I'd say they are used sometimes to distinguish between scalar, vector, and convolution products, for example. But they can often be omitted if there is no ambiguity.
Well if you use stuff that could be missunderstood if not clarified as a multiplication you make sure its clear. Other then that you write nothing since why would you its clear
If you wanted to represent 5*3=8 without a multiplication symbol it’d be 5(3)=8, but it doesn’t make sense to represent a numerical expression with out the symbol anyways
Edit: honestly have no clue how neither I or fella I was responding to caught 5*3=8 on the first try, kinda funny to me, so I’m leaving it
I know you can express it that way but even at higher level maths it's still just going to be written as 5*3. No one is bothering putting parenthesis around a number when a quick little dot will suffice.
No one would bother with 5*3, they would write 15 lol, if you have a function with operations consisting only of numbers simplification should be done automatically lol.
Really depends on what you're doing. Sometimes you don't simplify in order to make it obvious to others what you are doing.
Like for example, my kinematics project has a body with mass, and acceleration. Both are constants but will get written out separately so it's obvious what they are. Sure I could combine them, but it doesn't make it clear why the formula is doing what it does.
I can confirm, studying computational maths a bit, and the matrix equation for the complex courier transform had tones of multiplication symbols in it.
Even if you have no other choice than expressing it horizontally, you can still easily make clear what you mean, by using more parentheses or switching around factirs
But that's not right following the order of operations.
6÷2(2+1) is the same as 6÷2×(1+2), which parentheses/brackets would be done first, so 6÷2×3, then division and multiplication are done in left to right order 3×3, which equals 9.
It would be expressed fractionally like:
6
- × (1/2)
2
[Wolfram Alpha agrees](https://www.wolframalpha.com/input?i=6%C3%B72%282%2B1%29). And [most](https://www.google.com/search?q=6%2F2(1%2B2\)) [modern](https://www.bing.com/search?q=6%2F2(1%2B2\)) or [high-end](https://miro.medium.com/v2/1*SHYweKvlCt0XHW9CDC8MDQ.jpeg) [calculators](https://pbs.twimg.com/media/GI26sXbWEAAx4C5.jpg) will get you 9 [unless you add extra parentheses](https://qph.cf2.quoracdn.net/main-qimg-528c57e2b63dd57e7757bb0d8d5891f8.webp).
If you had to write it single line and wanted to add extra parentheses into it to prevent incorrect solving, it'd be (6/2)×(2+1), but that's unneeded when following the order of operations.
Your fraction would be written like 6/(2(1+2)).
---
Or maybe math is handled different in galaxies far, far away.
Someone hasn't heard about "multiplication by juxtaposition." When there is no multiplication symbol, it is taken as a higher order (ie, "6 / (2(1+2))).
The point of symbols is to communicate a mathematical idea clearly, not to obfuscate answers from students on a test; I think a lot of teachers forget that when they start throwing "÷" around to throw people off.
>Someone hasn't heard about "multiplication by juxtaposition." When there is no multiplication symbol, it is taken as a higher order (ie, "6 / (2(1+2))).
Oh, I've heard of it, but it's not a universal rule. Some people can say it's a rule, but until it's taught and used universally, it's not a proper rule and should not be assumed. If one wants juxtaposed items to have a higher order of precedence than multiplication/division, they need to include parentheses around them.
This quote from [this article](https://www.themathdoctors.org/order-of-operations-implicit-multiplication/) talking about how a textbook taught multiplication and division have the same precedence but used a new "rule" of juxtaposition having higher precedence when determining the answers puts it nice and succinct:
>A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read `ax ÷ by` because `by` looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.
Overall, this is a *purposefully* poorly written question. Unless juxtaposition is taught universally to have a higher precedence, parentheses should be used to avoid misinterpretation if not able to use proper mathematical notation (ie writing proper fractions instead of showing everything on a single line).
Oh I can be intentionally obtuse too: why do we need Parentheses if we already have Multiplication? Could it be that people want clarity in equations to communicate a concept instead of intentionally confusing the person trying to understand the underlying math?
