PEMDAS is wrong 

Are you saying compilers don't interpret mathematical e pressions correctly

@@Hclann1 They certainly do, if you make sure to tell them how.

​@@QuesoCookies Currently extending Java code. PMD keeps telling me I have superfluous brackets in my equations. Err, no I don't.

​​@@Hclann1 compilers do indeed use the traditional "left to right", i.e. multiplication and division are considered equal order of precedence. So in that sense, compilers (at least c and c++, the python interpreter too) are "incorrect". You cannot ommit parenthesis.

@@fernandogiongo I was not aware of this, thanks for letting me know.

Nobody uses the division operator

Totally agree. The fraction bar is much clearer and is a grouping tool like parentheses. Solve all numerator stuff, solve all denominator stuff, the divide top by bottom. Division signs are dumb

They literally have the exact same meaning. The division operator signifies the exact same thing as the fraction bar. The symbol is even supposed to represent a fraction, with the equation to the left replacing the top dot and the equation to the right replacing the lower dot. The reason why it falls out of use in higher math is solely because certain countries decided to use the symbol for subtraction, so they dropped the symbol entirely to avoid confusion.

@@GremlinSciences the use of the division bar is confusing because the fraction bar is so clear. 6÷2+3 could be clarified so easily is the fraction bar was used because this problem is NOT supposed to read (6÷2) +3 but actually the numerator 6/(2+3) When written out without the division sign (which keeps everything on one line) it reads much more clearly because the bar serves as a grouping symbol

@@swedbp1 You only have confusion because you are giving two symbols with identical meaning and usage two separate meanings. there is as much difference between '/' and '÷' as between '*' and '×' and '⋅' 6÷2(1+2) is exactly equal to 6/2(1+2) The only difference is an arbitrary distinction you are imposing that is not part of either symbol's actual meaning.

But there is no "÷" on the keyboard. I had to copy and paste yours.

And it doesn't matter anyway. The same issue remains with 6/2(1+2), or with the later story which is actually using / instead of ÷ in the 2x/3y-1.

@@irrelevant_noobon paper you should clarify if 6/2 as a whole is multiplying or if 2 is and the equation is under 6 that’s just an issue with keyboards not Order of Operations

ALGEBRA WILL KICK PEMDAS ASS ANY TIME ANY WHERE ANY EQUATION!

honestly the solidus (/) is really only used if an expression must be written inline - for absolute clarity the vinculum (--) is the notation to use - it's all I ever saw used from Algebra I onward

"we always wrote division operations as fractions, never in line". Really? Have you never had to enter an expression into a computer programming language, or even Excel? There are a few programs out there, like Wolfram Alpha that will accept expressions in your favoured form, but those are relatively rare. For the great majority you do have to be prepared to enter them in line form.

Same in physics.

@TheEulerID we only ever used Excell to makes charts and graphs from data. My college did try to heavily push Matlab while I was there, but that "program" was such hot garbage that it was easier to do it by hand.

Until you start writing entropy units.

It isn't about engineering schools.

I used to tutor prep classes for the GMAT and GRE. My examples always started with a simple problem students could follow all the way through, and then I'd show them the general form of the equation once they understood what it was for. One of my students' evals said that she'd taken Algebra in high school and college, but she'd never "gotten" it until I showed it to her my way. For example, I'd explain compound interest by showing an account through two two compounding periods, _then_ give them the formula with exponents and variables.

I mean, pemdas is just a tool to get students to the point where you can start discussing how the maths actually work. But, like, you have to teach the tools correctly or they'll never get to that point... I've known there were issues with how kids are being taught, but this is just BASIC. I can't even. I've run into your experience a lot and it makes me so mad. EVERY person I've ever talked to who professed to be "bad at math" were just taught poorly. Usually the basic concepts, and then they were rushed into stuff they had no chance of understanding and just had to struggle memorizing meaningless number combinations so they could pass tests. Every one of these people immediately "got" the math once it was properly explained to them - though I often had to work through years and years of conditioned response.

Math is like an onion, it's got layers (to quote Shrek). At the deepest levels, it becomes so abstract that even the brightest minds struggle to wrap their intuitions around it. In some ways, this is also where many advanced topics, such as abstract and linear algebra, topology, calculus, real and complex analysis,etc, tend to merge into each other. As the level of abstraction goes down, the fields drift apart and operations become more procedural. Reduce the abstractions further, and topics simply become impossible to express. Eventually, we have basic arithmetic presented merely as a few procedures to memorize. Precisely because of the increasing levels of abstraction, students tend to be taught from the outside in. For instance, you can teach complex numbers in two ways: 1) The common way, it to introduce i=sqrt(-1), and c=a+bi. This is the procedural way, introducing i totally without justifying it or explaining why its useful or how to build intuition about it. But it can be taught in an hour. 2) The intuition based approach, is to first teach group theory,then introduce the SO(2) group, represented by the matrix R=[a, -b | -b, a] and the relation to the O(1) rotation group. NOW you can introduce c=a+bi as just a shorthand for this, and work from there. But this takes the average student perhaps 1-2 more years of study, and before you reach c=a+bi, you lost half the students. Primary school teachers have the same dilemmas, just at a lower level. The smartest students tend to see most patterns without explicitly learning them, while the least talented would never understand those patterns. Those in the middle MIGHT gain a little from learning those patterns explicitly.

@@RWAKitty I was sitting in a statistics class when the prof put a formula for the line of best fit for a data regression on the board, then asked what the slope would be. I told him "Beta sub one" immediately. He asked if I could do the proof. I admitted I couldn't, but if we rearranged the formula slightly and renamed the terms it became Y = mx + b, which is just the slope/intercept formula from algebra 1. I can't depict the beta symbols, subscripts and carets properly in a YT comment, but the terms were in the order y = b + mx for some reason.

@haakoflo Your reference to primary school suggests to me that you were not educated in America. I think your statements are correct. But understand that in America, it is common for the teachers themselves to have little comprehension, of even basic algebra. They aren't executing pedagogy, to lay the groundwork for understanding. They are often just regurgitating what is written in their teaching materials, without really grasping it themselves. There is a reason this country ranks near last in the world, in mathematics education.

There is only 1 set of brackets in the equation. You got the answer 1 by injecting a second set of brackets which is not in the equation. I do not agree with PEMDAS. In fact, I never heard of PEMDAS until I was around 70 years old. The proper way to solve is left to right. Starting from the left the first step is 6/2. Because 6 is the first number, you deal with the 6 first. That is 3. The second step is 3x ( ) because (1+2) is the next number. In this case, 2x3. that is 9.

What if you typed in 6/2(1+2) without the Times symbol? Excel doesn’t even recognize it as a valid formula

@@jeromewaldemar6963 I haven't heard of PEMDAS either, mostly because I didn't grow up speaking English. Around here, the rule is "dot before dash" (because multiplication and division are written with dots). Yes, left to right, and that's also what PEMDAS seems to say, as explained in the video.

I got 1 also

@@jeromewaldemar6963 _"I do not agree with PEMDAS. The proper way to solve is left to right."_ Well that's just plain wrong, whether you accept the priority of implicit multiplication by juxtaposition or not. Are you saying that you believe 1+3×5 = 20?

These threads always attract a lot of order-of-operations trolls. It's weird, I didn't even know it was a thing.

And reportedly even Nobel laureate scientists have almost come to blows arguing about the Monty Hall problem.

Yeah, it's a bit like explaining tides and eclipses to flat earthers - pretty much pointless 🤣

@@Skank_and_Gutterboy It is a thing, and it should be utilized in linguistics as well.

@@justanotherguy469 Great, I'll go find some linguistics videos and start trolling their comments sections. Consider it done!

if you are an engineer you'd know this is not an equation, it's an arithmetic expression...

@@user-dc4xy7uk3b If you have to do crap like this to flex how smart you are, you're not.

Exactly. Anybody's engineering boss would have their red pen all over that submission. If your submitted equation can be interpreted >1 way, you're doing it wrong.

@@Skank_and_Gutterboy don't show the world how thick you are...could affect your job prospects, assuming of course you're worth employing...

@@user-dc4xy7uk3b I'm a senior engineer and have been doing that job for 24 years, I think I'll manage. Disagreeing with you makes somebody an automatic moron? Nothing arrogant about that. You're only proving my point.

Comletely agree. If, as an engineer you write expessions like this one that are ambiguous, you are a moron. This kind of video is absolutely ridiculous.

Then you missed the slight of hand she pulled. She went from an OBELUS (which is NOT universally agreed upon with respect to its grouping or non-grouping conventions) to a slash which is literally defined in the sources she mentioned as functioning like a fraction bar. And yes, brackets should be used. And an OBELUS should NEVER be used, and it pisses me off that she equated an obelus with a slash, especially because in at least the last two sources they are clearly used in place of a fraction bar to save space.

@@eliteteamkiller319 There is no functional difference between an obelus and a solidus. Both of those division symbols do not have grouping properties. Only the vinculum or horizontal fraction bar has grouping properties.

Thank god I finally saw this. I've always said that needing to use PEMDAS is the sign of a sloppy equation. I absolutely hate those "trick" math problems on the internet.

Totally agreed, especially the part about the responsibility of the equation writer to be clear. If one uses the leaning slash “/“ as you would when typing, then make the effort to toss brackets on there to convey the grouped terms of the horizontal divide sign.

Well, there is some notation corner cutting, like when you write ds^2 = dx^2 + dy^2 + dz^2 . It's `understood' that ds^2 is (ds)^2 , instead of 2s ds. Same corner cutting for the denominator of the second derivative.

@@trumanburbank6899 However, ds is not two symbols like the product of d times s, it is a single symbol for the differential of s so that there is no ambiguity in the application of the exponentiation when writing ds^2 as there would be if one were intending to write the squaring of the product d*s for which ds^2 might mean (d*s)^2 or d*(s^2).

@@luminiferous1960 Actually it is the product of the operator d times the variable s. For example d(r^3) = 3r^2dr is the the product of the operator d with r^3. Also, a = dv/dt = (dv/dx)(dx/dt) = vdv/dx = (1/2) dv^2/dx . ds^2 = (ds)^2 is accepted by convention as an exception to the rule. I do agree that PEMDAS probably wasn't designed with operator algebra in mind, but it would be on par with Parenthesis.

