Mathematics isn't about following instructions and memorising rules you don't understand; it's about creativity, ingenuity, curiosity, and puzzles. In my videos, I try to give you a glimpse into how my mind works as I solve problems. I was lucky enough to get a lot of extra problem-solving lessons through high school, and as a result, I got awards for top marks in maths in year 12, and first and second year university, and graduated with first class honours in physics.
PENDAS isn’t wrong, but there are two different things happening here. The first is the multiplication before the bracket, from memory it was called implicit multiplication, it comes first because a number outside a bracket means it had been initially factored out elsewhere. The second is the way we try typing something into a computer whereas in the day of writing it down by hand it was easy to put in the correct numerators and denominators but by PC you get a long string and as per other comments here, you should add extra parentheses to clarify.
PEMDAS is fine. Working and amateur mathematicians, engineers and physicists know how it works, so they don't write equations to go viral or confuse the mathematically illiterate online.
From about 2:13 to 4:13 in the video, the narrator shows some actual examples from mathematics, engineering, and physics textbooks in which the authors use expressions of the form A/BC to mean A/(BC), contrary to the supposed PEMDAS rule. Do you think her examples are cherry-picked? If so, I offer you a challenge, if you have the gumption to try it: (1) Pick a bunch of sources (books, articles, college class lecture notes, etc.) of the sort likely to contain mathematical formulas. Since the point in question is whether mathematicians, scientists, and engineers strictly follow the order-of-operations rule commonly given in high-school (and lower level) textbooks, I suggest focusing on material that is _not_ specifically intended for the primary or secondary education market and that is _not_ specifically about the topic of "order of operations" itself. (2) Look through your sources for expressions that would have different meanings depending on whether or not an implied multiplication (that is, multiplication indicated by juxtaposition) gets precedence over an inline division operator ("/" or "÷") textually to its left. (3) For each such expression you find, try to determine the author’s intended meaning based on the context, and not merely based on your view about "correct" order of operations. (4) See what patterns you notice. Do all, or almost all, authors who write expressions of the form A/BC (or A÷BC) consistently mean (A/B)C? Do (almost) all consistently mean A/(BC)? Do you ever find examples of both kinds of meaning in the very same document-and if so, do the authors seem to follow any consistent rule about when to give precedence to implicit multiplication and when not to? Are there cases where you find it hard to reliably determine the author’s intent? Do you notice anything else that seems worth mentioning? If you try this experiment, I'd be interested in learning what you find. Please note, however, that I am not interested in what you _think_ would be the result of the experiment _if_ you tried it, or in what kinds of expressions you _think_ you recall seeing in the mathematical and technical writing you've read over the years, only in what you find by _actually looking through documents while attending to the question at hand. Editing to add: To reduce the risk that the first few examples of relevant expressions you find might be atypical (as you may think is the case with the ones shown in the video), I suggest that you continue looking until you find examples of A/BC-type expressions written by, say, a couple dozen different authors and in works on a variety of subject areas.
It is already confusing enough without your completely nonsense video, The equation 6:2(1+2) can be written as 6/2x3 which is 9 You can verify this with ChatGPT, WolframAlpha or Symbolab, they all give 9 as answer. Please never post this crab again.
In another video on this same channel, titled "The Problem with PEMDAS: Why Calculators Disagree", the narrator demonstrates that some scientific calculators interpret 6/2(1+2) as 9 and others interpret it as 1. When people support their opinions about the One True Rule for order of operations by citing what a particular model of calculator or a particular computer program does with a particular expression, I wonder how they will react when a different calculator or computer program gives a different result for the same expression, or when the very same calculator or program they've cited gives an unexpected result for a different expression. Will they admit that the matter might not be as definitively settled as they thought? Or will they switch to saying that the unexpected behavior is "buggy" (even if it's documented, as is often the case for calculators), and that anyway calculator engineers and computer programmers don't get to decide rules of mathematics? Here are the results of some experiments I just did with the systems you've mentioned: When I typed 2x ÷ 3y into Symbolab, it interpreted it as (2x ÷ 3)y. But when I typed in 2x ÷ 3x it interpreted it as 2x ÷ (3x). It also interpreted 9x ÷ 3y as 9x ÷ (3y). Wolfram Alpha interprets 9x ÷ 3x as (9x ÷ 3)x and gets the result 3x². But it interprets 9x ÷ xy as 9x ÷ (xy) and evaluates it to 3/y (but displayed with a horizontal fraction bar). Also, Wolfram Alpha evaluates 6÷2(1+2) as 9, but if I write the expression, as you did in your comment, with a colon instead of an obelus (that is, 6:1(1+2)), it gets interpreted as 6:(1(1+2)). When I asked ChatGPT What is 6/2(1+2)? it gave me the answer 9, but when I changed the question to What is 6/(1+1)(1+2)? it gave me the answer 1. Do you find any of this surprising? Hmm?