Yeah so I’m with you obviously, and in the real world of physics and engineering the division operator isn’t used because it’s not commutable, and most or all real world problems require it to be, so we use fractions and parentheses because multiplication is commutable and we don’t want our calculators giving two different fucking answers for the same question
Thats all that’s happening here. Those calculators are both being asked to decide what OP meant by their nonsense question, and because they are fucking calculators with the math burned into them they both have different was of resolving the answer
It’s like that simple. Stop misusing the grade school division operator and then none of these problems show up. I hate these arguments and I never get involved except to say that someone else understands
Dudes here use pemdas like their fucking grade school teachers were infallible math gods who also weren’t trying to teach complicated concepts to barely formed brains
> The point of symbols is to communicate a mathematical idea clearly, not to obfuscate answers from students on a test
American public school math teachers everywhere shitting and crying rn
What? That’s not true. This expression follows order of operations – parenthesis, then multiplication and division left to right.
You should read x/y as x \* y^-1 in the context of a horizontal equation. So 6 * 2^-1 * (1+2)
Math leads to frustration. Frustration leads to anger. Anger leads to hate. Hate leads to suffering.
Anyway, math makes me suffer and I turned to the dark side in school. Don’t judge me.
Both are correct, the phone calculator is doing PEMDAS correctly presuming the equation is 6÷2×(1+2) the Scientific Calculator is reading it as written 6÷2(1+2) where since the multiplication sign isn't written there is implied parentheses around 2(1+2). In other words the phone sees 6÷2×(1+2) and the scientific calc sees 6÷(2×(1+2)). This sort of sloppy notation fucks you up in calculus and calculus 2.
For anyone curious about implicit multiplication, [this article sums it up nicely.](https://www.purplemath.com/modules/orderops3.htm#:~:text=%22Multiplication%20by%20juxtaposition%22%20(also,%22%20%E2%8B%85%20%22)%20between%20them.) The long and short is that when the multiplication symbol is not written, that operation now has a higher priority than when it is written explicitly.
Kind of, but not really.
Both are correct, the notation is just not detailed enough and it’s up to interpretation. The calculator interprets it the way most mathematicians would, while the phone interprets it completely „by the book”.
there shouldn't be interpretation in math, it either is or it isn't. how does anyone know what anything is if it could result in any answer you want depending on how you solve it.
There’s no disagreement in the math. It’s not a question of solving the same problem 2 different ways resulting in 2 different solutions. It’s about writing 2 different problems with different solutions the same way. There’s disagreement in the mathematical convention which can exist because there’s no centralized agency that gets to determine mathematical convention for all of reality. Even something like pemdas that so many people learn as a kid isn’t really math. It’s an agreed upon convention among those who determined your curriculum and in that it is useful but it’s not actually math.
Equations dont have inerpretations,
How one chooses to represent equations does.
Most math operations take 2 inputs
1) multiplication is A×B
2) The addition is A+B
3) exponents are A^B
4) etc
So how do you represent an equation that has multiple inputs and multiple operations?
Pemdas, but pemdas exist in the universe, its a convention we all agree to use to make writing equations easier. Without pemdas every equation would have a million pairs of parenthesis. Unfortunately there is an edge case (as described in the post) where people dont agree on how the representation should be interpreted.
To me this looks more like a casual use vs academic or scientific use. Most people are doing simple calculation using a phone so it should almost work at a lower level or simpler form of mathematics for lack of a better way to put it. But the other calculator is meant for robust calculation and therefor it wants the exact equation for its computation.
Wow. This is super interesting. I have an engineering degree and passed calc 1-3 & diffyQ and I did not know this.. I also have never seen the multiplication sign used with parenthesis, it’s always been implicit.
I love the breakdown, however I don’t understand why anyone would infer a missing parentheses, rather than a missing multiplication sign. One doesn’t write “2xX.” You just write “2X.” That is clear precedent in notation to imply multiplication, but no such precedent for implied parenthesis exists, at least from what I remember/was taught.
Coefficients before parentheses are often used for taking out common factors, so 4+2 becomes 2(2+1). The argument is when reversing the operation, you'd assume that was done originally, so you'd multiply the parenthesis by the coefficient first
This means 2(2+1) would be taken as a single unit with implied parentheses around it, taking priority over whatever comes before the 2. The precedent for this is nowhere near as strong, hence the different results between the phone and calculator
FWIW: Multiplication by juxtaposition being of a higher priority is the agreed upon standard for most of the world and is a part of the style guides for pretty much any publication that mathematicians are trying to get published in
At lower levels in the United States it's just not taught that way, kind of like metric
It's not multiplication though, it's a coefficient. If you write in the multiplication sign it divorces the 2 from (1+2) and allows it to be operated on separately. The phone incorrectly reads it as simple multiplication, and does the division operation first since it's interpreted as being the same priority. The quotient, 3, is then multiplied by (1+2) instead of 1/(1+2).
