mean I always use /, so I was like "does ÷ have a different rule or not?" (which is still silly, i know)
@@VieShaphiel Ah, of course! Phew, I was worried for a moment (" o.o)
2÷3÷4 feels like it goes left to right and 2/3/4 feels like it goes right to left. Same as 2^3^4 famously goes right to left also, and that's not what PEMDAS says either.
I always wondered about the quadratic formula - since PEMDAS does not include juxtaposition, over/under fraction notation, or root (and many other notations not taught before middle school), it seems that those notations should be ignored and the closest PEMDAS approximation inserted in their place. Now I know you calculate 𝗯² - 𝟰 × 𝗮 × 𝗰, take its root (even though not covered by PEMDAS), divide that root by 𝟮, multiply by 𝗮 and only then add or subtract all that that from -𝗯. In other words, the PEMDAS interpretation would be -𝗯 ± √(𝗯² - 𝟰 × 𝗮 × 𝗰) ÷ 𝟮 × 𝗮.
Or perhaps you could say that if a formula contains notations beyond 5th grade arithmetic, you need to look beyond PEMDAS to understand the proper interpretation.
Bottom line: If you are in 6th grade, you want to follow PEMDAS to conform to the simplified view. But if you are in the real world you need to recognize that juxtaposition customarily has a higher precedence in engineering, physics, and other fields involving higher mathematical. Time to drop the training wheels and set PEMDAS aside.
Just some small points.
Yes, PEMDAS doesn't contain implicit notation (multiplication or otherwise, like how Sin²x means (Sin x)²).
No, PEMDAS does take roots into account since roots are a form of Exponents. They are fraction powers, so are part of the E step.
If you have a two line fraction, that could be interpreted to represent division, which is in the order of operations, where the expression doing the dividing must be placed in brackets. It's best to resolve the top and bottom separately and after that look at the fraction as a whole for any final bit of simplifying.
a
--
b
is always (a)/(b) regardless of what "a" and "b" are.
In that simplest case, the brackets are not necessary but in a more complex one they are. If "a" = t²-1 and "b" = t+1 for example.
PEMDAS should only have an issue with the ÷2a part in the quadratic formula.
Years ago, I had a discussion with my father, who did not understand the following sum: ½:½=1. If I put it like that, it makes sense. But when I told him it sounded to him like half divided by half and that is ¼. His argument was that if you have half an apple, and you divide it in half, you have a ¼ apple. No matter how I explained it, he never understood and believes it.
I think it's the subtlety in the difference language-wise even that can cause confusion.
½ divided by ½ is indeed 1 but your father likely interpreted it as
½ of a ½ which is ¼
or maybe as
½ divided in ½, which is (½)/2 = ¼ also.
or as
½ divided by 2, which is ¼ also.
My guess is he interpreted it as
½ divided in ½ (which is ¼)
instead of
½ divided into ½ (which is 1)
A small change, into and in, makes a big difference.
I think the reason there are so many arguments about PEMDAS is that it *_seems_* like the order based on the acronym is supposed to be,
1. Parenthesis
2. Exponents
3. Multiplication
4. Division
5. Addition
6. Subtraction,
but it's not. It's,
1. Parenthesis
2. Exponents
3. Multiplication AND division
4. Addition AND subtraction.
The acronym, itself, is what causes the confusion and the arguments.
PEMaDAaS
i think that is simply people not understand the reason WHY we use PEMDAS. We could create our own system to evaluate expressions however we want, we just choose to use a standardised one so that people writing an expression dont have to clarify what order they would solve the operators in, so that we can universally share math... which is one of the first things humanity did centuries ago: finding an standard to communicate mathematical expressions. The reason why PEMDAS is not wrong as an acronym tho is that no matter what your expression evaluation order is, divisions are the inverse of multiplications, and subtractions are the inverse of additions, so they will both OBVIOUSLY be on the same level of precedence.
Think about it like this: you can do 6/2 which would be the same as doing 6 * (1/2) which reads out as "six times one half". They both give the exact same result, again, because division is the same as multiplication by the inverse of the number (one over the number).
Same applies to subtraction. 2 - 1 is the same as 2 + (-1).
In short, if you actually understand arithmetic, the name PEMDAS should not give you any trouble when it comes to understanding that there are certain operations that have the same precedence. The only problem is that most kids that learn pemdas in school pass without actually having a good grasp of arithmetic or understanding of what the operators really do, meaning that they will forever carry that mistake of misunderstanding the order of operations and PEMDAS.
I was taught to do it in passes, literally multiply and then divide.
It's nothing to do with that.
The ambiguity arises firstly because nobody who's done any maths beyond 11 yrs actually uses the ÷ symbol and secondly, more importantly, once you've done algebra, multiplication by juxtaposition takes precedence so anyone writing the problem as 6÷2(2+1) has either 1. forgotten the x purely by mistake or 2. is using the second 2 as a common factor in the same way as 6÷2(x+1).