@@UncleJim99 Surprisingly indeed. Conclusion: What is the reason for creating ambiguous questions? Just use some extra braces. I think we both agree on that.
yup correct those are ALL wrong ... BELMDAS is the one discovered by Thale with his student Pythagoras ... 2,500 Years ago and Pemdas is only Valid to grade 4 ... after that it is no longer viable btw pemdas also gets 1 ... because division is NOT done before multiplication ... it is only invalid once one starts into Logarithms ... which is learnt in grade 6 and 7 ... btw the equation 6 / 2(1+2) = 1 NOT 9 never 9 ... and the question isnt even ambiguous it is just using implied multiplication at the brackets with the 2(1+2) or 2 x 3 ONLY in America is Multiplication and Division or addition and subtraction even exponents and logs are NOT equal in order ... llogs division and subtraction are lower because they are INVERSE operations ... we are taught in school to add before we can subtract ... because Addition is higher order than subtraction ... same goes for Multiplication and division ... and even exponents and Logs ... we are taught the higher order then the lower order at each level because you MUST do the higher first to be able to do the lower order operation at all ... and as BELMDAS is both an acronym and Mnemonic ... you DONT change the order you do it as it is written ... it is NOT America States United ... it's United States of America ... or the Mnemonic or acronym is USA ... a/bc = 1/2a/b + 1/2a/c .. the BC MUST be resolved before the division can start .. if either B or c are 0 it is undefined immediately and NO answer is possible ... the texts yu are using are displaying it correctly BUT you are applying belmdas incorrectly and getting the wrong equation ... such as the pvm=rt ... when you divide BOTH sides by rt ... you get pvm/(rt) = 0 .. notice the brackets ... the text book is IMPLYING those brackets ... which you overlook secondly ... they are doing the higher order multiplication before they do the inverse operaton of division ... Polynomials such as 2x/3y -1 imply (2x)/(3y) -1 ... making it 2 ... because they unknown variable is directly connected to the actual value they MUST always be done together or split apart by doing an INVERSE operation such as dividing by x or by 2 the first expression and by y or 3 for the second ... BELMDAS ... by order from highest order down ... {} [] () Exp Logs Multiplication Division Addition Subtraction ... ALWAYS in that order ... like all acronym's and mnemonics (they are the same thing essentially BTW) you never change the order ... USA never is printed UAS ... or ASU ... it is written USA .. giving the exact order of the words or order to do things in
I grew up in Canada and have never encountered PEMDAS until a few years ago on the internet. The "natural order" of operations were as you describe in your video and makes generally makes good sense. When I took a computer science class I was then told that the computer order of operations may vary and to include extra brackets to avoid ambiguity. Here is what I see as the problem. The way equations are written in a text book, which seem to be natural, do not translate into a single line in computer code very well. And confusion entails.
You can always use parentheses to make a formula clear. Whether PEMDAS is "right or wrong" doesn't matter. People who rely on PEMDAS when they write an expression take the completely unnecessary risk that they will be misunderstood. People who violate PEMDAS when they write an expression also take a completely unnecessary risk. There are endless RU-vid videos that present a simple mathematical expression with the teaser comment that "95% will get this wrong". But almost nobody would get the answer wrong if a few parentheses were included. Do these people know anything about information theory? If a few extra symbols would reduce the error rate by 95%, it's like transmitting the original symbol stream over a noise channel without any error correcting code.
2+2=5 This is where we are in the clown world where we are currently living with this kind of stuff. Older calculators couldn't handle order of operations correctly so they provided the WRONG ANSWER. As technology and programming languages improved, they could. That is why there is a difference. It is VERY DISTURBING how many people watch this and believe it because an attractive woman is saying it.
Here's a different perspective for you! I'm an SLP, but before that, I studied Economics as an undergrad. We devoted nearly an entire class session to this question. We were asked to answer the question and the professor took count of how many people answered in a certain way. 49 of us answered 9. 6 of us answered 1. I don't remember how many answered something else, but there were other answers. We concluded that the problem has more than one answer and left it at that. I wasn't satisfied with that answer, but there it is. Fast forward a few years, and I have my masters in SLP. I finally understand the issue with the problem. It's got nothing to do with arithmetic. It's a sociolinguistic issue. Whoever wrote the problem wasn't clear about how it's supposed to be approached. A few years ago, as an Econ undergrad, I would have answered the question using PEMDAS, and according to my instructor, I would have gotten the answer correct. Another instructor may have said all 49 of us were wrong. The point is, this question created DISCOURSE for students everywhere. That's beautiful. I've also learned, maybe incorrectly so, that people who are getting 1 as the answer are following an older system of operations, whereas PEMDAS is a comparatively more recent system of following the order of operations. I don't know the answer to this problem, but I'm glad people talked about it. We don't often associate math with vagueness, but it truly can be at times.
well PEMDAS still provides the answer of 1 at the end there, IF you ignore the rule that multiplication and division are done in the order they appear and instead interpret PEMDAS more strictly and literally. Do the paranthetical first, then multiply where its found, then divide and its 1.