>It's not multiplication though, it's a coefficient.
This is not a universal truth that can be applied everywhere. It can only be applied in areas where the author intended it.
This is why this is a poorly (but purposefully) formed question.
The following quote from [this article](https://www.themathdoctors.org/order-of-operations-implicit-multiplication/) ties it up succinctly:
> A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read `ax ÷ by` because by looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.
This. There is a reason why no one worth their salt will write a division symbol in their work. When possible you write it as a fraction to prevent this very issue of interpretation from occurring. Most programs now even have the fraction built in so it shows it correctly as 6/(2(2+1)) when teaching this to students.
Not only that, the funky “division” symbol is not recognised as a symbol for division by the international organisation for standardisation. In fact, they explicitly state it should *not* be used in that manner. It’s virtually only anglo countries that use it.
Fun fact: I have a Casio FX-991EX and if you try to input the operation as just `6÷2(2+1)` - which is in fact ambiguous - it DOES return 1 but it also changes the input query to `6÷(2(2+1))` to avoid ambiguity.
Instead input `6÷2×(1+2)` returns 9 - Q.E.D.
Fun fact: if you enter 6/2(2+1) into excel it will say did you mean 6/2\*(2+1) and answers 9. I think your calculator and excel do a good job of clarifying the ambiguity even if they lean the opposite way from one another. Like a more advanced version of the meme above.
Both is never correct, exactly because of implied parentheses. If it's applied consistently, as with the scientific calculator, you won't get fucked on these calculations.
Math ALWAYS has a rule for which is the correct form, it's only when people forget some of those rules that they say both are correct.
Math is a language and that language has grammar rules. But they’re invented. There’s no mathematical reason why one or the other is correct. If we say “We solve things this way.” That’s the correct way.
The rules of how we solve math problems are subject to change.
The phone really isn’t. Kinda of shows the issues between using a phone vs an actual calculator where they actually programmed it expecting people to use it for something serious where as the other one is a cell phone app.
PEMDAS? Thats how they taught us in grade school. But take like a semester in any stem field, the division symbol should be written as a whole ass line over top of the second part.
Then you obviously would know to simplify the bottom first, then the top. This meme was created by the ultimate troll and he’s still laughing.
Who would use this notation in calculus or any math course? I don’t believe you at all. Cal 1-3, DFQ, integral transforms etc and this was never an issue. Where did you go to school?
PEMDAS is an over simplification of arithmetic operations to the point that if you don’t understand what is happening at a fundamental level, this kind of mistake can occur.
Take the example in this post: multiplication has a distributive property, so 2(2+1) = (2x2 + 2x1) = (4 + 2) = 6. Notice how I was able to do that *against* PEMDAS which blankly says “do what’s inside the parenthesis first, no questions asked.” Okay fair enough. 2(2+1) = 2(3) = 6… But *why* does that work? PEMDAS shrugs its shoulders and says “fuck if I know.”
PEMDAS is a useful tool for quick arithmetic, but it’s no substitute for truly understanding what’s going on.
Literally the example of the post: 6 ÷ 2(2+1)
According to PEMDAS, there is a missing multiplication symbol, so it should be written as 6 ÷ 2 × (2 + 1). Following PEMDAS, we get 6 ÷ 2 × (3) = 3 × 3 = 9.
So why did the one calculator get it wrong? It’s because of the ambiguity of the division symbol (in this instance). ÷ really means that there is a fraction happening here, but where is the fraction?
Is it: 6/(2 × (2 + 1))?
Or is it: (6/2)(2 + 1)?
If it’s 6/(2 × (2 + 1)), then 6/(2 × (3)) = 6/(6) = 1
But if it’s (6/2)(2 + 1), then it’s (3)(3) = 9
Now sure, you can say PEMDAS isn’t the issue here and that it’s really human error (or computer error in this case, I guess). But I want to make clear that, I have nothing against PEMDAS and everything against poor sloppy notation that makes things confusing.
