@@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.
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.
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.
@@lenonkitchens7727absolutely Dave and the boys at Microsoft are wrong in this case, because of juxtaposition! It really is PEJMDAS but the statement is ofcourse purposely ambiguous. But an engineer and mathematician will see the indirect multiplication result and (correctly) execute those first.
@@CallousCoder a mathematician will evaluate based upon the notation standard used - whether it is pemdas or pejmdas. pemdas and pejmdas are not rules of evaluating equations, they are in fact rules of notation.
@@steinanderson9849 that’s true, and therein lies the problem. It’s ambiguous. But juxtaposition comes before division. I can’t link to a great video by a lady who explains the history of pemdas. And juxtaposition and that’s how we’ve been taught. And than the answer is 1. Also because the obulus says that what comes to the right goes on the top dot and what goes to the left goes on the bottom dot. So 1 -- 2pi And thus you do juxtaposition before you do the division. Otherwise you would write (1 ➗ 2)pi SHARP calculators always just PEJMDAS and Casio change because daft American teachers who did pemdas. But now they went back to the PEJMDAS (like the rest of the world) Look for the video with the title: The Problem with PEMDAS: Why Calculators Disagree And you see why Dave (original poster) and I stand firm that implied multiplication comes before division.
@@CallousCoder stand firm all you want. at the end of the day the standard of notation is what prevails and depending on your environment implicit multiplication being of higher priority may or may not be the standard. simple stuff really.
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".
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.
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.
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.
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.
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!
@@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!✌️
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.
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
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 RU-vid... 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.
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)".
this assumes that every term after the / belongs in the denominator but that would require another parenthesis to make it explicit for the calculator n/(a(b+c))
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)
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).
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.
I have to take issue, because 6÷2(3) ≠ 6÷2×3 Distributing the 2 into the parentheses is part of evaluating the parentheses. The point of using implicit multiplication is that it's a common term that's part of the parens that was temporarily removed to simplify the contents of the parens. As such, it's wrong to separate it. Don't use implicit multiplication if that is not what is intended. (zx+zy) would be written as z(x+y). 6÷z(x+y) is not 6÷z×(x+y) It is 6÷(zx+zy) The 6 is not automatically grouped with the 2 without being written as six halves.
@@bfish89ryuhayabusa Exactly. Noone in their right mind would read 6 ÷ 2x as 3x rather than 3/x, so why should substituting x with a known value suddenly change that?
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.
@@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 ✌️
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
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.
@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.
The problem here is the laziness to i ignore the significance of implied vs implicit. We literally coded calculators around the fact that lesser intelligible teachers only wanted to teach simpler math. L->R multiplication division should've been excluded all together as it doesn't properly visualize problems. PEMDAS enforces all equations must be written lateral, PEJMDAS allows written equations to express multiple fuecets/dimensions/applications.
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.
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.
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
@@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
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.
"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.
@@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.
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.
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.
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.
@@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)
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.
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.
@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 RU-vid 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.
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).
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.
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.
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 🙄
It's 1 because as written it's the equivalent of 6 / (4+2) because it's a factor of 2(2+1) and factoring is part of the parenthetical section of pemdas. You only get 9 if explicitly add the multiplication sign which isn't present in the original problem.
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
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)?
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?
1 is the correct answer, as it preserves the integrity of the horizontal bar symbol to indicate to division, which also implicitly qualifies the order of operations, according to which you always do what's in parentheses first, so the expression would instantly become, "6 divide by 2 times 3. In real life we were told to qualify explicitly, by enclosing in parentheses. So, "6 divide by the quantity, 2 times 3", which yields 1. There would be no question of order of operations using the bar: 6/(2 times 3). The arithmetic order of operations specification seems clear that division is done last.
they don' t know math dude. This will be worthless to argue with them. they have no idea when we used such brackets 2(1+2) or 2(2+1). there knowledge is elementary not high school. cause your have to learn in starting around 5, 6 class, word to expression.
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.
I picked up an HP48 when at Uni in the 90s. Once you use RPN, it really is pretty natural. I now have the HP48 emulator on my phone and I use that for all calculator work I need to do
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.