@@trumanburbank6899 In your first example, d(r^3) = 3r^2dr, the operator is d/dr, i.e., the derivative with respect to the variable r. In the expression, d(r^3) is the differential element of r^3, not a product of d and r^3. In differential calculus, if s is a variable, then ds is the infinitesimal element or differential element of s, not an operator. The differential element is, as the name implies, elemental. Thus, ds is not a compound of d and s, i.e., it is not d times s, nor d of s. As such, no parentheses are needed around ds, and ds^2 = (ds)^2 is not an exception, and the parentheses are simply superfluous. Your example a = dv/dt = (dv/dx)(dx/dt) = vdv/dx = (1/2) dv^2/dx is an illustration of the fact that the derivative operator of one variable with respect to the other equals the differential element of that variable divided by the differential element of the other variable, and the fact that the differential elements are treated algebraically as variables. Thus, (d/dt)(v) = dv/dt, where d/dt is the derivative operator with respect to the variable t, dt is the differential element of t, v is a variable, and dv is the differential element of v. You also did not define your variables in this example, which is necessary since you use the fact that v = dx/dt when v = velocity, x = distance, and t = time. As a physicist, I recognize the equations relating acceleration a, to velocity v, distance x, and time t, but others may not.

​@@luminiferous1960 There's just a `d` operator tho when doing implicit differentiation.

I might need an example.

which you cannot express in a computer language. Only parens provide grouping in computer languages.

@@garymartin9777 You could probably make a language that it works for, though that doesn't mean it exists.

@user-ij3we9jp5j Well that won't work for the story at 4:29... since it doesn't mark where the parenthesis end. :-|

?

....were implied brackets around the parenthesis...

I'm discovering I learned math from mathematicians. Every teacher I had in middle school had their doctorate. The only bad math teacher I had after fifth grade was my geometry teacher who didn't know how to teach me. And give me enough but I somehow passed our state's very hard geometry assessment test with a 110%... A score that should have been technically impossible but because they say choose the best answer and then they had questions that actually had no correct answer I got the best answer correct so I got credit for the best answer and the question itself.

Yes, 2(1 +2) is a grouping. Sin 2(1 + 2) ≠ Sin 2 × (1 + 2). Just as 2pi and 2theta and 3.14 are grouped.

@@markprange2430 general substitution rules for integrating and differentiating by parts if f:(n) = a÷b(p+q) then let c= (p+q) therefore f: (n) = a÷bc if a=6, b=2, p=1 & q=2. a÷bc = 1 that MEANS , f:(n) = 1 The solution should be the same no matter HOW you work it. If I can work it 5 different ways to get 1, and only one way to get 9, then that way to get 9 must be wrong.

@@markprange2430 AND yes, when you have something like 2x, or 2y , if x = 2+1 , and y = 2+1 6÷2x = 1 6÷2y = 1 6÷2x does NOT mean (6÷2)*x

I agree, inline equations are sometimes interpreted this way. But, I have seen it both ways In the literature, it is important to pay attention to the context. PEMDAS is often not used in inline equations, but this changes when it isn't. I don't remember it not being used when it was a stand alone equation.

@@SpacePhys Implicit parenthesis after the division symbol is a bad convention. A math equation should not require any context. The context forms the equation, at which point it is pure math.

Hihi I have always enjoyed explanation and argument for maths and stuff ❤

Yeah that's true, but the point she touches on is also true. When I see these kinds of questions being answered on reddit, yt etc, even adults who did not go deep into stem or use advanced maths they would use pemdas. So pemdas/order of operations is a tool for younger kids to have a better grasp on how to do comparatively difficult questions a bit easier. So the point she touches on is valid

If the person seeking an answer gave us the problem to be solved, they would tell you how to solve it. In the given example 2x/3y-1, PEDMAS could be used if you will equate ÷ with /. In the video, the presenter uses the Division Slash / to represent Division when doing the Order of Operations, but while reading the problem she said 2x OVER 3y-1. That would necessitate it being a fraction of 3(x) ÷ (3y-1) or 3(x) / (3y-1).

According to the pemdas thing it should have been (6/2)(2+1)

Nope. The number adjacent to the brackets is part of the brackets sub-equation. PakShuMan read it as pemdas/bodmas reads. This is so weird!

Despite being from America, i also got 1, as i aways do the parentheses with any number next to it (mostly because thats also technically multiplication)

Same in Czech Republic. Textbooks were never ambiguous - fractions and brackets everywhere. It was so thorough I never realized this problem even could exist until now. We were thought a trained to use brackets and fractions quite early. For me 6 : 2 · (1+2) and 6 : 2(1+2) are not a same thing and I am pretty sure every Czech child is thought that in primary school. My old Casio fx-4500P is PEJMDAS. It had to be because PEMDAS would drive me nuts.

@@woodendoorgarage I say you are correct. Those 2 equations are completely different and have different answers. That is how I was taught during my 7 years of honors math classes in the USA. It's exactly the same as when writing a sentence, a comma can change the entire meaning.

It's not that people struggle with algebra. It's that there are inconsistent conventions. Writing something in an ambiguous manner and blaming others for misinterpreting it is the writers fault.

@@Andrew-it7fbNo, he saw errors in algebra.

@@jimbarrett5930 I'm sorry that you don't understand the difference between math and convention.

@@Andrew-it7fb I do. Intimately.

@@jimbarrett5930 If you did, we wouldn't be having this conversation.

In 1979 I was taught this in 7th grade and I guess I had a good teacher, for it was ever so simple! First 'in between the brackets', then what's 'attached to the brackets'. Then the rest according to EMDAS (we of course already used EMDAS within the 'P' part of the brackets too).

A(B+C) is A * (B+C) The A is not in parentheses and does not solve during P, in PEMDAS.

@@leighz1962 Incorrect. Guess you never heard of the Distributive Law. The Distributive LAW states: A(B+C) = (AB + AC). This is actually one of the fundamental laws of Math. So the facts are the number attached to the parenthesis should be moved into the parenthesis since that is actually a LAW in Math.

It's 100% NOT a law. It's called the distributive PROPERTY of multiplication and its no more powerful than the operation next to it. There is no division or subtraction really. All of it can be reduced to multiplication and addition (of inverses) so that they are closed under the respective operations. I'm actually a little annoyed that this presenter referenced the laziness of book writers to cite that Pemdas means nothing. There IS an order of operations and it's not helpful to have something like this go viral.

Except 6÷2(1+2) would = 9. If you wanted the multiply first you'd simply add the bracket or right it as a fraction properly. Her whole explanation is bullshit. Paraphrased as "people don't write things properly so order of operations shouldn't apply". Theirs absolutely no ambiguity with pemdas. People who argue ambiguity just don't follow the order. And people that write the equation and claim to want a different order should write it properly

These are people who have over-reliance in rules. They see the rules as something absolute and not conventions. Ask them why is multiplication done before before addition and they can only say PEDMAS. The answer is simply. If you buy 5 apples à 2 € and 7 oranges à 3 € how much does it cost. It is nice if you can write 5*2 + 7*3.

I did not see any example showing ÷ symbol. Either because they don't know how to write/type the symbol or real mathematicians or people who use mathematics professionally don't use the ÷ symbol. The examples showed if there is / that means group numerator and then denominator. 6 ÷ 2(1+2) ≠ 6 / 2(1+2)

The video showed nothing. The original expression contained ÷, which is the source of ambiguity, and then immediately pivoted to using the fraction bar. They are not the same thing.

And then there are English teachers who say, "Never begin a sentence with a conjunction." But good writers aren't held to such silly limitations, are they?

@@ILoveCunnilingus that symbol ÷ (called the obelus) is no longer of use according to ISO 80000-2, only "/" is used (a consequence of it being used in most programming languages and of keyboards not having the obelus symbol I think), and it was rarely used by mathematicians and scientists either in the first place, it was mostly used in some printed books). But ÷ and "/" were always the same thing. 1÷ab was also 1/(a*b).

YES!!!! No matter how many you have to use. Otherwise that spaceship might be lost.

@@aspenrebel The Guiness Book of World Records used to list the most expensive typographic error as a multi-hundred million-dollar launch that failed for similar reasons. Parentheses are cheaper.

@@cliffordschaffer5289 yup!! Did I ask this question, somewhere? What is 6 ÷ ab? Where a=3 and b=2. The Answer: 1. But the "woke", New World Math, Globalist will tell you it is 4. "ab" is one complete term. Just as "2(1+2)" is one complete term. I cannot see anyway to write 6 ÷ 2(1+2) =, under the PEDMAS-er's Rules, to make the answer 1. Yet under centuries old pure math, the answer is 1.

You didn't learn the Order of Operations until 8th grade?? Wow. _______ 2(1+2) two grouping symbols (2(1+2)) two grouping symbols q 6÷(2(1+2))=1 6÷2(1+2) does not equal 1

​@@RS-fg5mf Can you please tell me the difference between ÷ and /

​@@RS-fg5mf please

@@suineitling there is no mathematical difference. The obelus ÷ and the solidus / are both equal signs of division and neither one serves as a grouping symbol. The solidus / is more prominent these days when writing inline infix notation mainly because the obelus ÷ is used in some countries to represent subtraction instead of division. Anther reason is that prior to 1917 some text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math that were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses when dealing with inline infix notation so the ERROR was corrected post 1917. This ERROR i.e. misuse of the obelus by the text book printing industry to try and save time and money was wrong and means that 1 is not and has never been the correct answer... 6÷2(1+2) and 6/2(1+2) are mathematically the same. Both equal 9

@@suineitling Practically there is no difference. As a rule of thumb just do what the equation tells you. 6/2(1+3). This equation says 6 OVER 2 times 1+3. Or it says divide 6 by the product of 2 times 3. Practically, if you see a division sign or slash then everything to the left is the numerator and everything to right is the denominator. Let parenthesis tell you anything different.

*_Exactly._*

÷ and / are two different symbols entirely is the issue same as A*B and AB are different (those two are more similar though). ÷ means you are doing division of A divided by B. where / marks the right side OVER the left side, which can be broken up by a *. So A/BC and A/B*C are different, but A÷BC and A÷B*C are the same (in this case A/B*C is also the same). You almost never see A÷BC though because it's ugly and unintuitive.

1s is one second 2s is 2 seconds Are you telling me that 1/2s is not half a second but 0.5hertz?????

@@NunoRVOliveira "s" is not a constant or a variable. It is a unit of measure. Also, according to the International Bureau of Weights & Measures (the BIPM), there is supposed to be a space between the number and the unit. So one second should be written as "1 s". Two seconds should be written as "2 s". And half a second, if written as an in-line fraction, should be written as "1/2 s". If the rules are followed, there is no confusion.

​​@@MilescoI would still argue it's not clear. It's not uncommon to write units in the denominator in certain fields such as Physics. The context matters.