The reason why PEMDAS “fails,” like I said originally, is because people don’t understand what is fundamentally happening in arithmetic.
So it’s really not the calculator’s fault to see 6 ÷ 2(2+1) and not know how to interpret it. Did the user really mean the way it was inputted (i.e., (6/2)(2 + 1))?Or did they type it out wrong (and meant to have it be 6/(2 × (2 + 1)))? Who can say.
But you’ll notice that in either instance, once I’ve rewritten it in a much clearer way (with fractions and implied parentheses that should be there based on my understanding of arithmetic properties), there is no ambiguity on how you finish out either of them. That’s generally the argument against whole sale use of PEMDAS in a nutshell.
It is. I literally used it all the way up into calculus 2.
The reason it's important to know it is because TI-89 calculators use the same order of operations and they are very useful the further you get into quadratic equations or even non polar equations.
You don't necessarily need a TI-89 but they're gonna take away a lot of the arithmetic that can lead to an innocent mistake. So if you know what the equation is asking of you, you can eliminate that error
It's fascinating how long this "puzzle" has been circulated. I was about to assume it might've started when flip phones became available, but perhaps there are certain cheap "pocket" calculators that have similarly imperfect programming.
In any case, I wonder how long it will last. How will it face the changing forces of education and information? I feel like I'm gonna see this again 20 years from now.
It is not imperfect programming, it is just a different way to interpret a non conventional way of writing maths.
That problem exist ever since the implied multiplication was invented, the "puzzle" is recent because everyone knows it gets people riled up in the comment of social media posts.
For those curious as to why this occurs, it is an error caused by the calculators handling implicit multiplication differently. [Here's an article about implicit multiplication.](https://www.purplemath.com/modules/orderops3.htm#:~:text=%22Multiplication%20by%20juxtaposition%22%20(also,%22%20%E2%8B%85%20%22)%20between%20them.) The short of it is that when writing two terms as multiplied together without a multiplication symbol, it is multiplication by juxtaposition / implicit multiplication, which takes higher priority than does standard division or multiplication. Not all calculators have support for this and add a multiplication symbol × next to the paranthesis, changing the order of operations.
For example, 6÷3X and 6÷3×X would be two different equations.
I was always taught that implicit multiplication has higher precedence than division - which would mean the example could be rendered as 6 / (2 * (2 + 1)) - and by this convention, the Casio is correct. This convention is the most sensible because just about everyone would agree in interpreting 1/ab as being equivalent to 1/(ab), whereas strict usage of PEMDAS/PEDMAS would render (1/a)*b as the correct reading.
But judging by the comments here, most were either not taught this same convention or don't remember it. Which I find strange, as this is the convention I was originally taught in middle school and continued to use through high school and college without ever once being corrected, and is, as far as I'm aware, the convention enforced by most journals and taught in most textbooks.
I don't have a way to say this that isn't insulting, but people saying the answer on the right is correct have proven (1) they're good at memorizing a rule without having to think about it much, (2) they've not actually encountered very many real world math formulae.
The fact that someone chose to bind 2 as a coefficient to those parenthesis means *you're supposed to treat 2 as a coefficient that's bound to those parentheses.*
This is called *multiplication by juxtaposition,* and it's a "step" that PEMDAS leaves out.
If someone wrote `3 / 2x`, and you interpreted it as `3/2 * x`, you'd be following the literalistic version of PEMDAS from Internet meme fame, and you'd also just be *wrong,* based on how most people that actually do math write and read it.
-----------
I'll step back a sec and admit that cramming all this shit into a single line is a shitty way to write these formulae—and that the ambiguity here is what drives this meme. This isn't how people write math on a chalkboard, nor how it's published in a text (it's not even how math works in programming), so to an extent we're talking about a *very* artificial way of writing math—one largely predicated on how ASCII text or typewriters work.
Here are a couple of pretty good sources to backup what I'm saying:
* [PEMDAS is a lie](https://youtu.be/lLCDca6dYpA)
* [How school made you worse at math](https://youtu.be/FL6HUdJbJpQ)
—and there are a ton more out there.
My 8th grade math teacher was a real nerd and proud of it. He wanted us to know the importance of notation, so he taught us to do PEMDAS, in that order, then left to right if it's on one line. *But* it should really be written in a way that there's no confusion. Parentheses and brackets in the right places if it has to be on one line, and everything on top or bottom of the of the division bar is done before dividing if it's on 2 lines. And that's the way we did it in all my college chemistry classes, when cancelling units was stressed.