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.
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.
It was 1973 when I was in college and co-oped at the local electric utility. Its engineering group had just spent the $400 to get the latest and greatest HP-35 calculator that uses Reverse Polish Notation (RPN). My first job, since I was "in college", was to teach the engineers how to use it. Only a few of them got it, so the calculator was given to me to use during my co-op workblock. FYI, Bing's Copilot gives the correct answer of 9, with a correct explanation of the PEMDAS steps.
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))
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.
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.
If you are asked to calculate the curve of a thrown ball in a medium you need to use the formula which includes the medium. The result from the simplified formula in a vacuum would be wrong. If you are asked to evaluate an equation which uses implied multiplication you need to use a rule-set which includes implied multiplication. PEMDAS doesn't and gives the wrong answer. Easily seen by the fact that calculators using PEMDAS either make you or automatically insert an explicit multiplication sign. But 6/2(2+1) =/= 6/2*(2+1). The calculators just interprete implied multiplication as if it were explicit multiplication, changing the question. Correctly translating the question to a system without implied multiplication results in 6/(2*(2+1)). With implied multiplication it will evaluate as 6/2(2+1) = 6/(4+2). A scientific law always has to describe reality: Is there a medium or are we in a vacuum? The reality is that implied multiplication is used in the question. PEMDAS is a simplification which fails to describe this reality. PEJMDAS better describes the reality of how this equation will be evaluated.
There is a medium, it's just insignificant for the simple situations. It also turns algebra into differential equations, when it comes to solving projectile motion. It's not necessarily wrong to ignore air drag, it's just a simplification.
The discourse around this often forgets that these notational conventions are fairly arbitrary and don't always agree. For instance, if we replace (1+2) with x most people would probably agree that 6/2x is best read as 6/(2x) and not (6/2)x. At least, that's the closest approximation to what one normally writes on paper with pencil - which typically bypasses the ambiguity altogether. Regardless, this notational ambiguity means that PEMDAS is better understood as a rule of thumb, not a prescriptive formula for the correct way to do arithmetic written in a computer-friendly notation.
There is no problem with the order of operations. The expression is ambiguous. The mathematical term is "not well defined". The problem is with the expression.
For the record I was explicitly taught that multiplication by juxtaposition should be treated as regular multiplication. I definitely question the logic of how anyone could think 6/2x would be any different from 6/2(x).
@@JonathanGray89 your math teacher was dumb-ass and tried to tought you the incorrect logic. The only correct logic nowadays is that this expression is unambigious and avoid unambiguity in exact sciences at all costs.
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.
@@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!
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.
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.
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 😁
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.
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 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.
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.
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.
@@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!
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.
Came back to this video to check whether Dave had made a follow up or something, seeing how his reasoning was flawed and people in the comments pointed out the flaw. Well, if he ever does let me know, I guess.
You'd be completely ignoring the Left to Right of equivalent operations rule to get this answer. So you're still not following the rules properly. Going left to right without solving parenthesis first gets you 6/2 first 3*(2+1) second Even if you want to multiply each term individually you still get 9. 3*2 + 3*1 Anyway you cut it, its still 9.
@jackinthebox301 @jackinthebox301 The left to right is followed after the terms are completely worked out. See the example above. The 2B is the 2(2+1) term that must be completed. Inside the term PEMDAS is applied, and once that and all other terms are resolved, then PEMDAS is applied to the equation as a hole. Hence, 2B is resolved to 6.
@@michaellavalle3843 But then by your own admission you've simply ignored 6/2 as a term and placed 2(2+1) above it in importance, when they are in fact equal in weight because multiplication and division are the same on the hierarchy. You don't perform M and then D like it is in PEMDAS, they are completed left to right after Parentheses and Exponents are taken care of. Like he says in the video, if you are putting the proper operator in place you will always end up with 9. 6/2*2*(2+1) is the same thing as the 6/2*2(2+1). Edit: your original comment is likewise incorrect. You are lumping together 2B as though that is one object, but it is just shorthand for two different numbers. So the proper expression would be A/2*B. So left to right still applies.