*This* 100%. The expression 2(3) has not yet been resolved as a parenthetical operation, and is not equal to 2*(3). Just because people misinterpret the rules, does not in any way make the rules wrong... just improperly taught/understood. Ignorance of the Law is no excuse.

have seen a lot of this misinterpretation of PEMDAS and was extremely shocked to find out that even Google algorithms use the misinterpretation Its probably necessary to overload any equation or expression with parentheses to avoid ambiguity Strangely enough when I learnt BODMAS i learnt that the O stood for both Order (Exponetial) as well as Of which meant evaluate completely any expression within or adjacent to parenthesis so in this case 2(3) has to be evaluated before the division. Reading the expression would be 6 divided by 2 of 3. 2 of 3 means you have two sets of 3 but these guys have turned this completely around and decided that this statement should read 6 divided by 2 times 3 which is not quite the same thing

@@drift_ah1518Yep. This is why I used the Distributive Property in my example. This is an actual mathematical "law" where as PEDMAS is only a procedural ranking based on a set of conditions and in this case based on the "grammar" of writing it in one line rather than long form on paper.

@@drift_ah1518 adding in additional (and needlessly complicated and completely irrelevant) parentheses into a mathematical expression, only to hold the hand of people who never understood the concept loses some of the elegance math is capable of. When leading Trolls and Ogres through a Magical Forest, they are bound to mash some flowers into the mud. Better to teach them the *right* way to do it, as some of these RU-vidrs try to, with the assistance of others. And maybe fix the education systems that pass people ho don't actually know what they ere taught.

@@corwyncorey3703 lol - am having quite the debate elsewhere with someone that just wont see the logic

To a degree yes you do need PEMDAS but you also have got to remember you need law of mathematics supersedes PEMDAS first so in like this equation is the distributive law will be applied first and then PEMDAS then the answer is still one. Nobody ever remembers that..

@@nathanlarson1033 I have never learned PEMDAS at all. I have managed just fine.

What is really stupid is the new CORE math my kids had to learn, what a dumb confusing mess of nonsense !

​@@drinkoldcokeIs that the same one that required us to do black out poetry during my Physics class back in High School?

@@drinkoldcokeCommon Core is what educators came up with when they analyzed what all the countries that had high math scores and successful math programs, had in common.

I'd love to claim those mad skills, but I'm using a stylus on a yoga 260 thinkpad :)

@@THaWoM ohhh I see! 😝 Still very satisfying handwriting nonetheless! Also love your voice 🤭

No one asked

EZc1apz Rude

I'm a bit concerned that she writes the letter "x" by using two mirrored "c" characters.

And if you attempt to use a computer or calculator to assist computation for such endeavor you had better be aware that that device is likely to use PEMDAS to parse expressions. Simply do not write ambiguous expressions -- use parens to coerce an exact parsing.

It is a "fun little brain teaser" though because NO ONE writes an elementary expression like that. It's completely academic. Furthermore, I hope you meant to say you were taught PEMDAS in the Naval Academy. "If you are designing a bridge or a building, or worse - a more complex dynamic system, math matters." I agree. That's why mathematicians, scientists, and engineers don't write stupid little "trick" problems like this. Nevertheless, even as written, the expression is NOT ambiguous. It's 9.

@@garymartin9777 The expression isn't ambiguous. It's 9.

​@@garymartin9777100%. This is the actual correct answer.

That is an answer by computation alone. The correct answer is 1. If we read the equation plainly, it is simple to see how the answer is derived. 6 is being divided by twice the sum of 1 and 2.

not the best thing to dumb it down. Example 5:5+5=? vs 5/5+5=? If dumbed down to "use fractions" one can interpret as 5 divided by the result of 5+5 when 5 divided by 5 then add 5 needed.

Correct

Absolutely. Sometimes we try to write, or type equations out on one line, and it confounds things. I definitely found writing the equation out long form on multiple lines for fractions made this make much more sense.

@@AudriusN For the question 5/5+5=?, the fraction would be with the 5 in the numerator and the other 5 in the denominator, and then the +5 would be completely outside of the fraction. That or if you did want to do 5/(5+5)=?, putting the 5+5 in the denominator would also send the message that you wanted the 5+5 to be done first before the division. It is super useful and concise to use fractions in engineering math problems because sometimes you need to divide a number by the result of a bunch of variables that need to be added and/or subtracted together and it's nice to show it on 2 lines, like 10/(1+b(60/a)).

Where did Donald Knuth argue this? (Just asking for the source, not disputing it--it is certainly true.)

@@pandoorloki1232 the textbook to one his programs, METAFONT. It’s basically a programming language for building fonts, but the rules for building expressions include multiplication by juxtaposition, not generally seen in either typical high level languages or Excel. And multiplication by juxtaposition binds more tightly than multiplication by *. It’s in his METAFONTbook.

that complicates parsing and was probably rejected early on for that reason.

Mathematician: mn is mathematically equivalent to m*n! Programmer: That's a totally different variable. You could do mn = m*n. Mathematician: That's illegal! Programmer: You telling me dy/dx is three different variables?

As a programmer, I apologize for our abuse of the equals sign. As annoying as it (sort of) is to write := or similar for assignment in the languages that use it, it's probably a saner choice.

I was also taught this method

We were taught BODMAS (as you can see D comes before M) but still 6 ÷ 2(1+2) is obvious to do the juxtaposed multiplication first. Mainly because if it wasn't meant to be first then they would have written it in a way that doesn't suggest that it is first.

I was also taught multiplication first

Same here. Which makes it seem kinda weird when she says PEMDAS is wrong and yet the right rule follows PEMDAS exactly as it was taught to me. Makes me think the thing of doing multiplication and division in the order written must be a newer thing I think and PEMDAS the way we learned it was older? So it's not so much that PEMDAS is wrong, but that it's being taught to mean different things.

Same. I never knew people did the blindly following the order of division and multiplication until the internet.

It seems that if you have to 'add' parentheses to an already stated question you actually change the question. But if you actually use PEMDAS as written and not worry about the left to right issue, then in this question presented the answer still comes to 1(I seem to remember learning both ways in HS and then the traditional left to right order in college)

Look up PEMDAS rules. Ties(MD or AS) are done left to right. The answer is 9 and other answers are not correctly done using PEMDAS. NYT has an article from 2019 covering this. 8÷2(2+2) = 16.. not 1.

PEMDAS is a mnemonic for the order of operations. The order of operations must always be equal otherwise two people would get two different answers. 8+2*4 = 8 + 8 = 16 using the order of operations. 10 * 4 = 40 just going left to right. Which is correct? The order of operations is not just some random process someone came up with. Why the order of operations? P - Parentheses - groupings. If someone emphasized a group then that should be done first. a + (b - c), b-c should come before adding a. E - Exponents - exponents are just a group of multiplications. 3^2 = (3*3) M - Multiplication is just a group of additions, 2*4 = 2 + 2 + 2 + 2, thus 8 + 2 * 4 = 8 + (2 + 2 + 2 + 2) = 16. D - Division is just a group of subtractions, 12 / 4 = 12 (- 4 - 4 - 4) = 3 -Multiplication and Division are inverses of each other thus they are of the same precedence, 12/4 = 3, 3*4 = 12. Addition and Subtraction are just the last operations, the most basic so they're saved for last. In this issue 6/2(1+2), the 1+2 are first, then it comes down to 6/2(3) which is only division and multiplication, the same precedence, so they're done from left to right, 6/2 = 3, 3(3) = 9. This could also be written as 6 * 1/2 * (1+2). Multiplication can be done in any order thanks to the communitive property, this 1/2 of 3 is 1.5, 6 * 1.5 is 9. The only answer for 6/2(1+2) is 9.

@@leighz1962 PEMDAS does not treat juxtaposition properly, so it gets the wrong answer, as the video demonstrated.

@@leighz1962 Damn bro never knew the new york times, a bunch of journalists, know more than actual mathematicians! Might as well remove mathematics as a major and replace it with more courses in journalism as they clearly know more!

in the sciences the brackets that should be there are removed for laziness and the assumption that everyone who reads it should know that it is implied.

parentheses, exponentiation, adjacency, (multiplication | division), (addition | subtraction ) isn't even really different than what we generally teach primary school students. merely an extension thereof. primary school students, if my memory serves, simply never use adjacency to mean multiplication. Elementary students always use operators between terms. Adjacency doesn't hit till algebra for students since it makes no sense till you have variables. The algebra teachers merely should mention that algebra extends PEMDAS with the knowledge that the adjacency of coefficients binds tighter than operators, while any exponentiation, which isn't an operator, will still apply only to the exponentiated term.

Computer languages don't use juxtaposition because it changes the name of a variable. If you have variables x and y, then xy is not x*y, it is a compiler error caused by an attempt to reference the non-existent variable "xy".

>> you can never have too many brackets

@@FrostSpike machine code and assembly language.. Fast and efficient. no brackets.. lol just a pain in the ass professor that made sure there were no mistakes so my engineers understood everyone has to understand there work./

Actually you do not assume ....you were just taught correctly. As 1 is the correct answer.

@@JennaGibbens9 is the correct answer, if they want 1 to be the correct answer add brackets.

The "2" is part of the "(1+2)" expression, so "2(1+2)" has to be completed first. THEN 6 ÷ 6 = 1.

​@@JennaGibbenscorrect!!

​@@shiijei26389 is INCORRECT!! Brackets are not necessary, there are akready parentheses. 6 ÷ 2(1+2) is different than 6 ÷ 2 x (1+2). In the first, the 2 is part of the parentheses expression, so 2(1+2) MUST be completed first. In the 2nd, the 2 is not part of (1+2). I'm sorry that you are wrong, and I'm sorry that you were taught wrong. And I'm sorry that you have been brainwashed. But that is your problem not mine/ours. 6/2 x (1+2) = 9 is diff than 6/2(1+2) = 1. That is same as 6/[2(1+2)]. I hope you don't work in some important engineering, medical, or scientific field, and use your kind of math. Because if you do, I'm very scared right now.

Unfortunately there is no central authority to make official rules. This implicit vs explicit rule that a lot of people try to insert into these kinds of meme internet problems don't work. If there was a difference between implicit and explicit multiplication then 2x and 2*x would be different. Likewise 2(1+2) would be different to 2*(1+2). And if they were different then contracting 2*x into 2x at any point would be a terrible idea and we would have a 'rule' not to do that etc.. Our 'rules' have ambiguity, they always have and likely always will because the 'rules' are often not rules and are actually just conventions that some people make up to try to make it work better. Even the video called these 'unofficial rules' that engineers and stuff use when writing and interpreting formulae. The reason we have parenthesis and why that is evaluated first is because this ambiguity was a known problem and the parenthesis was a way to address it, but it only works when the person writing the equation uses them correctly. If a person writing the original doesn't see how it could be interpreted differently then they might not be clear enough when writing it down, which can lead to different people getting different answers. But the only person who knows the 'correct' way of evaluating the formula is the person who originally wrote it. No amount of arguing on the internet about what the correct order of operations is will get you the 'only and only absolutely correct and everyone else is wrong' answer.