That's a bad idea though as you could then get 10 - 5 + 5 = 10 - 10 = 0 doing A before S when 0 is clearly wrong. You would have to keep them equal priority: A first: 10 - 0 = 10 S first: 5 + 5 = 10 Same answer regardless.
What you are showing from textbooks is complete nonsense. What you see is not what mathematicians nor physicists would write but rather what publishers would write. Why do they write it this way? Simple, because they can write it on one line and hence save paper.
I'm skeptical of your claim. I can understand that publishers might often prefer to write an expression with "/" or "÷" instead of using the space to write it with a horizontal fraction bar, especially if the latter would involve putting an equation or expression into a display instead of keeping it in running text. But I doubt that publishers would often begrudge an author the cost or space to include a pair of parentheses and write an expression in the form (A/B)C or A/(BC), whichever they mean, instead of A/BC, especially if they believe that the standard interpretation of A/BC without added parentheses would be opposite to the author's intent. I have a challenge for you, and I'd be interested in knowing what you find _if_ you take me up on it. We are living at a time when there are lots of documents available on the web written by mathematicians, scientists, and engineers without the interference of penny-pinching publishers. Examples include online open-access textbooks; preprints on sites such as arXiv, engrXiv, and preprints (dot) org; and lecture notes that many professors have made freely available online. My challenge is to try the following experiment: (1) Skim through a bunch of documents of the sorts I've just described and look for expressions of the form A/BC (or A÷BC). (2) For each such expression you find, try to determine whether _the author's intended meaning_ was (A/B)C or A/(BC). Judge based only on your ability to determine which interpretation makes sense in context, not based on assumptions about "correct" order of operations for expressions involving implied multiplication. (3) To guard against the risk that the first few examples you find will be flukes of chance, continue until you've discovered relevant expressions in at least, say, a couple dozen documents by different authors (or until you've examined a sufficiently large and varied sampling of documents with lots of math but no A/BC-type expressions to be convinced that such expressions are extremely rare). (4) Tell us what you found. Please note that I am not interested in what you _think_ would be the result of this experiment if you tried it or in what kinds of expressions you _think_ you recall seeing in the mathematical and technical writing you've read over the years, only in what you find by actually looking through documents while attending to the question at hand.
You're bringing up a great point; however, I would be interested to know if the preface, or an appendix, of the texts you're referring to discuss some kind of notation preference; if so, while you're statements are not necessarily incorrect about PEMDAS, there would be validity to how they each write their respective statements. While my degree was in Mathematics, I studied COSC, Phys, Engr as well, and for the math involved with any of these, fractions were generally written as fractions to alleviate any confusion, the elementary division (÷) symbol was never used, and a forward slash (/) is used for division in programming, where btw, PEMDAS is rigidly adhered to ... meaning, no parens means that _each_ numeric value or variable is interpreted according to the very rules of PEMDAS without question. As a software engineer, what I've stated continues to hold. To be fair, it actually seems like the rules of PEMDAS are fixed and true ... it's the professors in higher level math that abandon some of the rules in favor of shortcuts and assumptions. Interesting topic ... thanks for the video!
It matters if you are in engineering building a bridge or a skyscraper. The Algol compiler that l used, you had to put in the multiplication operator or it would give a syntax error. I can’t remember what the Fortran compiler did.