As soon as I saw this I knew what it was going to be. Sorry, Dave, on this one you are wrong, as is your calculator. The issue is implicit vs explicit multiplication. When PEMDAS/BODMAS/BEDMAS/whatever combination you were taught (for me it was BDMSA, which makes me wonder if my maths teacher should be on some sort of register) was created (initially described in a Danish book on gunnery in the early 19th but not codified as a guideline till a maths book published around 1920), as a guideline only, it’s not a rule whatever your teacher in school said, the convention was that you ALWAYS include the symbology so you would never write 6÷2(2+1), it was always 6÷2×(2+1). Multiplication was always explicit. In those circumstances PEMDAS worked fine. Then engineering and other numerate sciences (the STE of STEM) got involved with big equations where dropping unneeded symbols, like the explicit multiplication sign, would let you get more equations on a page. The convention in STE became that implicit multiplication, a number followed by either a variable (e.g. 2x) or a calculation in parentheses (e.g. 2(2+1)), would be calculated with higher precedence than explicit multiplication or division unless otherwise indicated. This, when scientific calculators come out that could have calculations entered with implicit multiplication, was the convention adopted everywhere except North America where a caucus of math teachers kicked up a fuss. When the memes came out I spent some time researching this, hence knowing the history. First off, there is no supreme ruling body in mathematics to set rules so everything you learn is assumptions that have been found to work and guidelines derived from those assumptions that work in the situations they were derived for. PEMDAS, therefore, is only a guideline and one that was derived to handle an environment where multiplication was always explicit. Secondly, 7 billion (people outside the North America) who are taught that implicit multiplication is higher precedence) is bigger than 400 million (people inside North America who are taught dogmatic adherence to PEMDAS). Thirdly, but possibly most importantly, It’s more important that things like rockets work than we soothe the feelings of some teachers who apparently don’t realise that the subject they teach is mathematics, not mathematic. Fourthly Things would be a lot easier if LaTeX was the default way to type mathematical equations. The generally accepted convention is: where possible you should use symbols such as the multiplication sign and parentheses, or layout (LaTeX, you’re on) to make precedence explicit; if the text is in a field where implicit multiplication is king (most STE but more and more M) then give it higher precedence; if in doubt ask the person who wrote it to make it explicit using the methods previously described. Since there’s more calculators used in STE than M they will generally go with implicit multiplication being higher precedence. Unsurprisingly, the Windows 10 calculator pulls the same trick of adding in the multiplication sign the user didn’t enter to make it explicit. Excel complains of a typo then asks you if you want it to insert the multiplication sign. The Microsoft Math app, however, applies implicit multiplication at higher precedence. The problem with your pull it down solution is that in most fields it’s equally or more valid to multiply the 2 into the parentheses to give 6÷(4+2), essentially applying FOIL and treating the 2 as (2+0) so the I and L evaluate to 0. In summary, make things explicit. I suspect that if you were to enter the explicit calculation into your Sharp calculator then it would apply PEMDAS as you want it to. That’s what my Casio calculators (I’ve had to upgrade for additional functionality a few times since the mid 1980s, yes, we’re about the same age, the latest one I have can run Python) do.
As an engineer myself I am also solidly in the "implicit multiplication takes precedence" category. This is essential when you are dealing with engineering units such as mA and kV. The implicit multiplication by the units and their scaling prefixes takes precedence over everything else. For example 10kV÷10mA=1MΩ if you ignore implicit multiplication and try to apply BODMAS you would incorrectly get 10x1000xV÷10x0.001xA=1VA.
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 ☺
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.
‘The Why and How of Mathematics’ has a couple of good videos about PEMDAS and Multiplication by juxtaposition. She says that PEMDAS is the oversimplification taught to children but PEJMDAS (J = Multiplication by Juxtaposition) is what is expected when getting to University/Academia and what you’ll find in scientific papers. Calculator manufacturers ask teachers how they like the order of operations to work, apparently they go back and forth from year to year.