@@Subjagator Let me quote myself: "I've learned it as [...]". I did not claim it's "more betterer" or in any way shape or form the LAW.

@@marian-gabriel9518 Of course. It is just a pretty common thing that comes up on these kinds of videos. Yours was the first comment that I came across which mentions different types of multiplication with different rules etc.. so I added my two cents. It is a public forum after all. I was in no way trying to straw man an argument or anything and if that was the impression I gave then I can only apologise.

@@Subjagator 2x is one expression, 2*x is an operation. Imagine 2pi would be read as 2 x pi everywhere. Then we should effectively use “tau” in stead.

@@Subjagator _"If there was a difference between implicit and explicit multiplication then 2x and 2*x would be different."_ And it is! _"Likewise 2(1+2) would be different to 2*(1+2)."_ And it is! 6 ÷ 2(1+2) = 1. 6 ÷ 2 × (1+2) = 9. The operator (the multiplication sign in these examples) serves to separate the coefficient from its variable so that it doesn't get the priority that implied multiplication gets. _"And if they were different, then contracting 2*x into 2x at any point would be a terrible idea and we would have a 'rule' not to do that, etc."_ In real life, engineers and, well, just about everyone I've ever come across, rarely if ever use actual multiplication signs, so the idea of "contracting 2*x into 2x" isn't an issue. If it were ever necessary to insert a multiplication sign to avoid the priority that implicit multiplication by juxtaposition creates (e.g. "2 × x" or "2 · x"), most likely the person writing the expression would use parentheses in that case. But contrary to your assertion, the universal and ancient convention of the priority of implicit multiplication by juxtaposition is so, well, universal and ancient, I believe it can fairly be considered a "rule". Indeed, the creator of this video has referenced two authoritative sources that say so. I don't think there's a mathematician or engineer or similar professional in the world who would say that a/bc is anything other than "a" divided by the product of b and c.

I agree, perfectly good logic.

I always saw the (x) as a way to show that the number within the brackets is seperate from the others.

There is a multiplication symbol between them it's just implied rather than written.

@@nekogod I was told to only omit multiplication symbols if the values belong together Why are you omitting symbols if there is no relation between the values?

That's multiplication by juxtaposition. It's just tha juxtaposition also applies to numbers and not just variables. I always interpret 2(3) like 2x (but x=3).

no. the inbetween operator extends to the next operator.

Correct. Parenthesis must be cleared first.

Nein.. there are technically hidden parentheses around everything. 6÷2(1+2) is also ((6)÷(2)((1)+(2))) If you are solving using PEMDAS, the answer is 9. 6÷(2(1+2)) would be 1.

If you're distributing you have to distribute the entire value, it's not 2(1+2), it's 6/2(1+2), thus 6/2 * 1 + 6/2 * 2 = 6/2 + 12/2 = 18/2 = 9.

@@J-kd2qc you've not handled juxtaposition in that solution. As the video explained, PEMDAS cannot be used when an equation uses juxtaposition - you need PEJMDAS.

It is still wrong! DM can be MD

You have to deal with it from left to right with DM because we read from left to read and if you don’t then you end up with different answers, this is not the case for AS because the order doesn’t change the overall answer, it’s the same regardless of whether you do A first or S first.

I learned multiplication and division are in the same hierarchy so you do left to right, not multiplication then division, just left to right. think about 4/6*3, it wouldn't be 4/18, it would be 2, so the rules do infact matter. also it was only for 6/x, where x is 2(1+2), and the new rule states "when slashing a fraction, this is the accepted order of operations: raising to a power, multiplication, division, addition and subtraction" it's far different than the way we would do other equations without slashing fractions because pemdas/bodmas always do parenthesis, exponents, multiplication and division, addition and subtraction. left to right also only apply for multiplication and division, and addition and subtraction.

So with 6 ÷ 2(1+2) we get rid of brackets (parentheses) to get. 6 + 21 + 2 = ? Correct? Which, accordung to, PEDMAS, would give 6 ÷ 21 = 0.2857, then + 2 = 2.2857

@@aspenrebel no wrong, you are not using BODMAS/PEDMAS, you need to follow the order. I have no idea where you get 6 / 21 + 2 from 6 / 2(1+2) = 6 / 2(3) = 3 x (3) = 9

I somewhat disagree. Overusing brackets can make an equation a lot less readable, which is why in papers and textbooks in fields utilising a lot of maths you will find xy/zw rather than something like (xy)/(zw). As far as the maths itself is concerned there is no real "proper order". So long as everyone agrees with the notations and conventions used in a field, it's all good.

@@adamlindstrom5750 there is no such thing as "overusing" brackets. you either need them or not. they solve a problem. left to right is the proper order is brackets are not used.

@@adamlindstrom5750 unfortunately, the key problem is no one can agree! Everyone was hot different methods, and I don’t think we should call anyone method wrong. I may have been taught pemdas, but people might’ve been taught different. The issue comes with the fact that since people aren’t at the same things, we need to be more clear about what the problem wants using parentheses/brackets/braces/and an actual fraction, not just a slash cause that gets confusing too. Because the slash was sometimes taught to people as a division, but for other people it was taught as a fraction (division using grouping).

@@adamlindstrom5750 You're essentially saying it saves time by not writing extra brackets. However, what if you factor a fraction? Let's say you have 20 and want to factor 1/2 from it. You would now have to write (1/2)40, instead of 1/2(40). Either way you do it it'll require the use of brackets somewhere, so this doesn't save any time, or readability.

@@mariuspuiu9555 brackets are not needed when you have juxtaposition, so using them would be an overuse. If you use brackets to communicate to people who were not taught the correct order, you should also consider that your audience might not understand brackets.

Is it a casio calculator? Because I know casios do that autocorrect thing, but it'd be very interesting if other manufacturers are adopting that behaviour.

@@THaWoM yes it is. Would be interesting to know if the other manuefacturers do it too as well as the programmable models, like Texas.

@@THaWoM All proper calculators follow the PEMDAS (etc.) rules these days - these rules are built into them, and they use a stack to make it possible to delay intermediary results until they are needed. What you have to watch out for is old and / or very cheap calculators without these features, which can only handle two operanda and one operation at any one time.

I did the same.. I just assumed I'd make an easy mistake.

Yeah, I'm not very well-versed in math but wouldn't PEMDAS literally make the answer 1? Parentheses first (3), then Multiplication (6), then Division (1). I don't see how PEMDAS could possibly lead to the answer being 9. It seems like willful wrongness.

This is why I hate PEMDAS. Such an awful thing that in order to have multiple factors in a denominator we need parens. If it were up to me, I would just go with prefix or postfix notation. If, for example, you have an expression that is definitely the sum of two smaller expressions, then prefix is good because you can just slap down a plus sign before jotting down the smaller expressions. If you have a bunch of small expressions that you’ll be somehow combining into one large expression, postfix allows you to jot down the small expressions first and then tie them up with the necessary plus/minus/multiply/divide symbols at the end. Prefix is also natural for programming because we call functions in prefix style. However, I’m also partial to postfix because postfix math naturally evaluates from left to right while prefix evaluates naturally from right to left. To explain evaluation direction, if you tried to evaluate prefix notation from the left, you would need a stack of operators and operands as you scan right, waiting until there’s enough operands ahead of an operator to perform an operation. If you evaluate prefix from right to left, you only maintain a stack of operands, and every time you run into an operator you immediately perform the operation, as if there aren’t enough operands available for the operation, you just have an incomplete expression. So the evaluation order for post fix makes more sense if you want to evaluate left to right. If you’re lisp, you’ll still need parentheses because of functions with an arbitrary number of args, but if you require functions to have a fixed number of args, you don’t need parens.

I just use parenthesis, to avoid possibility of error and especially, method of documenting my code.

when I deal with math equations in programming, I will do it across several lines. Each line doing an assignment that will be used in subsequent lines. I take a different approach in Hastell. Within each line of the calculations, I try to make everything as unambiguous and succinct as possible. Indeed, to avoid using too many parens, I may resort to throwing in a few extra lines.

@@J7Handle I also like postfix in a calculator. But really, you're explicitly defining the order of evaluation : you're doing the parsing mentally. It's not the use of postfix that gives the right answer, it's that you're parsing the expression rather than the calculator.

@@theelmonk I can trivially write code to evaluate a postfix expression. Read from left to right, putting operands on a stack. If you encounter an operator instead of an operand, apply the operator to the operands on top of the stack (note that the operator must have a fixed number of operands, so the binary minus sign and unary negative sign can't use the same character), then replace the operands on the stack with the result of the operation. So 1 2 3 + - would evaluate as 1, 2, 3, then + takes the 2 and 3 and replaces them with 5, so 1 5 - follows, then the - takes the 1 and 5 and makes -4. Prefix is evaluated in the opposite direction, that is, starting from the right side of the expression, which is a little weird, but is standard for functions in both math and programming languages (we do f(x) and print("Hello world!") instead of (x)f or ("Hello world!")print). This is because reading prefix from left to right, while a bad idea from an evaluation perspective, is really good for giving us the big picture of the expression rather than having to dig through the nitty gritty details first. Infix is awful in so many ways. It only works when you have the superscript exponents, multiplication by juxtaposition, proper divisions that physically put the numerator and denominator over each other, and subscripts to allow more variable names without running out of symbols to use (since juxtaposition kind of prevents variables with more than one character in their name). In simple text with none of those features, prefix and postfix are absolutely superior to infix.

Yep; "2(1+2)" is analogous to "2a."

I was taught that the divisor symbol is a relic of the type-writer era. The fact is that we didn't have tools like LaTeX so we needed an easy way show above the line and below the line for division. Feels like this point is keeps getting skipped over by everyone explaining the division symbol on youtube. Also, the example at 5:30 is given by a teacher that shouldnt be teaching. 2x/3y should be written as 2xy/3 if the answer has any hope to be 11.

Or, to get the answer of 9, just put in an explicit multiplication: 6 / 2 * (1+2) No need for the extra parentheses

@@JK-tq5oe You arent making things less ambiguous with the * operation. Write the damn equation better. 6(1+2)/2 or 6/(2(1+2)). Super simple.

​@@RaspKNo it is not, because there are no variables, and the parentheses' purpose (to state a quantity) is fulfilled after resolving the quantity within them. Therefore, they are to disappear and no juxtaposition is applicable.

As a programmer I would say that parentheses remove all doubt.