How can't people know 3rd grade stuff?! To determine the value of the expression 6÷2(1+2), we need to follow the order of operations, often abbreviated as *PEMDAS/BODMAS:* P/B: Parentheses/Brackets E/O: Exponents/Orders (i.e., powers and square roots, etc.) MD: Multiplication and Division (from left to right) AS: Addition and Subtraction (from left to right) Let's break down the expression step by step: 1. Parentheses/Brackets: Evaluate inside the parentheses first. 1+2=3 So the expression becomes: 6÷2(3) 2. Multiplication and Division (from left to right): Perform the division and multiplication from left to right. Here we have two operations: division and multiplication. According to the order of operations, we proceed from left to right. 6÷2×3 First, perform the division: 6÷2=3 Then, multiply the result by 3: 3×3=9 Therefore, the expression evaluates to 9. *Explanation:* The confusion often arises because some might interpret the expression differently, grouping 2(3) together before performing the division. However, according to standard mathematical conventions (PEMDAS/BODMAS), multiplication and division are performed from left to right as they appear. Thus, the correct and mathematically logical interpretation of the expression 6÷2(1+2) is: 6÷2×3=3×3=9 *So, the answer is 9.*
This is not a consensus viewpoint. Yes, older books used this convention, modern books with modern type setting don’t write fractions like a/bc. Algebraic solvers and most math professors have abandoned this rule, you can check by putting your problem into google, maple, Mathematica, WolframAlpha, or any modern symbolic calculator
There may not be consensus for giving elevated precedence to implied multiplication over inline division, but there certainly isn't consensus for the strict PEMDAS rule either. To illustrate, I offer for your consideration a few excerpts from _The Princeton Companion to Mathematics_ (Timothy Gowers, ed., Princeton University Press, 2008). I hope you won't claim that 2008 is too long ago to be considered "modern". Note: Various expressions I've attempted to quote below include subscripts, superscripts, Greek letters, and/or other mathematical symbols outside the basic 7-bit ASCII character set. If any of them don't get properly rendered in a RU-vid comment, I'll be happy to explain how I intended them to appear. On page 3: "Then if |h| ⩽ 𝜖/6x, we know that |3xh| is at most 𝜖/2." [In this sentence, "𝜖/6x" clearly means 𝜖/(6x), not (𝜖/6)x.] On page 64: "... and let X be a set with n elements. ... Now the number of subsets of X of size k is n!/k!(n − k)!, ..." [Here "n!/k!(n − k)!" means n!/(k!((n − k)!)), not (n!/k!)((n − k)!) as it would if the division and the implied multiplication were to be done in left-to-right order.] On page 218: "−|x₂−x₁|²/4(t₂−t₁)" [The quoted expression is an exponent within a formula related to the heat equation and while I haven't followed the details of the math, I'm confident that the denominator of the "/" is the entire product 4(t₂−t₁).] On page 220: Given two vectors v and w, the angle between them is defined by the fact that it lies between 0 and π (or 180°) and its cosine is ⟨v,w⟩/‖v‖‖w‖." [Of course "⟨v,w⟩/‖v‖‖w‖" means ⟨v,w⟩/(‖v‖‖w‖), not (⟨v,w⟩/‖v‖)‖w‖.] On page 285: "(Physically, the quantity |p|²/2m = ... represents kinetic energy, while ..." [Here "|p|²/2m" means |p|²/(2m), as I know both from my knowledge of physics and from the fact that the equivalent expression occurs in a nearby displayed equation, where it is typeset using a horizontal fraction bar with "2m" in the denominator.] On page 334: "In fact, since [the sum for n⩾1 of] 1/n(log n)² converges, we see that ..." [The expression "1/n(log n)²" must mean 1/(n(log n)²), not (1/n)(log n)², since the series whose general term is (1/n)(log n)² diverges.] On page 346: "... never greater than 2x/φ(q)log(x/q)." [The next sentence refers to the logarithm of x/q as "the log in the denominator".] On page 358: "We have a fairly concise understanding of how many zeros [of the Riemann zeta function] there should be up to a given height T. In fact, as already found by Riemann, this count is about (1/2π)T log T." [Consultation of other sources on the web confirms that the meaning is (1/(2π))T log T, not ((1/2)π)T log T.] There are a number of other expressions of the form "A/BC", both mixed in with the ones I've quoted above and in the remaining 600 or so pages of the book, and as far as I can tell, the great majority of them have the intended meaning A/(BC)-that is, the writers intended the implied multiplication to have precedence over the division indicated by "/". An exception is an expression on page 422 of the form "1/|G|𝚺...", where "𝚺..." is a summation over the elements of set G. Without having followed details of the math, I'm pretty sure the intended meaning is (1/|G|)𝚺..., which would be the average value of the expression being summed.
@@UncleJim99 mathematicians have always read 1/2x as 2x on the bottom. The question is really 1/2(3). I think we will agree it is a poor way to write it though.
@@profribasmat217 In choosing the expressions quoted above, I've picked some that resemble "1/2x" in that the part to the right of the "/" consists of just a numeric coefficient juxtaposed to a variable or a symbolic constant, namely, "𝜖/6x" from page 3, "|p|²/2m" from page 286, and "1/2π" from page 358. But I also have intentionally included some that I think are more like "1/2(3)" or "1/2(1+2)", viz., "n!/k!(n − k)!" from page 64, "−|x₂−x₁|²/4(t₂−t₁)" from page 218, etc. I will agree that these "viral" problems about expressions such as "6÷2(1+2)" and "8/2(2+2)" are meant to provoke dispute, and that I'd try to avoid writing that sort of thing if I had some mathematical idea that I wanted to communicate clearly.However, I'll note that even if I'd be inclined to clarify various expressions above by adding parentheses or by using a horizontal fraction bar instead of "/", they all come from articles contributed by highly qualified and accomplished mathematicians.