That puts academia at odds with accepted computing conventions. To be honest, until today I had never even heard of assigning a higher precedence to juxtaposed parenthetical terms. I can see the utility, for example in engineering we often have transfer-function expressions such as [(s+1)(s+3)]/[(s+2)(s^2+10s+100)] and the juxtaposition rule would allow it to be written without the brackets. However, I believe the rule just causes confusion and misinterpretation (not to mention argumentation) and I'm now wishing it never existed!
@@marianneoelund2940 Except that in computing, you would never interpret 4x ÷ 2x as 2x^2, you of course treat it as 2 (oops, I had 2x before! Edited to fix). That is the same concept as the implicit multiplication described above. PEJMDAS keeps us consistent with how we interpret statements with variables ... as it must, because mathematics depends on being able to substitute portions of an equation with arbitrary equivalents, including variables defined to be equivalent.
@marianneoelund2940 except implicit multiplication doesn't really appear in computation. Like Dave had to explicitly put it in himself for most of the video.
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!
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.
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).
I don't remember what channel I saw the video on, but the correct answer is 1 according to engineers, scientists, and mathematicians. It's pretty much only American math teachers who say 9. According to the same video, most calculators made prior to 1990 or so, and those made after about 2000, such as the Casio fx-300MS, say 1, if I remember the dates correctly. When the multiplication symbol is left out, as in the example, it's called implicit multiplication, or something like that, and it's usually giving a slightly higher precedence. 2/3y would be another example.
The problem is only in America you are allowed to put a number next to brackets with out an operator, in Germany that alone is illegal to write, which solves the problem all together. But pls watch the video at at least 0.75 speed or slower!😜
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).
What a nice video, pleasant, entertaining, and well explained. (electrical engineer, Curt here). And I will never get tired of your dry sense of humor, Dave! 😊
Not even one mention of the distrubitive law of mathematics? The one that says any time you have a number multiplied by a sum in parentheses like 2(2+1) has to have the same anser is if it was calculated as (2*2 + 2*1). It seems like that fact from 8th grade algebra really needs to be discussed here, as it's fundamental to the issue. As you said, if math isn't something you should play by your own rules, the distrubitive law or distributive property should be included. And if zi sound angry, I assure you I'm not, it's just that I've been agonizing and losing sleep over this for months and when I saw this video from you show up, I thought today would be the day I can finally rest again. I love your channel Dave. Please help me understand.
The thing is you're confusing two different things. PEMDAS is separate from where you use distribution. Distribution is typically useful when dealing with unknowns and simplification.
@joshknight1620 correct but only if the 2 is not involved in a higher priority expression such as the 6/2 in this example. The 2 cannot be distributed because of that.
It is more than possible to use distributivity for 6 / 2 * (1 + 2). You just need to also acknowledge commutativity and that "right-to-left" is bullshit. What do you want me to distribute? I can give you 6 * (0.5 + 1) or / 2 * (6 + 12), but that last looks weird with nothing before the division so lets fix that: (6 + 12) / 2. Much better. Or I can just distribute both factors and get (3 + 6). Now to apply the parentheses rule having done everything else in exactly reverse order and get 9.
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.
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.)
@@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.
When I was in school, they simply taught us that multiplication went before division ;D It took a while for schools in NL to adopt the new order, with the last Math books still mentioning this order, being printed in the 90s. So it's not a question of "Right" vs "Wrong" but more about which convention you are using. Modern convention dictates it should be 9, while some older conventions or science conventions dictate it should be 1. Your Casio is not wrong. It simply uses a different convention.
When I went to school I was also so taught. In the US we have the American Mathematical Society [AMS] and the American Physical Society [APS]. They each have different order of operations. PEJMDAS for the AMS or EPMDAS for the APS When you enter university level math, either formally, or by your instructors methodology, you will be directed to the proper style-sheet to follow. PEJMDAS Juxtaposition multiplications 1st then [M or D] L to R AMS EPMDAS Multiplications always before division. APS Also, look up the international standard for mathematics ISO-80000-1 Within this standard is the prohibition of a division followed by a multiplication or another division, as indeterminate or ambiguous.