The implied multiplication is not connected to the brackets. Sure brackets allow its use: you can write 3(2) but not 32 to mean 3*2. You also can use implied multiplication with variables. If you had x=3 and did 6/2x you still would get the difference even though parenthesis are nowhere.

@@okaro6595 you actually can't write 3(2), implicit multiplication must always contain a symbol (both from usage, as you can see in the examples she provides, there are always symbols/variables in implicit multiplications and also from the ISO 80000-2 which defines the rules for mathematical and scientific writings). This internet meme is just improper writing and the answer should be "no value" :). But of course if somehow you *have* to decide what value it should have, any mathematician would reluctantly interpret the 2(2+3) as an implicit multiplication and answer 1.

Thank You ! 53 here and this was junior high math class for me as well !

I've had same rule implemented, PEMDAS but not until you have parenthesis eliminated and that pretty much solved the problem

I agree completely. I graduated high school in 2007 in the increasingly digital age, all our tests and exams were written on PC and printed out but everything in math was formatted correctly to avoid this problem such as 1 ------ = y 2x but I always understood 1/2x=y to be the same thing because otherwise it would be written as x/2=y

In mathematics you avoid things like "a/bc". Not a correct way of writing down a mathematical expression! This is creating confusion because in the early days printing of the correct way was often not possible and took too much space. So pemdas etc. is mathematical correct and so you write a ------ b c

OK, but isn't 5 (6 + 3 x 4 / 2) reallly PEMDAS instead. It gets much worse. How about 5 (6 + 3 x 14 / 2 (3x5) / 5)

​@@jeanpauldelauw7716It's incorrect only in the eyes of the undereducated.

Get lost! @@sqlexp

Actually when you see a parenthesis preceded by a number; It mean this number has been removed from the parenthesis and to properly evaluate the parenthesis it must first be distributed back. 2(1+2)=(2+4)=(6)=6 Or 2(1+2)=2(3)=(2*3)=(6)=6

Nowhere in PEMDAS order of operations is outer multiplication of parentheses stated as being done with P. It is the the MD portion. 6÷2(1+2) = 6÷2×(1+2) = 9 6÷(2×(1+2)) = 1

@@leighz1962 PEMDAS does not cover juxtaposition. When there is juxtaposition you need to use PEJMDAS.

So did you answer 1 or 9

I've been thinking of doing a follow-up video on the various ways that calculators and computer algebra programs deal with expressions like this, because the variety of rules they use is quite interesting. Some don't allow implicit multiplication, some put it at the same precedence as explicit multiplication, some put it higher, and some give implicit multiplication a higher precedence in some situations but not others (wolfram alpha for instance treats 1/2x differently to 1/xy). I've found that the Casio and HP calculators available to me locally evaluate implicit multiplication before division, but the TI one doesn't, and I'd be interested to find out why TI switched over given that they state on their website that implied multiplication has a higher priority when writing on paper (see epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 ). I'm familiar with Cajori's statement, but note that he was only talking about expressions that involve the × symbol, not ones involving implied multiplication by juxtaposition. I know your stance is that it's ambiguous and I'm forced to agree with you because there is debate around the answer to 6/2(1+2), but the only reason there is ambiguity is that primary school teachers are pushing an oversimplified version of the rules widely used by those in STEM, and the real rules are rarely stated explicitly. We don't have to stop at asking "How is it interpreted?" but can also ask "How should it be interpreted?" and then adjust teacher training and textbooks accordingly.

That would be lovely. Can not wait to see that.

@Oliver Knill I have no problem agreeing with you that Florian Cajori is the best authority in the HISTORY of Mathematical Notations. Too bad he is not the highest authority in setting standards in mathematics in the USA where the confusion as to whether implied multiplication should be given a higher priority in calculation than division has become an epidemic. As far as I know in the tradition of science and engineering calculations, implied multiplication by juxtaposition has always been given a higher priority than division. That means in the expression 6÷2(2+1), 2(2+1) should be considered as a product with the factors of 2 and 3. That also means in the expression 6÷ 2(3), the number 6 should be divided by either the product of 2 and 3, i.e. 6, or by both factors 2 and 3 instead of divided by 2 only. Therefore, all following calculations are correct: 6÷2(2+1) = 6 ÷ 2(3) = 6 ÷ 6 = 1 6÷2(2+1) = 6 ÷ 2(3) =6 ÷ 2 ÷ 3 = 3 ÷ 3 = 1 6÷2(2+1) = 6 ÷ 2(3) =6 ÷ 3 ÷ 2 = 2 ÷ 2 = 1 because a ÷ b(c) = a ÷ (b*C) = a ÷ b ÷ c = a ÷ c ÷ b In other words 2(2+1) is always equal to (2 (2+1)) However 6 ÷ 2(3) ≠ 6 ÷2*3 In 6 ÷ 2(3), 2(3) is the divisor. In 6 ÷2x3 , only 2 is the divisor. In other words, 2(3) means more than multiplication of 2 and 3. The use of parentheses implies the 2(3) must be treated as a product that would become a divisor when there is a division sign before it. Remember 2(3) is always equal to (2(3)). But 2*3 or 2x3 does not equal to 2(3) or (2(3)) when there is a division sign before each one of them. *It is a wrong idea to think that parentheses is for representing multiplication only when it also has the function of grouping.* In the expression 1÷2π, 2π is the divisor instead of the number 2 only. When the Common Core teachers do not follow the long tradition of giving higher priority to implied multiplication by juxtaposition than division, they treat 1÷2π as 1÷ 2 x π, their result would be in conflict with the order of operations being used in science and engineering. 2π is always equal to (2π) or 2(π) but 2*π or 2xπ does not equal to 2π, (2π) or 2(π) when there is a division sign before each one of them. Again implied multiplication by juxtaposition like 2π has the function of grouping as well as multiplication. Of course, implied multiplication by juxtaposition without the use of parentheses cannot be used on two numbers like 2 and 5 or 2 and 3.1416 because 2(5) is not equal to 25 an 2π is not equal to 23.1416. When an engineering project is worked on by different engineers, some uses SI units but some use the British units, the result would be very confusing. However, as long as each engineer specifies the units being used instead of just mentioning the numbers, they can still convert the units to the same ones agreed on in the final phrase. However, when a crooked leader tried to sabotage a project has caused some engineers to disobey the traditional rule of giving higher priority to implied multiplication by juxtaposition than division in interpreting the data given and in calculation while some other engineers follow the traditional rule of giving priority to implied multiplication by juxtaposition, there is no way to ensure that no mistakes will be made because there is no way to bring their works together when we don't know which order of operations they had used. Even if each engineer states which system of operations he has used in his calculations, we still need to know if the data supplied to the engineers were from someone who follows or disobey the traditional order of operations. For example, if one of the data for calculation is 1÷2π, we don't know if it means 1÷(2π) according to the traditional order of operation or it means 1÷2 x π according to the new instruction of the crooked leader. That means we cannot be sure if our calculation is correct according to our preferred order of operations if the preferred order of operations of the person who gave that expression of 1÷2π is not known. The fact that different models of calculators, even from the same manufacturer, may be programmed with different order of operations, without specifying the order of operations being used in the calculator makes the bad situation even worse. Since almost all science and engineering documents mandate giving higher priority to implied multiplication by juxtaposition in calculations, the only way to correct the present confusing situation is to ban the common core nonsense of not giving a higher priority to implied multiplication by juxtaposition than division because there is no way we can change tons and tons of science and engineering documents existed before the common core nonsense to conform to the stupid and wrong PEMDAS ONLY operation that ignores the rule of giving a higher priority to implied multiplication by juxtaposition than division. The leftist politicians who hate the USA know that a unified country is a strong country and a divisive country is a weak country. That's why they tried to force the the American children to use PEMDAS without teaching them about giving a higher priority to implied multiplication by juxtaposition than division, which is still being used in engineering and science, in order to create confusion and conflicts among the different generations and different groups of people for the purpose of weakening our strength and unity. PEMDAS without juxtaposition should be history just as Obama's other Common Core nonsense should be history in the USA.

Earlier I found out that Google Chrome has partially hidden my comment on Oliver Knill's thread but it is not censored on Firefox.The Common Core cohorts promoting the PEMDAS ONLY nonsense who have also hijacked my connection to Google are scared of the truth being exposed.

@@THaWoM Here is a note from Wolfram alpha, you should read it, it's about the division symbol when used in 'in-line math' : mathworld.wolfram.com/Solidus.html "Special care is needed when interpreting the meaning of a solidus in in-line math because of the notational ambiguity in expressions such as a/bc. Whereas in many textbooks, "a/bc" is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages such as the Wolfram Language, it means (a/b)×c. For clarity, parentheses should therefore always be used when delineating compound denominators." So even Wolfram alpha say that YOU should check if parenthesis are needed for the expression to be calculated properly. It also say that they know that for such expression the result they give may be wrong !! You may read too that document from a Berkeley professor math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html It is a document about the order of operation and the expression 48/2(9+3) On a french site, I found out a reference to this document (discussion in french writen by a french scientific researcher, from the CNRS [ french acronym for National Center of Scientific Research] about the need about a new convention of calcul or 'Does the mathematician still have a say in the evolution of the calculation conventions of the secondary school?') images.math.cnrs.fr/Une-nouvelle-convention-de-calcul.html In that page, the author talk about his exchange with Professor Bergman, who said that prof. Alan Schoenfeld, « who is the intersection of the Math Department and the School of Education here, also agrees that such expressions are simply ambiguous ». And Alan Schoenfeld is a math professor at Berkeley gse.berkeley.edu/alan-h-schoenfeld Concerning the calculators, you may find some online casio calculator manual. In them you may find their inner 'order of operation'. You'll find that for some implied multiplication is to be evaluated at step 7 and general multiplication/division at step 10 but in other, they are both evaluated at the same step... I know that when I was at school (in France), one of the check to have an 'school approuved calculator' was if 1/2Pi was interpreted as 1/(2*Pi) (Good) or (1/2)*Pi (Wrong) But mainly a special course is present in the math book (still now) to check how 'YOUR' calculator respond to several cases, just to be sure HOW to enter YOUR math expression in it later to have the result that YOU want. (and not re-interpreted differenly by your calculator)

"Nah, I am lazy and identify as right. You are wrong and need to change."

Adding parentheses changes the equation. 6/2(1+2) =/= 6/(2(1+2))

Just swap the divided by 2 with a multiplied by 1/2. U r welcome.

​@@om-qz7kpIt is so simple. We are just following the rules. Thanhs

@@harrymatabal8448 that is what I said. Have u understood my comment?

On a keyboard, curved lines ( ) are parentheses. Cornered lines [ ] are brackets.