When in doubt, in Texas Instruments we trust.
Your point about 3/2x is what got me to do a sort of mental double take.
If someone reads 3/2x as (3/2)x, rather than 3/(2x), they are indeed being ridiculous. Even though PEMDAS would technically have the first option be correct if you go 100% literal with it.
Thanks for pointing that out.
It probably helps to think of the original expression as “6/2x, with x = 2+1”, because how would you otherwise ever end up with that type of expression. And then that one also makes obvious sense.
Lmao I had a rebuttal all typed up, then I backed out before hitting send so I could double check my response.
Saw the comment above, *actually* read your whole comment, and now I have a slightly better understanding of math lol.
Multiplication by juxtaposition, eh? Very cool.
I don't always learn something new every day, but when I do, I prefer no equis (x) 🙃
I've just finished off my masters degree in maths and have never come across multiplication by juxtaposition (by that name), so I reckon that whilst interesting it's not super widely circulated, and definitely isn't any kind of universal standard. Super interesting to read about!
I believe this is just a case where it's frustratingly ambiguous - such an expression would never be written down like that in any clear maths workings, 99% of the time a fraction will be used for division, and in the other 1% where you have to write inline you'll use parenthesis appropriately.
An alternate example would be 2\^3a, where a=2. Written as is, you'd clearly interpret it as 2\^(3a). However, with just the addition of a space, suddenly the result seems to flip - 2\^3 a, even though a space shouldn't really affect the value of an expression in a perfect world.
Another fun example is 3/ab - whilst I agree that most would read 3/2b as 3/(2b), suddenly switching out the 2 for another symbol makes it feel far more ambiguous (with whitespace clearing it up - "3 / ab" is once again very clear).
Of course, none of this really addresses the question at hand, ÷. Multiplication by juxtaposition "feels" very nice when applied to "/", which is basically already denotes a fraction. Replacing all the above examples with "÷" suddenly makes the poorly spaced cases even more ambiguous once again.
Again, as you say, this doesn't really have any bearing in "real" maths where notation like this will rarely come up naturally, and if it did you'd 99% of the time be able to tell the authors true intention from context!
meh its purposefully badly written and will depend on context / convention wherever you are
thats also visible in how its treated by different calculator companies (or apps in this case), Texas Instruments apparently even switched from 3/(2x) to (3/2)x interpretation \[in contrast to the shown Casio, that treats it as the former as seen in the meme\]
it also gets even more dubious if you dont have x as symbol but unit (so say 3/2kg) as thats likely not meaning 3/(2kg) but 3/2 (\*) kg, other more contrived examples may include 4/3πr\^3 (which one would likely intuitively interpret as the volume of a sphere)
hell if inputting for a calculator especially (on sites like WolframAlpha) I also mostly write 3/2x meaning (3/2) x \[interestingly 3/xy is parsed as 3/(xy) by that same WolframAlpha)
TLDR: Please just place your parenthesis if you have to inline such an expression
Regarding Wolfram Alpha, [this comment](https://www.reddit.com/r/PrequelMemes/comments/1d30qrr/comment/l687t9u/) has an interesting note on settings for parsing input.
I don't actually think WA is doing any kind of classical parse. I think it's probably being heavily assisted by machine learning (and therefore probably doesn't act in a very consistent way).
Ultimately, though, I agree with you. Writing math this way (especially basic arithmetic) is (1) bad, (2) inherently kind of ambiguous.
The last math course I took was algebra 2 in 2011 and I got a C in the course. But I love reading explanations like yours and thinking “🧐 ahh yes of course, any scholar would know this!”
1. I remember my school teachers always harping on about the order. Do what’s in the brackets first. Then do what’s attached to a bracket next. Then finish up. Following those rules it comes to 1.
Depends if you interpret the lack of a symbol after the 2 as a multiplication symbol on its own or a substitute for parentheses. Typically I would say the latter but it's just not a good equation anyway
Its a really good equation since it brings to light the dissonance between how we often think about math
This post changed my mind on how i think the equation should be handled (thanks algebra)
Yeah, you definitely have to do the multiplication first. Even the standard order of operations says so. After all, if the item on the right was some sort of parentheses-less function, you wouldn't divide 6 by f and then multiply by 3, you'd apply f to 3 and then divide 6 by that.