Nobody would evaluate 6÷2(x+1) as 6÷2 * (x+1). Its ambiguous but people just know if means 6/(2*(x+1))
The confusion may also come from the authors of PEMDAS. When they wrote their books, they wrote a/bc as "a" in the numerator and b*c in the denominator and calculated the results based on that structure. Otherwise, they would have just wrote ac/b.
Interesting, I was curious what the "Algeo" app I like to use on my smart phone would do with this since it supports implicit parenthetical multiplication syntax. It basically behaves like a TI graphing calculator where the whole expression gets written out. What I found is that when you do division it always does it as a horizontal line with numerator on top and denominator on bottom. So in similar fashion it forces a more explicit expression where you either include it in the denominator under the numerator line, or you force it to the side of the entire fraction making it obvious that the fraction is done first.
That's a really cool table with your Mercedes AMG wheel! I got my first AMG not long ago to go along with the rest of my collection, and those wheels do look great. Also, great video! Was cool seeing you try it "over the years" in Windows versions.
Thanks for explaining this correctly several different ways. The programing is good on the calculators that force you to define the problem more clearly, the programmers were clearly aware of the confusion.
We never used "÷" in math past grade school, it was always "/", and make it clear what is the denominator.
Why oh why do Americans not learn the juxtaposition rule 🤔 juxtapositions come after P and E (on their equivalent script level) but before D or M!
PEDMAS and BODMAS are aids yes but the more correct version is PEJDMAS and BOJDMAS.
The slash vs the obelus sign has nothing to do with the underlying problem. 6/2(2+1) is the same thing as 6÷2(2+1). The underlying problem is whether juxtaposition has priority over division.
@@carultch Would 6÷2*(2+1) remove the ambiguity? That'd just be left to right evaluation (after the parentheses of course), or am I still confused?
@@davidg4288 You are correct. It would remove the ambiguity, by using an explicit symbol for multiplication.
@@carultch So *does* juxtaposition have priority? I don't remember that it did, but I haven't taken an Algebra course in over 40 years!
I was always taught BODMAS, B being Brackets. I would always treat that left side as having brackets too as the equation seems to be poorly written on purpose.
Multiplication and division are equal operations, so they can be done in whatever order they appear, left to right. Same with addition and subtraction. So it doesn't matter which order those two sets are in for the mnemonic.
This is something argued for years, people seem to think math is a fact. Math is like language ... if you speak the same language you can communicate other wise there's confusion.
@@brocka.6479multiplication and division are not really equal in the sense that one is commutative and the other is not.
PEMDAS etc. are used for basic school arithmetic, not by mathematicians.
2(2+1)=(2+1)2
Therefore
6÷2(2+1)=6÷(2+1)2=1
Otherwise you get
6÷2(2+1)=9
And
6÷(2+1)2=4
Mathematically both of the above must have the same answer therefore the only unambiguous way to evaluate it is with the result 1.
This guy gets it. (johndorian) Math is a language and if you use ambiguous language you get ambiguous interpretations. The whole argument is silly. Write your expressions in an unambiguous manner or people will interpret them based on context. Implied operators are generally done BEFORE any other operators, but they are generally only written as implied operators if the intent is clear to the intended reader.@@johndorian4078
Brackets and parentheses are the same for the purposes of mathematical expressions. Same with exponents and ordinals.
I had the SAME Sharp calculator back in my engineering days. Could store 49 commands and had PB (playback) so you could check/edit long formulas. Awesome calculator.
There is a way to drain your Sharp calculator's battery....quickly. Give it a long equation composed of a lot of very large calculations. I forget what the max factorial was on the calculator. Say it was 47! So the equation would be 47!/47!*47!/47!*47!/47!*47!/47!*47!/47! etc. until you had used up the 49 commands allowed. Then hit equals. My longest/slowest would take almost a minute. Since the calculator retains the last mighty equation, you can hit equals again. etc. All I know is I changed my batteries at least once in the three years I had the calculator.
I bought the Sharp instead of an HP RPN calculator because of the guy in the UBC bookstore. He said, simply, "This is what the Chinese are buying". I'm eternally grateful that I never had to suffer with an RPN calculator.
While most people consider PEMDAS to be the truth in its highest form, it's actually an approximation. Here's a quote from easily findable article:
The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. So, 6/2(2+1) is NOT the same as 6/2*(2+1).
And using the same logic, 1/2s doesn't mean half a second, but rather half a Hertz. Wait a minute...
Thank you. I have a different perspective now ! But, I ALWAYS prefer brackets throughout :
(6/2)(2+1) vs. 6/ (2)(2+1)
Brackets MUST be taught right from the beginning ... it brings clarity to the problem.