@@anchorskid We didn't have computers when I went to school in the 1950s here in the UK. And we called them brackets at the times.

@@anchorskid in the UK we use BODMAS - Brackets, Orders, Division, Multiplication, Addition, Subtraction. So they're definitely brackets here, and the wikipedia article for Bracket agrees. Even though BODMAS specifies division before multiplication, and juxtaposition is not included, we always multiplied juxtaposed symbols as a priority.

@@shaunpatrick8345 We use parentheses for these symbols ( ) wherever they're used. These [ ] or these { } are brackets.

Has grade school math changed this much? Because this is how I was taught in grade school in the 80's.... I never even heard of PEMDAS until I was an adult.

@@antonioliles5027 as someone who went through 2000s grades school, math at an elementary level has been slowly made easier with different dumb and lengthy methods. Nowadays time tables arent even memorized because it takes memorization and teachers don't spend enough time worrying about helping that be memorized. From what I see what most grade school teachers use to teach math is some weird method where you put numbers into boxes and somehow multiply them/divide them. Math on an elementary level has basically become something entirely based on how a teacher teaches instead of actual formulas or methods.

@@geodom85yt40 You just described common core... yes it is horrible.

This I how I was taught 40 years ago and I went on to get an honours degree in maths. The answer is 1.

I remember getting taught this ruling in primary school(UK) during the 90s. This was when we were learning BODMAS. It was explained at the time that you had to do the multiplication at the same time as removing the brackets as it was considered higher on the order of operations than regular multiplication. I also remember very distinctly as a ruling because they purposely gave us questions with stuff like 2( and 2*(, just to make sure we were paying attention.

Different languages have different rules of precedence for example C vs. Pascal. In anyway PEMDAS does not even begin to address the complexity to operations in programming languages.-

The added set of parentheses really is the key to knocking off the ambiguity.

FORTRAN and other languages supporting mathematical expressions in a similar way will not accept an implicit multiplier anywhere. You must put in a multiplication operator (typically *) or the example will not be accepted. Once you do, it becomes easy to see how the expression is evaluated left to right with a "PEMDAS" precedence. If you intend different grouping, you need to show it with additional parentheses. FORTRAN furnished a method of stating any explicit calculation unambiguously, which made the engineers happy. It did not provide for a mix of implicit and explicit calculation operators and would give you an error diagnostic if you tried to compile that into a program, so did not make lazy algebraists (intending a nod to blackboard style notation) happy. Having a masters degree in computer science, my take on this is "What's the big fuss. Write what you mean, using the rules of the language, and if someone gives you an incomplete expression, tell them so and don't proceed without clarification." This is not being passive aggressive or obstructionist. This is being a computer, a machine, which promises nothing but to be a machine.

A well formatted paper will arrange to format a blackboard style equation when intended, even if it has to incorporate extra space to do so. The viral example is simply an unhappy mix of the two.

PEMDAS is a subset of PEJMDAS. Computers use PEJMDAS just like all the professional bodies she cited, but because they can't use juxtaposition PEMDAS works too.

Great video. And very thorough going into some official sources. I would also add that the parentheses/grouping do not necessarily have to occur first, though I agree that there's nothing wrong with presuming that they be carried out first. For example, in 2 + 4(3-1) + 5*7, there's not actually anything wrong with multiplying the 5*7 first. And it's even okay if you want to then add the 35 to the 2, all before you do anything inside the parentheses. The important thing is just that, whenever you do get around to the 3 and the 1, that you realize they are grouped together (i.e., you can't just multiply 4 by 3). The points you made are more important, because they actually change the interpretation of the expressions themselves, but I also wanted to point out the parentheses issue because I see it very often in school math that students think you absolutely have to follow a certain order to the steps in mathematics when really there is more flexibility than they realize.

Not

Never. Bodmas only

I foresee if bodmas doctorine tinkered with,it can entail far fetching adverse implications in wherever math is applied. Technology in wherever form n manifestation is dependent on math n can fall prey to such whimsical and imaginary ordained doctorine 3×2=6;=3(2); 6÷6=6÷3(2)pemdas 6÷3=2×2=46answer. Isn't it farce 4r math otherwise a p0le star subject. 2×2=4 unimpeacheably can never be 5

@The How and Why of Mathematics Thank you so much for your "PEMDAS is wrong" video which gives very convincing reasons and examples from engineering and scientific documents on why PEMDAS is wrong. I have been suspecting that this confusion and the conflicts associated with it caused by PEMDAS which does not give priority to implied multiplication, such as 2π in the math problem 1÷2π and 2(2+1) in the math problem 6÷2(2+1), was part of the sabotage on the American education system with implementation of Common Core enforced by the leftist politicians led by Obama. The support of that wrong practice by many commenters who happen to share the trait of having totally empty channels only makes my suspicion deeper. If you click on the usernames of the commenters who support the wrong practice of not giving priority to implied multiplication even though it is still being used in science and engineering, you can see that almost all of them have empty channels, which is a sign of them being shills who are not interested in anything other than shilling for the Common Core nonsense that the leftists politicians have forced on the American children. Here are the names of the shills for the Common Core nonsense under this video: Brijmoham Sharma, Daniel Klein, Harpoxy H. and Maithily Tandabany. These shills should be ashamed of themselves for being cohorts of the leftist politicians who used Common Core as their political weapon in sabotaging the American education system by forcing American children to accept their Common Core nonsense.

As long as you know the bias, which tries to take over, you can simply ignore that bias and concentrate on the brackets first. You'll never get wrong outcomes again :)

people are nitpicking with their strict convention but yes when you read 1/2x you probably read 1/(2x) without thinking about it or it would be 1/2 x or x/2

this is correct. Am just shocked but not entirely surprised that Google algoriths also follow the wrong interpretation of PEMDAS as "explained" in the video. I will never trust Google or Android calculators and will have to insert unnecessary parentheses to avoid wrong answers

Mathematicians don't write "1/2x" like that for precisely that reason. It's not ambiguous, but it's too easy to misinterpret. As written you have 1/2 of x, not the reciprocal of 2x, despite what you may have been taughtt.

@@drift_ah1518 The expression 1/2x is half of x, not the reciprocal of 2x. This is the only interpretaion of the order of operations.

@@herbie_the_hillbillie_goat Technically, if you can interpret it a different way, a different interpretation exists. Whether it's misinterpreted can only be determined by the author of the problem, perhaps they're the ones who miswrote it. I find 1/(2x) to be a more natural representation, because implicit multiplication seems like scaling. A scaled value is still a single value, not a multiplication. It's a "thing that's twice the size of x", but not "two times x". If it was the latter, it should be explicitly written as "2*x". But then again, everyone uses the fraction representation to avoid misinterpretations. It's ambiguous, neither is right or wrong, it depends on consensus and there is no such thing in this case.

@@Ignas_ The debate around the expression 6÷2(1+2) often boils down to the interpretation of implicit multiplication. According to the widely accepted oerder of operations, multiplication and division should be performed from left to right. Therefore, 6÷2(1+2) would be evaluated as 3(1+2), which equals 9. I understand the temptation to view implicit multiplication as a form of scaling. You suggest that 2x feels like a single, scaled unit rather than 'two times x.' However, this subjective interpretation doesn't override the existing mathematical framework. In this framework, implicit multiplication is no different from explicit multiplication: 2x and 2*x are the same. Moreover, you question why 2 should be grouped with x but not 1/2. In both cases, the numbers are being used to scale x. From a mathematical standpoint, there's no reason to treat 2 and 1/2 differently. They are both numbers that scale x, whether implicitly or explicitly. It's true that ambiguity can be resolved by using fraction representation or parentheses, but the need for such workarounds highlights a lack of clarity rather than a genuine ambiguity in the mathematical rules themselves.

No. answer 9. even with variables.

That’s why there arent variables

@@adamwalker8777 it's 1 and 9

@@random22453 9

PEMDAS is a mnemonic, it is no more of a "Lie" as is E=MC^2 because obviously, PEMDAS doesn't include the exceptions and neither does E=MC^2 include resting mass. So, of course there are a lot of exceptions depending on how a person decides to handle implicit multiplication and implied parenthesis for fractions vs pure left to right... A*B / C*D is handled differently when it appears as a fraction because there is an implied parenthesis for the denominator, as opposed to A*B÷C*D where there is no implied fraction and order of precedence is simply left to right. And depending on the teacher, some include that in their lessons and some don't. I don't really see this as a disaster for PEMDAS or any other mnemonic, it simply means that teachers and students need to be aware that there are subtleties in the rules and depending on the level of math (ie the scope) of knowledge, they might have to adjust the order of operation. The smart move is to ensure that parenthesis are in places where you want your precedence to be highest, but basic math questions leave a lot of that as part of the question to gauge the student's level of understanding for the basic concepts. I think this video points out some flaws with PEMDAS, but the bigger issue is integrity... the thumbnail claims PEMDAS is a lie... as usual in order to boost views everyone shouts "This is a lie"... no, it's not a lie, it's click bait to the rescue.

IT does not matter. The problem is that people are using PEDMAS wrongly. Some are coming up with a 6/2*3 . There is no 3. It is 6/2(4+2) or 6/2(2+4). In 2(1+2) the 1+2 is commutative and can be swapped to 2(2+1).The 2 gets distributed to the 1 and the 2. It is 6 /2(4+2) or 6/6

@@detroittigersandotherbaseb7220 wtf are you saying there? 6/4+2 = 3.5, and 6/2+4=7, neither of which are the correct answer...

@@irrelevant_noob Oops, I left off the 2() and just had the additions. I fixed it.

Not to put a fine point on it, but maybe lazy instruction at the starting level is part of why our education system sucks. It's easy to learn something when you're a child but unlearning something or relearning something is very difficult. IMO, math equations are a precise set of instructions for calculations and should never be a question about it. As you said, use appropriate parentheses.