@@profribasmat217 It's definitely a poor way to write it, yes. Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are *required* to remove ambiguity. So even the standards know it's ambiguous notation and both conventions are still in use.
It's arbitrary. Maths is about arbirtrary axioms, followed by logical procedures. If a group of people agree on Bodmas, then Bodmas it is. If another group agrees on something else, then something else it is. One group might then see the problem as ambiguous, the other might not. Dogma doesn't help. Relax, kids.
Correct answer is 1, not anything else. 6 ÷ 2(3) is equal to 6 ÷ (2*3). You cannot simply substitute bracket with multiplication. You have to do it within brackets and after its value has been obtained, then only can the brackets be removed.
Attempt to plug 6 ÷ 2(1+2) into an Electronic Calculator and it will be changed to: 6 ÷ 2 X (1+2) by the Calculator. Attempt to delete the X and the Calculator with tell you: Incorrect Format. In the '50s the numeral next to the ( ) was "owned" by the ( ) as a Multiplier. Thus: 2(1+2) equaling 6 was the complete Divisor.
Wrong in the first 2 minutes. Devision and multiplication order are EXTREMELY important and make the biggest difference between PEMDAS and BODMAS. The only time things are done left to right is when there is no CLEAR distinction between the functions.
Since M and D have equal priority the order doesn't matter so you can always do M first if you like or D first. It doesn't matter if you treat it as 6 steps or 4 steps or L to R or not as you get the same answer regardless. 8×4/2 for example M first: 32/2 = 16 D first: 8×2 = 16 10 - 5 + 1 A first: 10 - 4 = 6 S first: 5 + 1 = 6 No difference. Viewing it as 4 levels and using the L to R version reduces errors though which is why it's a better version. Many people add incorrectly with 10 - 5 + 1 and say 10 - 6 instead of 10 - 4 or 11 - 5. With L to R you get 5 + 1 = 6 instead. Many people multiply incorrectly with 10/5×2 and say 10/10 instead of 20/5 or 10×0.4. With L to R you get 2×2 = 4 instead. The problem here is the use of multiplication by juxtaposition which, academically, can imply grouping so the question becomes 6/(2×(1+2)) before you start. Once you get to 6/(2×3) M first: 6/6 = 1 D first: 3/3 = 1 or 2/2 = 1 If you use the programming interpretation, you get 6/2×(1+2) which is 6/2×3 M first: 6×1.5 = 9 or 18/2 = 9 D first: 3×3 = 9. Both are widely used interpretations so both are valid. Instituions like the ISO and the American Mathematical Society say it is ambiguous notation and multiple professors and mathematicians agree. It's just terrible writing. The order of operations like PEMDAS etc. are all oversimplifications for what's allowed anyway. Minutephysics did a great short video called "the order of operations is wrong". Worth a watch.
Multiplication is to Division what Addition is to Subtraction... Just like 10-7+2 does not equal 10-(7+2) but rather 10+(-7)+2 likewise 6÷2×3 does not equal 6÷(2×3) but rather 6(2⁻¹×3) = 9 Multiplication and Division absolutely share equal priority and can be evaluated equally from left to right as they appear as they are inverse operations by the reciprocal... 2(1+2) is not a parenthetical priority. Only (1+2) is a parenthetical priority despite the false and misleading information, subjective opinions and willful ignorance people have about parenthetical implicit multiplication... The correct answer is 9 not 1
Sorry, if you put it in the windows calculator or any other calculator it gives 9. It cannot be interpreted in an alternative way. However it is best to avoid the inline notation and use the fraction symbol for division.
Depends on the scientific calculator but here are some that give one or the other: These give 1: Casio FX 83GTX, Casio FX 85GT Plus, Casio 991ES Plus, Casio 991MS, Casio FX 570MS, Casio 9860GII, Sharp EL-546X, Sharp EL-520X, TI 82, TI 85 These give 9: Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES, Casio 570ES, TI 86, TI 83 Plus, TI 84 Plus, TI 30X, TI 89. Microsoft Math gives both answers on screen. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation which implies grouping (1). Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation (1). TI later changed to the programming/literal interpretation (9) but when I asked them were unable to find the reason why. Some commenters have said it was pressure form American teachers but I've no confirmation of that.