Other things that beginners MUST be taught, is the use of :
Multiplication by 1 = 3/3 = .....= A/A = ...... EG ➡️ A(.......) = A(.......)
Addition of 0 = 3-3 = A-A = ....... ➡️. (........) +A = (........)+A
You seem to be grouping with spaces, not brackets, in your second example.
How about just use reduction. It leaves you with 3 x 3 = ?
@@brostenen because reduction isn't necessarily correct. Using strict PEMDAS, yes, the answer would be 9, but if you're using multiplication by juxtaposition (implicit multiplication) which has a higher precedence that multiplication/division, this would be re-written as 6/(2(2+1)), which is indeed 1. The difference in answers depends on if implicit multiplication is to be used or not (and to drive engagement from people arguing).
Multiplication and Division share the same tier. Making 6/2*(2+1) into (6/2)*(2+1) leads to the same answer but is not necessarily equal:
6/2*(2+1) = 6/2*3 = 6*3/2 = 18/2 = 9
(6/2) * 3 would force 6/2 to be evaluated first.
Not too important in small equations like here. In large terms, you might want to refactor terms for simplification.
See my answer above. The 2( portion is PART OF THE PARENTHETICAL! It ALSO must be solved BEFORE continuing the PEMDAS rules. Multipliers adjacent to parentheses MUST be computed first. Replace all those numbers with variables and solve it algebraically and you'll see what I mean. The TI engineers knew this, but apparently schools are failing this as outlined in math books going back centuries.
I'm younger than you and I started with RPN in college. Couple years back my HP on/off key broke and have been using R ever since. R, by the way, requires parethesis and gets the correct answer.
The answer IS 1, the "rule" is called adjacency, it's not violating PEMDAS it''s just lazy human shorthand for an actual valid equation:
3 ÷ 2x
is adjacency shorthand for
3 ÷ (2 * x)
So, when we unravel
6 ÷ 2(1 + 2)
in the same fashion, we see the PEMDAS compatible long hand is actually
6 ÷ (2 * (1 + 2))
NOW you can actually evaluate the expression, and not a moment beforehand.
The answer is 1.
I am also a decades-long ex-Microsoft FTE software engineer with a math minor.
Would always add the multipler symbol to avoid ambiguity
Of all people I wasn't expecting YOU to be this wrong, Dave. What you are missing is PEJMDAS, where "J" stands for "Juxtaposition" (meaning, "multiplication by juxtaposition", i.e. when you do not put in the multiplication sign). This beats the MD level. Let me put it this way to make it simple. If I assign x=1+2 and then tell you to compute 1/2x do you still seriously get 9?
Basically the error you repeat is putting in the explicit multiplication sign, which indeed results in the answer 9. You are testing it with broken calculators that cannot handle juxtaposing multiplication, which has higher priority.
PEMDAS is a rule for small children. THe true rule is PEJMDAS. Any ACTUAL mathematician would write this expression with the 6 on top, a horizontal line under it, and 2(1+3) under, so for them this problem doesn't occur. The only reason we have this nonsense discussion is that regular humans can't write LaTeX :)
See but you're also assuming that it's 6/ all of that not 6/ 2* the rest. That's the problem with this notation. It's just imprecise because we have the division in there. You really should add more context. People like to add the whole juxtaposition thing but when you look at equations that use the juxtaposition it still typically follows PEMDAS because anybody who's using juxtaposition typically is writing the equation without ambiguity. As soon as there's any debate over the equation, it's a bad equation.
@@actually_it_is_rocket_science There is no assuming, and no, it is not "6/all of that". The longer expression "6/2(1+2)+4-2" would not have the "plus four minus two" part in the division. Its the fact that the 2 is coupled to the parenthesis that makes it stronger than a simple 2*parenthesis. When written like that anyone experienced with math would think of it as a single unit, and the unit is 2parenthesis.
A good example is a/bc which would ALWAYS mean a over bc. a/bc+5 would mean a over bc then add 5.
@@johanlarsson9805 Exacatly. Go back to my example where we assign x = 1+3 and we then do 1/2x ... I say that is distinctly different to 1/2*x ... you can't just take a multiplication-by-juxtaposition and replace it with a multiplication sign, without wrapping the juxtaposed entities in parenthesis. I.e. everybody agrees 1/2x is 1 / (2*x).
When you unjuxtapose, the juxtaposed pair gets parenthesis implicitly.
@@ZapAnderssonIt's not quite as simple as that. a/bc² = a × b⁻¹ × c⁻²
@@0LoneTech Agreed that would mean a / (b * c *c) which is equivalent to what you wrote. But the PEMDAS people here want it to be (a / b) * (c * c), which is obviously wrong. They are all simply wrong, @dave included