Juxtaposition does NOT indicate grouping! Juxtaposition simply indicates multiplication between the two quatities being juxtaposed, as if a multiplication symbol (e.g. "*") had been explicitly written between them. So 3x means "3*x" not "(3x)" and a/bc means a÷b*c = (a/b)*c not "a/(b*c). However, the horizontal fraction line, called a "vinculum," DOES indicate grouping, both above and below the horizontal line. Thus, mn/ab simply means m*n÷a*b = m(n/a)b. This is NOT the same thing as writing a horizontal fraction line (vinculum) with mn in the numerator and ab in the denominator, which means (mn)/(ab). The essential misunderstanding is that the slash symbol used for division in command line interfaces is NOT equivalent of the horizontal fraction line (vinculum) symbol commonly used when students do math with paper and pencil: The former is NOT a grouping symbol; the latter IS a grouping symbol. It was the widespread adoption of command line computer interfaces - wherein vinculums are not available as symbols (as they are when doing math with paper and pencil) - that screwed everything up. Graphing calculators just solved this problem, recently. Newer editions of the Ti-84 graphing calculator operation system, using "Math Print," now allow the on-screen display of vinculum symbols to indicate division (and fractions) exactly as students have traditionally done using paper and pencil. Doing so has completely removed the ambiguity, confusion, and errors caused by non-equivalent use of "/" in place of a vinculum to indicate division. en.wikipedia.org/wiki/Vinculum_(symbol)

@@chrisborland4972 While that may be the case in the area you're from or the circles you frequent, that is not necessarily the case everywhere, as the multiple examples provided in this video illustrate. Among multiple professionals, the usage is obviously different from the one you describe. I do personally agree with your interpretation, but the only proper answer is: the usage of the symbols is not uniquely determined, and therefore the "instruction" is a priori ambiguous. As with any text, the meaning is contextual, and you must learn to become acquainted with the author's meaning from the surrounding text. But, as you mention, an author who is not certain of their readers' abilities should therefore use parentheses or the horizontal fraction line to avoid ambiguities of interpretation and thus confusion for uninitiated readers (such as elementary or secondary students).

@Flutesrock9800 Thanks for your thoughts. However, I strongly disagree, and I think I can prove you're in error. Without a universal standard as precisely what mathematical symbols do and do not mean, how are we to cooperate on anything mathematical? How are we to relate mathematical ideas and results? How are we to understand each other and collaborate? There would simply be no way to do so. In fact, mathematical symbols (like “/“) and conventions (like juxtaposition and the order of arithmetic operations) ARE standardized - and those who do not conform to these standard and well-established usages of math symbols and conventions are just doing their work incorrectly - whether or not they happen to be “professionals.” The simplest way to test this is to use any computer programming language to assign values to the variables in the expressions I listed, ask the computer to evaluate those expressions with the given values, and see what value the computer returns. In each case, the computer will return the value that I have stated is the correct value. For instance: If m=2, n=3, a=4, and b=5, mn/ab (or the equivalent m*n/a*b) will be calculated by any computer language as m*(a/b)*b and the value returned will be 7.5. It will NOT be calculated as (mn)/(ab), which would instead return 0.3. If it weren't for agreed-upon standards defining correct usage, it would be impossible for different computing machines programmed according to different "usages" of mathematical symbology to share calculations productively (since each would return different values under identical conditions), and the information age would never have arrived. Those who do not employ standard usage of symbols and conventions are being unrigorous, unprofessional (even if they happen to be “professionals”), and certainly unhelpful. At best, non-standard usage exposes a habit of sloppiness and laziness on the part of the writer. Mathematicians and computer scientists have decided upon standard conventions and clear definitions for mathematical symbols. These standards define correct usage. All other usages are incorrect, as demonstrated by the simple programming experiment I’ve suggested above. The verdict is in. There is one - and only one - correct way to employ mathematical symbols and conventions. No matter what “multiple professionals” may or may not do, “alternative” usages of mathematical symbols and conventions are just wrong. Full stop.

@@chrisborland4972 As an engineer with a PhD, mn/ab≠m*n/a*b. mn/ab=(m*n)/(a*b).

Formally?

Nonsense. Mathematicians are communicating with likewise educated (at least college grad in the sciences) people using a convenient standardized notation, and that notation has a lot more symbols and conventions than the few you learned in elementary school. Even if mathematicians tediously parenthesized everything, making it far less readable, mathematically uneducated people still wouldn't be able to make heads or tails of it.

Other interpretations are only PLAUSIBLE when rules aren't CLEAR! I don't see how the fuck this kind of shit is so hard for people to understand! But when influencers do stupid shit and convince morons that the correct way is wrong everything goes to hell!

That is not needed. As long as you understand that a(b+c) = ab+ac then you will never find any ambiguous equation ever again.

@@ihadtochoosethisuser a*(b+c) = ab + ac True? If True then: a*(b+c) = a(b+c) True? If True then: 2x/a*(b+c) = 2x/a(b+c) Still true? Or do we need to start adding in more brackets at some point to clear up some ambiguity? 2*x = 2x? True? If True then: 2x/a*(b+c) = 2*x/a(b+c) = 2x/a(b+c) = 2*x/a*(b+c) True? Unfortunately our 'rules' have ambiguity, especially when the person initially writing the problem is sure there is only a single way of interpreting it. Anyone who claims that there is only one true and correct way of evaluating an ambiguous statement needs to stop because those are the types of people who will end up causing ambiguity by failing to see different potential interpretations when writing down their own formulae.

@@SubjagatorYou're intentionally breaking up perfectly implied multiplication in order to introduce unnecessary ambiguity. No one who is trying to actually do arithmetic or algebra would ever do this; you're just trying to win points on the Internet. Do you need a cookie that badly?

The obelus and solidus have the same meaning, both mean divide the number to the left with the number to the right. You don't divide more than that number unless the next character is a group. a/bc = a/b*c = ac/b = a*c/b.

Was it University of Illinois Committee on School Mathematics? I just want to make sure I'm finding/researching the right material. I couldn't find anything under a search for University of Illinois Course on Secondary Mathematics. When I searched for UICSM it returned results for the former, not the latter.

@@phoenix2634 The Internet Archive contains complete texts for the 9 volumes of the 1959 edition of _High School Mathematics_ by the University of Illinois Committee on School Mathematics (Max Beberman, Director; Herbert E. Vaughan, Editor). A description says: ---------- Each unit comprises Student's ed. and Teachers' ed., interleaved Unit 1. The arithmetic of the real numbers.-- Unit 2. Generalizations and algebraic manipulation.-- Unit 3. Equations and inequations.-- Unit 4. Ordered pairs and graphs.-- Unit 5. Relations and functions.-- Unit 6. Geometry.-- Unit 7. Mathematical induction.-- Unit 8. Sequences.-- Unit 9. Elementary functions: powers, exponentials, and logarithms ---------- The archive also has editions of some of the volumes from other years. It appears to me that they do _not_ say that "lacking parentheses, multiplication ALWAYS takes precedence over division, ..." In the 1959 edition of Volume 1, on the white page [for students] numbered 1-39, Example (4) shows 8 ⨉ 2 ÷ 4 ⨉ 3 being interpreted as [(8 ⨉ 2) ÷ 4] ⨉ 3 and on the green page [for teachers] numbered 1/50, it says: ---------- Example (4) on page 1-39 may contradict conventions given in other texts which tell you "to do all multiplications first, then divisions next". Please follow the convention given here. The parallelism between addition-subtraction and multiplication-division is too neat to want to give up. ---------- [So we see that at least as late as 1959, the order of operations for mixed multiplication and division was not yet a settled matter, contrary to claims of some that applying them together in left-to-right order has been the agreed rule for hundreds of years, or at least since not long after 1917.] I have not yet checked whether the authors actually follow the strict left-to-right rule in cases involving implied multiplication (if they write any expressions of the forms A÷BC or A/BC at all). I also haven't looked for the place where they first introduce implied multiplication to see whether they introduce an additional order of operations rule (or, as they might put it, an "unabbreviating" rule) for it at that time.

Look into the origins of the obelus as a symbol for division-- it worked via line division.

The “÷” symbol used to mean the same thing as line division? Does that mean it used to group both sides of the operation?

@@drednaught608 the text book printing industry tried to push the obelus as a grouping symbol to be used like the vinculum because the vinculum took up to much vertical page space for the printing methods at that time...BUT this was in direct conflict with the Order of Operations just as parenthetical implicit multiplication is in direct conflict with the Order of Operations. The obelus ÷ and the solidus / are equal signs of division and nothing more... Parenthetical implicit multiplication does NOT have priority over division despite the misquided and confused notion of many people.... Math is not based on opinion or popularity it's based on rules and the rules do not support parenthetical implicit multiplication having a higher precedence...

Every equation came from fraction in case u don't know.

That is true

Me too. Only came across the pemdas/bodmas mnemonics and their fanatics as an adult when I stumbled on these questions. I was horribly shocked at how terrible math education is. And how they seem to think this childish oversimplification is the top "law" of math 🤦‍♀️

The way i see it is that 2 is tied to the parentheses and you gotta distribute that first or you’re breaking the order.

You did pemdas Wrong ….

@@kurtshaw229 Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial Factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9

@@TheJocelynrae BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....

The 2 is not multiplication by juxtaposition... it is simply parathesis.

@@Dead_Goat ÷ was invented to mean "fraction". as it was too hard to write the textbook in question with everything written like we do on a board ( 1650s ) - so it was all written in one line to make typesetting easier. Hence why it looks like a miniature fraction. So the issue really is people confusing it with / . Note that * and () are not the same, either. The rules work IFF you understand this. Mix and match the terms, and, well, I hope you're not teaching math to kids as they are in for a world of hurt when they get into college.

PEDMAS is not necessary. Doing the paranthesis is not necessary. People are getting 6/2*3. I ask where do you get the 3? There is no 3. Using the distributive and commutative rules, the 2 is distributed to the 2+1 for 4+2 or 6 for the denominator.

2(3) implies that it is a factorisation such that you can factor an ammount of 2 from an unknown ammount, which you can find out to be 6 if you multiply the factor in again. same thing if you factor for instance a from a^2-a which you would get a(a-1) the original ammount was a^2-a which you should take care of before anything else since it is in brackets only then the division

That's how I understood it. You could replace the (2+1) by x, giving 6/2X, which would give 1.

I was taught that 2(3) is the same as 2x3, you don't have to have a multiply sign

@@danielzhang1916 if it's easier for you to solve 2(3) by writing 2 x 3 then you have to write it correctly. Which is (2 x 3).

Pemdas is not wrong. This is entirely an ambiguities in grouping problem, not order of operations problem.

@@linsqopiring6816 Yes. So it is. Shown in another video here at RU-vid by a maths teacher.

@@linlasj name?

To be more precise, it's woefully inadequate and simplified such that it breaks with algebra and most more advanced math classes. x(stuff inside here) is one such exception that it does not cover correctly. So people and teachers.. well, maybe the teacher certification tests need to be harder. lol.

There are no ambiguities in math. If you ever find an ambiguity it's because you did something wrong. 1 will always equal 1, it will never sometimes equal 2.

In pemdas multiplication is not always supposed to go before division. In the same way addition and subtraction or equal presidents, so is multiplication and division. You decide to do one or the other based on which is first when you read the problem from left to right. Personally, I think pemdas is a really good tool and if people wanted problems to be interpreted in other ways, they should just use parentheses, the issue comes in when people are taught pemdas differently or are even taught other things.