PEDMAS. Answer is 1. Hold my soda Do not simplify 1 side of the equation. PEDMAS is a strp by step process. When you do a "letter" stop. Go to the next "letter" of PEDMAS Enjoyed the video
In this case your calculator is correct. Implied multiplication by juxtaposition has higher precedence than division; the answer really is 1. PEMDAS is an over-simplification of the rules. To help illustrate why this is the case, the expression in parentheses can be substituted with a variable, n, in which case the problem becomes 6 + 2n, something you can't treat as (6/2)n, the 2n must be evaluated first.
Wikipedia says that there is no universal definition here. Some textbooks say implied multiplication by juxtaposition takes precedence some don’t. So really one should use parentheses to avoid ambiguity.
@@sebastiang7394introduction of ambiguity was a result of only introducing PEMDAS for elementary education. PEMDAS completely ignores the importance of implied vs implicit, doing so deteriorates the application to complex formulas commonly used in several significant fields. PEMDAS is just a lazy procedure all together, that we programmed around to make "lateral" equations the only way, whereas PEJMDAS propagates more of a 3D visualization.
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.
If I were doing this by hand, I would distribute the '2'. So 6 / 2(2+1) would become 6 / (4+2), or 1. If the multiplication symbol was supplied; 6 / 2 * (2+1) would become 6 / 2 * 3, which is, of course, 9. If I were doing it on a computer, I'd probably assume that it doesn't use the distributive property, which I find primitive.
You can distribute only after you have agreed on the order. If it was 6 / 2 * (2+1) you could not distribute the 2. You are adding an unnecessary step to hide the decision you made.
@@okaro6595 Extra step? The missing operand means distribute. It means the 2 goes with the (2+1). Otherwise, you're just missing an operation. If it's not distribute then why does the missing symbol mean multiplication? Why not division? It could just as easily be 6 / 2 / (2+1).
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.
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 + /
@@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.
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.
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.
Sorry Dave, but the correct answer is 1, Juxtaposition takes Priority over Multiplication and Division. It should be called PEJMDAS, not just PEMDAS You should always put in the Missing Brackets in the equation. 6/2(2+1) should be expanded to 6/(2x(2+1)) Always put the missing brackets in where there is a Juxtaposition.
If the equation was instead 6/2( X+Y) you would get 6/(2X+2Y) expanding the Juxtaposition with Brackets you would expand it to 6/(2x(X+Y)) which equals 6/(2X+2Y ) So doing the same with 6/2(2+1) you expand and get 6/(2x(2+1)) which is the same as 6/(2x2+2x1) = 6/(4+2) = 6/6 = 1 PEJMDAS
@@wyldanimal2 I would have answered this as 1 in 5th grade and gotten it correct. I agree, but think there is not a universally recognized convention for evaluating this expression.
My wife brought an algebraic calculator into the household. I couldn't use her calculator to balance the checkbook, so in 1978, I bought her an HP-16C, prgrammer's edition that supports binary, octal, hexadecimal, and floating point, that uses postfix (RPN). It still works to this day. Since it looks like H-P has exited the calculator, I bought the 17B, 12C, and 12 Platinum.
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.
Neither answer is wrong.* It's the question that is wrong. Usually, the point of communication is to be unamibiguous. In this case, communicating the steps required to figure out a number. These stupid PEMDAS/BODMAS puzzles are merely bad communication in a context where there are other less ambiguous ways of asking the same question. It is like saying "I like listening to Dave and eating". Do I like both listening to Dave and listening to eating? Do I like listening to Dave while eating? Do I like listening to Dave and also like eating? Since it's unclear, I'll choose a better form of words. Whichever calculation is intended, formulating the question to remove ambiguity is trivially easy, so do that, and we shouldn't treat this as any more profound than that. (* - but the only possible right answer is 9 - as Dave says, it's silly to have two semantially distinct versions of one operation, yet only one version of the other three.)
The calculator is not wrong, it is simply using a different convention (PEJDMAS) to interpret the expression, because it is ambiguous. There are a lot of videos on RU-vid by mathematicians that explain what is actually happening. A bit more research wouldn't hurt...