@@holretz1 Most programming languages don't allow implict notation. So, it still has to be interpreted *before* they can write down what the calculation is, and that's where the ambiguity happens.
@@GanonTEK I see that in a comment above, and in several of your other comments about order of operations, you refer "the academic interpretation" vs. "the programming/literal interpretation". I'm curious. Are those terms, "academic interpretation" and "programming/literal interpretation" your own coinages? Or did you get them from some other source (and, if so, what was that source)? Note that at the moment, I'm not asking or commenting about your general conclusions. I'm just asking about your terminology. Thanks.
@@UncleJim99 Good question. I got "academic interpretation" from it being very common in academic writing and in academic writing style guides to use that interpretation so it made sense to call it that. "Literal" I took probably as a synonym from the American Mathematical Society's official spokesperson Mike Breen who said in PopularMechanics' ambiguity article that following "strict" PEMDAS you wouldn't give implicit multiplication higher priority (not those exact words (he did say strict) but that's the implication. So, they are somewhat made up but connected to the evidence for either interpretation.
I would suggest that PEMDAS is right, but is being interpreted wrongly 6 / 2(1+2) Evaluate inside parentheses:- 6 / 2(3) There are still parentheses, but this time they're external. They still outrank division, so evaluate them:- 6 / 6 Evaluate the division:- 1
Parentheses i.e. grouping symbols only group and give priority to operations INSIDE the symbol not outside the symbol. 2(3) is not a parenthetical priority and is mathematically the same as 2×3 Parentheses also serve to delimit the TERM outside the parentheses from the TERM or TERMS within the parenthetical sub-expression. The correct answer is 9
@@RS-fg5mf Thank you, I'm well aware of how PEMDAS is often interpreted. I posit that it is usually interpreted wrongly, and that once the inside of parentheses are evaluated, the outside should then be evaluated. Then, all the ambiguities go away, everyone's answers are consistent, and we don't get never-ending threads like this. The pre-existing PEMDAS rules broke with the advent of the printing press, when the horizontal line for division was replaced with a division symbol or an inline slash. With that, the implies precedence of the horizontal line was broken. Evaluating external parentheses restores the previous status quo.
@@peterthomas5792 Ummm WRONG Please answer some questions 6 ---------- how many grouping symbols do 2(1+2) you see grouping operations within the denominator?? 6÷(2(1+2)) how many grouping symbols do you see grouping operations within the denominator... The P in PEMDAS is being misinterpreted by you... Grouping symbols... If you're standing on one side of a wall and your friends are standing on the other side are you grouped together?? NO 6÷2(1+2)= 6÷2×1+6÷2×2 by the Distributive Property which I'm sure you fail to understand correctly as well...
@@RS-fg5mf Just stop. You're arguing for one set of rules that has been broken since Gutenberg invented the printing press in around 1440, I'm arguing for a revised and more-widely accepted interpretation. You know that, I'm not playing your games.
@@peterthomas5792 you can argue for a REVISED INTERPRETATION all you want... As it stands unless and until you CHANGE the basic RULES and PRINCIPLES of math the correct answer is 9. Giving priority to parenthetical implicit multiplication is in direct conflict with the Properties and Axioms of math...
I don't like to use this “÷”. I'm a fan of the fraction line. Because this entire term 6÷2(1+2) is not logical. That's why everyone is confused. People buy calculators that have the ability to calculate in fractions. In my interpretation it is "six half times three"
There is nothing wrong with the expression. Yes, the correct answer is 9. Failire to understand and apply the basic rules and principles of math correctly as intended doesn't make the expression ambiguous and isn't a valid argument against the expression... The vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are REQUIRED to maintain the grouping of operations within the denominator ... 6 ---------- how many grouping symbols 2(1+2) do you see grouping operations within the denominator?? 6÷(2(1+2)) how many grouping symbols do you see grouping OPERATIONS within the denominator?? It's really as simple as that but people won't let go of their misguided beliefs, subjective opinions and willful ignorance about parenthetical implicit multiplication... People confuse and conflate two different types of Implicit multiplication .... One without a delimiter and one with a delimiter.. Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed i.e. juxstaposed WITHOUT a delimiter and forms a composite quantity by Algebraic Convention... Example 2y or BC This type of Implicit Multiplication is given priority over Division and most other operations but not all other operations... This can be seen in most Algebra text books or Physics book. Physics uses this type of Implicit Multiplication quite heavily.. Type 2... Implicit Multiplication between a TERM and a Parenthetical value that have been juxstaposed without an explicit operator but WITH a delimiter...The parentheses serve to delimit the two sub-expressions.. Parenthetical implicit multiplication. The act of placing a constant, variable or TERM next to parentheses without a physical operator. The multiplication SYMBOL is implicit, implied though not plainly expressed, meaning you multiply the constant, variable or TERM with the value of the parentheses or across each TERM within the parenthetical sub-expression. Parentheses group and give priority to operations WITHIN the symbol of INCLUSION not outside the symbol. Terms are separated by addition and subtraction not multiplication or division. The axiom for the Distributive Property is a(b+c)= ab+ac but what most people fail to understand is that each of those variables represents a constant value OR a set of operations that represent a constant value... A single TERM expression like 6÷2(1+2) has two sub-expressions. The single TERM sub-expression 6÷2 juxstaposed to the two TERM parenthetical sub-expression 1+2. The lack of an explicit operator implies multiplication between the TERM or TERM value outside the parentheses and the parenthetical value or across each TERM within the parenthetical sub-expression... The parentheses DELIMIT the TERM 6÷2 from the two TERMS 1+2 maintaining comparison and contrast between the two elements... Implicit multiplication is always by juxstaposition but not all juxstaposition is Implicit multiplication. Example 2½ = 2.5 not 2 times ½... There is “implicit multiplication” WITH delimiters and there is “implicit multiplication WITHOUT delimiters. Two different types of Implicit multiplication and mathematically different. 6÷2y the 2y has no delimiter.... 6÷2y=3÷y by Algebraic Convention. 6÷2(a+b) has a delimiter... 6÷2(a+b)= 3a+3b by the Distributive Property... 6y÷2y = 6y÷(2y) = 6y÷(2*y) 6y÷2(y)= (6y÷2)(y)= 6y÷2*y 6y÷2y(y)= (6y÷(2y))(y)= 6y÷(2y)*y= 6y÷(2*y)*y ÷2y the denominator is 2y ÷2(y) the denominator is 2
@@RS-fg5mf This is why we in germany uses the explicit system. For devisions we used this : and this is a ratio. A term with a ratio is always solved from left to right. In german this term looks like that 6 : 2 ⋅ (1 + 2) = 6 : 2 ⋅ 3 = 9
@@tom091178 you're confusing ratios and division... 6:6 = 1:1 NOT just 1 As a ratio symbol 6:2(1+2)= 6:2×3= 6:6= 1:1 nothing more is done with it as it isn't 1 divided by 1 it is a ratio of 1 to 1 ..... One part this to One part that... As a symbol of division then 6:2(1+2)= 6:2×3= 3×3= 9 which is correct ... With an explicit multiplication symbol or an implicit multiplication symbol, the answer is still 9
@@tom091178 We don't write like this either. (6/2)(1+2) would be 9 6/(2(1+2)) would be 1 Better yet, two line fractions are best practice which remove ambiguity and reduce brackets. That's the solution here, not arguing over arbitrary notation conventions.
The better solution is write notation unambiguously so it won't matter which one you use. (6/2)(1+2) for 9 6/(2(1+2)) for 1 At least on scientific calculators most have a two line fraction button by now, which removes the issue. Just never use the ÷ button.
@@tom091178 It actually depends on which interpretation of multiplication by juxtaposition you use. Academically, multiplication by juxtaposition implies grouping (1 is correct). Literally/programming-wise, multiplication by juxtaposition implies only multiplication (9 is correct). You'll see many scientific calculators give 1 for this expression, for example. It's bad writing.
@Shaun Patrick Thanks for your response: "if you want google to decide, try 1/2π. It says it's not the same as π/2." I did try that. Unfortunately, Google won't calculate 1/2π (at least not on my Mac laptop, using Safari). But Wolfram Alpha will, and gives the same result (about 1.57) for both 1/2π and π/2. You'd think Wolfram engineers would be able to get it right. I realize I was wrong in my original post (mind blown). I did research on this, and found that not all forms of multiplication are "multiplication" as understood in PEMDAS. There's an exception in step three (MD). Normally, when multiplication and division exist consecutively, you should work left to right. But there's an exception: juxtaposition always happens first, in step three (MD) of PEMDAS. That is, the correct acronym to use, the one commonly used by engineers and mathematicians, would be: P E J MD AS. Five steps. Still, why does Wolfram Alpha, arguably the most technical site online, apparently use the technically incorrect acronym, P E MD AS? Maybe to avoid spending all their time answering incredulous emails from non-engineer high school math teachers like me? FYI: I will NEVER divulge the truth to my students. Secondary school math causes quite enough difficulty and confusion as it is, without rocking the boat at such a fundamental level. Learning P E J MD AS rocked my boat almost into capsizing; no doubt many students would flip out.