@@nightshade8538 PEMDAS is a subset of PEJMDAS, where the J stands for juxtaposition. PEMDAS is incomplete, and should not be taught; making people use parentheses is not a suitable alternative.

No, it's you and the lady in this video... The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y. This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2 All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. There are no coefficients in this expression... 6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b is 3 not 2 Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property, when FULLY applied, is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property, when FULLY applied, REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2 no parentheses required 3×1+3×2= 3+6= 9 You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression..... When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses. TERMS are separated by addition and subtraction not multiplication or division. 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 6÷2(1+2)= 3(1+2) no rules have been broken 2×2×4(a+b) partial Distribution 2×2(4a+4b) However the TERM outside the parentheses when simplified equals 16 and 16(a+b) 2×2(4a+4b)= 2(8a+8b)= 16a+16b which is the same as 16(a+b) 2×2×4(a+b) when fully Distributed is 2×2×4×a+2×2×4×b and the LIKE TERMS can be factored out of the expanded expression. The LIKE TERMS being 2×2×4 So... 2×2×4(a+b) 6÷2×1+6÷2×2+6÷2×3-6÷2×4= 6÷2(1+2+3-4) as the LIKE TERM 6÷2 was factored out of the expanded expression... Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board...

@@RS-fg5mf Brilliant! Now, why do the rules change if logic is always consistent?

@@forgottenpeopleplacesandol4258 Convention when it comes to directly prefixed coefficients and variables... 2y there are no parentheses by which you can Distribute... I can explain it to you but I can't understand it for you...

@@RS-fg5mf Somehow I managed to get through an engineering program and also managed those skills to achieve substantial retirement benefits while never drawing a paycheck from anyone and I am enjoying my accomplishments just fine despite my lack of understanding. I don't know how I made it without your abilities. But I did.

@@forgottenpeopleplacesandol4258 🤣🤣🤣🤣🤣🤣 if you say so... LMAO You never worked in your field and never recieved a paycheck?? Hmmm. And this supports your wrong answer, how exactly?? And FYI.. Any career field that relies heavily on math would NOT use the obelus by rather the vinculum so you're argument is lacking...

BODMAS works fine if only explicit notation is used and for equal priority operations you go L to R (just to reduce errors. Like 10/2×5 going L to R is 5×5 = 25 and from R to L is 10×2.5 = 25 or 50/2 = 25, so the same answer regardless, but people often make mistakes going R to L doing 10/10 = 1 giving M higher priority by accident, so it's not advisable unless you're experienced with it.) When you have implict notation you have to interpret that first and that's where the issue is in writing division on one line with implict multiplication or division directly after. Modern international standards like ISO-80000-1 mentions that situation and says brackets are required to remove ambiguity. (6/2)(1+2) or 6/(2(1+2)) resolve the issue and are good notation writing.

@@GanonTEK I agree. We need the brackets to avoid the possible ambiguity of a PEMDAS solution. Better not to use PEMDAS or BODMAS, therefore. It’s lazy and problematic.

The line is called a vinculum, and yes, it acts the same as brackets. I've read a lot of comments saying that whether the obelus (÷) or solidus (/) is used should affect the precedence rules, but I don't think there's much official support or consistency to this. Many authors use yc÷rb to mean yc÷(rb) and the same with the solidus. See my latest video ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-4x-BcYCiKCk.html for examples. And some calculators treat 1/2x as (1/2)x (wolfram alpha for instance) so the solidus is no guarantee of grouping.

@@THaWoM The biggest problem we have here is the confusion and misconception that a convention given to variables in an algebraic expression/equation applies to parenthetical implicit multiplication..... 6÷2a the 2 is the coefficient of a by algebraic convention NO parentheses (grouping symbols) necessary. It forms a coefficient/variable compound quantity... 2(3) is not a coefficient/variable compound quantity. Real numbers (constants) can be coefficients but real numbers do not have coefficients.... Grouping symbols only give priority to (OPERATIONS INSIDE) the symbol not outside.... Do you people lack the intelligence to know the difference between inside and outside?? 2(3) is not a parenthetical priority and is not a coefficient/variable compound quantity... It is called implicit multiplication not implicit grouping... Implicit as in the physical operator is implied as if it were there when a constant, variable or TERM is juxstaposed next to the parentheses without a physical operator.... TERMS are seperated by addition and subtraction. When you have an algebraic expression like 6÷2a when you replace a with a constant or value such as (1+2)=3 we know the value is 3 so we would then write 6÷(2×3) NOT 6÷2(3) OR we could write 6÷(2(1+2))... When you replace a variable with a constant value proper grouping symbols are required to maintain the grouping that WAS the coefficient/variable compound quantity... To give priority to parenthetical implicit multiplication over division is to break the Order of Operations and the various properties of math. NOT to be confused with the simplistic use of PEMDAS. BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a parenthetical priority and is exactly the same as 2×3 M not P or B in PEMDAS/BODMAS. Brackets/Parentheses only give priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property states to multiply all TERMS inside the parentheses with the TERM outside the parentheses not just the factor next to it.... 6÷2 is one TERM juxstaposed to the parentheses... Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator... Parenthetical implicit multiplication is nothing more than multiplication without the need for a physical multiplication sign and the only correct answer is 9 not 1 Algebraic proof against this BS.... The reciprocal of a is a^-1 6a(1+2) = 6÷a^-1(1+2) Operational inverse by the reciprocal. 6a(1+2)=9 6a(3)=9 6a=9÷3 6a=3 3÷6=a a=1/2 or 0.5 0.5^-1= 2 6÷0.5^-1(1+2)= 6÷0.5^-1(3)= 6÷2(3)= 3(3)= 9 6÷(2(1+2))= 6÷(2×1+2×2)= 6÷(2+4)= 6÷6= Distribution is a method of eliminating the need for parentheses by bringing the terms INSIDE the parentheses to the outside.... You can NOT take 6÷2(1+2) and write 6÷2×1+2×2 You can write 6 --------- 2×1+2×2 because the vinculum is a grouping symbol and is the same as writing 6÷(2×1+2×2) I'm saddened by the fact that so many people share the same misconceptions BUT I'm even more sad and concerned with those who want to blame Common Core or politics.... Common Core has absolutely nothing to do with this debate. It is a teaching method and does NOT change the rules or the answer.... Those who think that this debate has anything to do with Common Core or a political conspiracy are just flat out CLUELESS and STUPID

@@THaWoM : First off there should be absolutely no reason for a grade school kid to ever be introduced to PEDMAS. Why are they be given ambiguous equations in the first place? This just means whoever is supplying the equations is lazy or sloppy and don't know what they are doing. An equation is essentially a sentence and as with all sentences you should employ correct mathematical grammar and punctuation. They should be taught the exhaustive use of parentheses and brackets. Their understanding of what they want an equation to say increases immeasurably. There should be no ambiguity. Secondly this cavalier use of the obelus wreaks havok with trying to perform algebra. I was taught that everything to the left of the obelus is the numerator and everything to the right is the denominator. 2x/3y-1 SAYS Two ex DIVIDED BY Three why minus 1. That is (2x)/(3y-1) And that is how it should be carried out. Any variations should be indicated by parentheses.

The part of the quadratic formula under the square root symbol has a name. It is called the discriminant. And sorry to say, you have it written incorrectly. It is b squared minus four ac. Not 2 bc. It is for finding the roots, or factors, of an expression which has the form y = ax^2 + bx + c I can still remember exactly how it was always written and said out loud, and I learned it over 43 years ago. Minus b plus or minus the square root of b squared minus 4 ac, all over two a. The plus or minus means that there are two solutions, also called roots, and the sign of the discriminant tells whether the roots are real or not real. Another way to say it is it says how many zeroes there are, zeroes being the places on a Cartesian coordinate where the parabola crosses the x-axis. If the discriminant is zero, there is only one solution, or root, or zero, or factor, (all mean the same thing, basically IIRC) meaning the two factors are the same, and the parabola just touches the x-axis in exactly one place. IOW Something like (x +2)(x +2). If the discriminant is positive, there are two real roots. Physically this means the parabola has it's vertex below the x-axis and there are two y intercepts, or zeroes, or factors. If negative, there are no real roots, as negative numbers do not have a real square root, only an imaginary one, some multiple of i, which is the square root of negative 1, which does not exist. These are called complex numbers. c is where the parabola intercepts the y-axis. All of this has meaning that applies to the physical world. When you point out that the expression yc/rb is taken to mean the y and c are multiplied together, as are the r and the b, and then the division is done, is exactly correct in real world usage, at least in the world of science. Because if the order of operations rule that is at issue here applied, this would be the same as saying yc/r multiplied by b. But that could be very clearly written as ycb/r. And no one ever writes ycb/r as yc/rb. It simply makes no sense. There is no logical reason to write it out that way if what is meant is to have the b in the numerator.

@@williejohnson5172 well to begin with 2x implies that 2 has been multiplied to x Equivalent to 10 implies 5 had been been multiplied to 2 Next 2x/3y-1 is an ambiguous if presented without context. Based on the problem you solve 3y-1 may or may not make sense and with a some spacing it will make sense in both Interpretation. Here's another 1/5x²+6/5x-6y Equation is ambitious and solely depends on context of what you are solving. With a bit of brackets, it will make sense in bodmas for anyone without context.

I've always used a curly x. I assumed it was fairly standard. It does let you write multiplication as × then but usually juxtaposition is common to remove the need for the × symbol, in my experience.

@@GanonTEK yeah, it does seem fairly standard, despite my impression that people look down on the use of the × symbol, so how one writes the x variable doesn't seem to matter much

There is no need and no excuse to use the vulgar and incorrect BEDMAS "rule" which does not exist in Math! When I was taught, this was never used and I heard it only once whispered to a student during a test! It is fake and along with all of the other fake "crutches" used to "teach" math which work perhaps less that 100% of the time have no place in Mathematics education, but unfortunately are often used INSTEAD of teaching Mathematics properly!

Same here. I have a BS in math. We never learned this. We just understood what to do.

So was I. Implicit first.

There really is no problem with PEMDAS! It is a question of notation at different levels of math. PEMDAS works at the foundational level of math where expressions are non-composite. As the level of math advances, mn/qs "implies" taking mn and qs together. It can no longer just simplify as a × n/q × s ! There are many more operators that cannot follow PEMDAS (factorials, summations, even simple trigs --> sinAb is not bsinA, etc.). In short PEMDAS for basic order of operations, like LIATE for integration by part are general and convenient ways of manipulating tasks. They do not work for every situation. As complexity of a task increases, mathematical notations take over to guide the order of operation.