@@ThomasVWorm You're so close to getting it. You're just confused about which "rule" is new and which is old. I put rule in scare quotes because none of this is actually rules, it's all notational conventions. PEMDAS originated as a mnemonic to help remember the convention, but it left out a step. Oops. And now there are a lot of people who learned the convention from the mnemonic. Which means there are two conventions. One is used by mathematicians and scientists. The other is used by schoolteachers, schoolchildren, and people younger than 50 whose mathematical education ended at that level. If that sounds like it's a mess, that's because it is.
@ThomasVWorm The question is what are the rules? Clearly PEMDAS isn't the end all be all of else some mathematicians wouldn't use PEJMDAS. People were taught different methods in different areas. Certain things are absolutely true in math A=A, 1+1=2, the area of a circle is πr². If there is a proof that sets the order of operations then this wouldn't be something people still squabble over. I think the biggest issue with this is notation. 6÷(2(1+2)) is clearer than 6÷2(1+2). Also there's ÷ vs / to some people everything after / is a denominator. So I do think mathematicians should hold a council and figure out official definitions. If they can get together to decide Pluto isn't a planet they can decide an official order of operations.
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.)
It's so sad that we live in a world where it took about 14 minutes for someone to mention this. Back in my day, this would have been the very first comment.
Yes, this was the way it was taught back in the day (at least in some countries) and it was correct and is correct when following that rule. However that rule has since been changed and now implicit and explicit multiplication are equal in priority. I was also taught the old way.
You're right in saying x/2x is equal to 1/2, but only because the 2x element is a reciprocal. The difference in / and ÷ is that elements affixed to ÷ have to be converted to ×(reciprocal) to work with algebraic rules.
@@casse82 And it still is a valid way of doing a problem. The problem with this specific problem is that there's no way to know which convention that they are intending on using. Generally most people you stop using the obelus before you even get into order of operations and start using the fraction bar and visual location to make it more obvious.
@@markthompson2874 Well the consensus nowadays is that the default is to follow PEMDAS. The old way is generally now valid only if it's specifically defined to use that OR if it's used as a convention in the context that you are in (your workplace, your basement etc.).
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
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.
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.
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I haven't forgotten PEMDAS, because I never learned it… in Germany. We here only learned (the correct, mathematically) "Punkt vor String" (dot before line). And we write division as ":", not as "÷" (and multiplication is a single "·"). So, I never "learned" that you first multiply then divide, which is of course wrong (and people wonder why the US education system has a bad reputation…)
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.
@@akdm82 Fair. Here is the complicated answer. While the numbers are different, it's the same equation with the same argument. Order of arithmetic operations; in particular, the 48/2(9+3) question. A problem that hit the Internet in early 2011 is, "What is the value of 48/2(9+3) ?" Depending on whether one interprets the expression as (48/2)(9+3) or as 48/(2(9+3)) one gets 288 or 2. There is no standard convention as to which of these two ways the expression should be interpreted, so, in fact, 48/2(9+3) is ambiguous. To render it unambiguous, one should write it either as (48/2)(9+3) or 48/(2(9+3)). This applies, in general, to any expression of the form a/bc : one needs to insert parentheses to show whether one means (a/b)c or a/(bc). In contrast, under a standard convention, expressions such as ab+c are unambiguous: that expression means only (ab)+c; and similarly, a+bc means only a+(bc). The convention is that when parentheses are not used to show the contrary, multiplication precedes addition (and subtraction); i.e., in ab+c, one first multiplies out ab, then adds c to the result, while in a+bc, one first multiplies out bc, then adds the result to a. For expressions such as a−b+c, or a+b−c, or a−b−c, there is also a fixed convention, but rather than saying that one of addition and subtraction is always done before the other, it says that when one has a sequence of these two operations, one works from left to right: One starts with a, then adds or subtracts b, and finally adds or subtracts c. Why is there no fixed convention for interpreting expressions such as a/bc ? I think that one reason is that historically, fractions were written with a horizontal line between the numerator and denominator. When one writes the above expression that way, one either puts bc under the horizontal line, making that whole product the denominator, or one