People confuse and conflate two different types of Implicit multiplication .... One without a delimiter and one with a delimiter.. Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed i.e. juxstaposed WITHOUT a delimiter and forms a composite quantity by Algebraic Convention... Example 2y or BC This type of Implicit Multiplication is given priority over Division and most other operations but not all other operations... This can be seen in most Algebra text books or Physics book. Physics uses this type of Implicit Multiplication quite heavily.. Type 2... Implicit Multiplication between a TERM and a Parenthetical value that have been juxstaposed without an explicit operator but WITH a delimiter...The parentheses serve to delimit the two sub-expressions.. Parenthetical implicit multiplication. The act of placing a constant, variable or TERM next to parentheses without a physical operator. The multiplication SYMBOL is implicit, implied though not plainly expressed, meaning you multiply the constant, variable or TERM with the value of the parentheses or across each TERM within the parenthetical sub-expression. Parentheses group and give priority to operations WITHIN the symbol of INCLUSION not outside the symbol. Terms are separated by addition and subtraction not multiplication or division. The axiom for the Distributive Property is a(b+c)= ab+ac but what most people fail to understand is that each of those variables represents a constant value OR a set of operations that represent a constant value... A single TERM expression like 6÷2(1+2) has two sub-expressions. The single TERM sub-expression 6÷2 juxstaposed to the two TERM parenthetical sub-expression 1+2. The lack of an explicit operator implies multiplication between the TERM or TERM value outside the parentheses and the parenthetical value or across each TERM within the parenthetical sub-expression... The parentheses DELIMIT the TERM 6÷2 from the two TERMS 1+2 maintaining comparison and contrast between the two elements... Implicit multiplication is always by juxstaposition but not all juxstaposition is Implicit multiplication. Example 2½ = 2.5 not 2 times ½... There is “implicit multiplication” WITH delimiters and there is “implicit multiplication WITHOUT delimiters. Two different types of Implicit multiplication and mathematically different. 6÷2y the 2y has no delimiter.... 6÷2y=3÷y by Algebraic Convention. 6÷2(a+b) has a delimiter... 6÷2(a+b)= 3a+3b by the Distributive Property... 6y÷2y = 6y÷(2y) = 6y÷(2*y) 6y÷2(y)= (6y÷2)(y)= 6y÷2*y 6y÷2y(y)= (6y÷(2y))(y)= 6y÷(2y)*y= 6y÷(2*y)*y ÷2y the denominator is 2y ÷2(y) the denominator is 2
The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division. The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them... 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... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication... The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral... A variable can represent a single value, a set of operations that represent a TERM that represents a single value or a solution set... 6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property...Parentheses removed... 6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property... Inner parentheses REMOVED This can be further demonstrated using the vinculum.... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ------------ = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.... ________ 2(1+2) how many grouping symbols do you see grouping OPERATIONS within the denominator?? ÷(2(1+2)) how many grouping symbols do you see grouping OPERATIONS within the denominator?? That over bar (vinculum) is a grouping symbol __________ 2×1+2×2 how many grouping symbols do you see grouping OPERATIONS within the denominator AND what was REMOVED?? ÷(2×1+2×2) how many grouping symbols do you see grouping OPERATIONS within the denominator AND what was REMOVED?? Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation... 6. 6 -------- = 6÷(2(1+2)) = 6÷(2×1+2×2) = -------------- 2(1+2) 2×1+2×2 If you choose to Distribute the 2 into the parentheses by itself you have to do one of two things. Either take the division symbol with it, as division is right side Distributive or change the division to multiplication by the reciprocal... ÷2= ×0.5 So... 6÷2(1+2)= 6(1÷2+2÷2) still equals 9 Or... 6÷2(1+2)= 6(0.5×1+0.5×2) still equals 9 Variables can represent more than just a numeral and it's important to understand that when you replace a variable with a constant value or a set of operations that represent a constant value that you apply grouping symbols where called for by the Order of Operations and the basic rules and principles of math... example 6÷a does not have parentheses BUT a= 2+4 so 6÷a = 6÷(2+4) not 6÷2+4. BUT if a=2×3 and we have a÷2 we can write 2×3÷2 because we evaluate Multiplication and Division equally from left to right... a(b+c)... a=12÷3, b= 2×3, c= 2^2 we have... 12÷3(2×3+2^2) = 4(6+4)= 4(10)= 40 ab+ac = 12÷3×2×3+12÷3×2^2= 4×2×3+4×4= 8×3+16= 24+16= 40. <<< same answer What most people don't understand is that 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÷2×3^2-6÷2×4= 6÷2(1+2+3^2-4) as the LIKE TERM 6÷2 was factored out of the expanded expression. I hope this helps you understand the issue a little better...