just makes b the denominator and puts c after the fraction. Either way, the meaning is clear from the way the expression is written. The use of the slant in writing fractions is convenient in not creating extra-high lines of text; but for that convenience, we pay the price of losing the distinction that came from how the terms were arranged horizontally and vertically. Probably another reason why there is not a fixed convention for order of multiplication and division, as there is for addition and subtraction, is that while people frequently do calculations that involve adding and subtracting lengthy strings of numbers, the numbers of multiplications and divisions that come into everyday calculations tends to be smaller; so there is less need for a convention, and none has evolved. Finally, the convention in algebra of denoting multiplication by juxtaposition (putting symbols side by side), without any multiplication symbol between them, has the effect that one sees something like ab as a single unit, so that it is natural to interpret ab+c or a+bc as a sum in which one of the summands is the product ab or bc. Without that typographic convention, the order-of-operations convention might never have evolved. When one has numbers rather than letters, one can't use juxtaposition, since it would give the appearance of a single decimal number, so one must insert a symbol such as ×, and there is less natural reason for interpreting 2 × 3 + 4 as (2 × 3) + 4 rather than 2 × (3 + 4), but I suppose that we do so by extension of the convention that arose in the algebraic context. Likewise, because addition and subtraction constitute one "family" of operations, and multiplication and division another, and perhaps also because the slant "/" doesn't seem to separate two expressions as much as a + or − does, we are ready to read a/b+c etc. as involving division before addition. But when it comes to a/bc, where the operations belong to the same family, the left-to-right order suggests doing the division first, while the "unseparated letters" notation suggests doing the multiplication first; so neither choice is obvious. It is interesting that in the 48/2(9+3) problem, the last element was written 9+3 rather than 12. If the latter had been used, it would have been necessary to insert a multiplication sign, 48/2×12, and I would guess that a large majority of people would have then made the interpretation (48/2)×12. Perhaps we will never know where this puzzle originated; perhaps it was cunningly designed so that one interpretation would seem as likely as the other; or perhaps it came up as a real expression that someone happened to write down, not thinking of it as ambiguous, but that other people did have trouble with. From correspondence with people on the the 48/2(9+3) problem, I have learned that in many schools today, students are taught a mnemonic "PEMDAS" for order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. If this is taken to mean, say, that addition should be done before subtraction, it will lead to the wrong answer for a−b+c. Presumably, teachers explain that it means "Parentheses - then Exponents - then Multiplication and Division - then Addition and Subtraction", with the proviso that in the "Addition and Subtraction" step, and likewise in the "Multiplication and Division" step, one calculates from left to right. This fits the standard convention for addition and subtraction, and would provide an unambiguous interpretation for a/bc, namely, (a/b)c. But so far as I know, it is a creation of some educator, who has taken conventions in real use, and extended them to cover cases where there is no accepted convention. So it misleads students; and moreover, if students are taught PEMDAS by rote without the proviso mentioned above, they will not even get the standard interpretation of a−b+c. Should there be a standard convention for the relative order of multiplication and division in expressions where division is expressed using a slant? My feeling is that rather than burdening our memories with a mass of conventions, and setting things up for misinterpretations by people who have not learned them all, we should learn how to be unambiguous, i.e., we should use parentheses except where firmly established conventions exist. If expressions involving long sequences of multiplications and divisions should in the future become common, then there may be a movement to introduce a standard convention on this point. (A first stage would involve individual authors writing that "in this work", expressions of a certain form will have a certain meaning.) But students should not be told that there is a convention when there isn't. Incidentally, it is worth noting that in certain cases, no convention is needed. The meaning of a+b+c is unambiguous even without the "left-to-right" convention, by the associativity of addition, and similarly abc by associativity of multiplication. By further properties of the operations, the values of a+b−c and ab/c come out the same whichever order one uses. In contrast, a−b+c and a−b−c require the "left-to-right" rule, while in the absence of a corresponding rule for multiplication and division, a/bc (as discussed above), and likewise a/b/c, are ambiguous.