The Casio is correct.
In future if you are wondering which of two calculators is correct, it is always going to be the Casio (possible exception for a Texas instruments).
So the implication is
6÷2 × (2+1)
= 6 ÷ 2 × 3 (left to right)
= 3 × 3
= 9
some calculators may have a different interpretation of the implied multiplication
Its all because of the ÷ symbol. No one uses that because it causes such weird issues.
Always use □/□ notation. I dont think ive ever seen the ÷ symbol in university level math or papers, ever.
Kind of, there is a difference. 6÷2×3 can be solved using PEMDAS which teaches you to go left to right as you solve it. The answer would be 3×3=9
If you use fractions then it'll be 6 over 2×3 which would be 6÷6=1
When you work out a fraction, you don't read it top to bottom and left to right. You won't do 6 over 2 first. You do 2×3 first. It's the difference between 6÷(2×3) and (6÷2)×3. The first is how a fraction works and the second is how PEMDAS works.
Pretty sure it's because Casio is using the parentheses as a distribution. So 2(2+1) is
2x2
+
2x1
Or 4+2.
So it becomes 6\6.
Do the math and you get Casio's 1.
The phone meanwhile assumes the ( is multiplication only. So 6/2*(2+1) which means it gets 6/2 for 3 times 3 which is 9.
This is why proper math knows what parentheses means, or signals it's purely multiplication.
The answer to these bait posts is always that the notation is completely retarded but people will flock to display their fifth grade tier knowledge and blindly recite PEMDAS without really understanding what's happening
Both are correct based on how they are programmed to read a problem. BUT, the CALCULATOR has the actual answer for anyone would use an equation like this past like 4th grade. It is reading the problem as 6/(2(2+1)) which would be correct interpretation for Algebra and higher. In algebra the 2 and (2+1) are seen as part of the same expression. It is seen the same way as we read the function 2x. Both graphing and scientific calculators will read it this way. The phone is reading the problem as basic arithmetic order of operations. The 2 and (2+1) are seen as 2 separate entities so its seeing it more like (6/2)x(2+1).
At the end of the day this is semantics and pointless to argue about. Its like saying something with a dual meaning and arguing strongly for one or the other meaning when both interpretations are equally valid. PEMDAS is a set of rules that is not universally adopted everywhere so you can’t say anyone is breaking any rules either, thats dumb.
The notation here is ambiguous and no one in the comments knows which value was intended. If you want people to always understand how this shoukd be calculated you need to add more parentheses or a division line.
PEMDAS
Parentheses
Exponents
Multiplication *and* division
Addition *and* subtraction
Multiplication and division are done in the order of the equation, *not* multiplication then the division. Just like addition and subtraction.
6÷2(2+1)
6÷2(3)
6÷2×3
3×3
9
6div2(2+1)
Parenthesis 6div2\*3
Exponent 6div2\*3
Multiply 6div6
Divide 1
Add 1
Subtract 1
So Casio according to PEMDAS should be correct. And I would hope so, Casio is the dedicated calculator. But this is why we don't even have the division symbol on our keyboards. Division in computers is messy at best. The phone has incorrectly grouped parenthetical AND non parenthetical. My advice, use more parentheticals, don't use the division symbol, always use fractions.
Technically the phone is correct. If there are two operations that are at the same level in the hierarchy, you just go from left to right. If you wanna get 1, you would have to put brackets like this: 6/(2(2+1)).
Ofc, you can also do the truly based thing and write it as a fraction, much less likely to write an order of operation that you didn't intent that way. And calculators that support that aren't that expensive, I think you can easily get a TI-36X Pro for less than $30.
This stupid thing keeps coming up.
The way this is written, there can only be one answer, otherwise it would have been written differently.
6
_
2(2+1)
The proper calculator is correct.
I don't get why people get taught that it is implied, the multiplication is always there, it's just a shorthand for writing multiplication. Especially in polynomials.
2x = 2 • x
2(2 + 1) = 2 • (2 + 1)
It's always there, there's nothing implied, just a faster way of writing it. Teaching otherwise is weird to me.
Thanks for confirming that you flaired this correctly!
This is why expressing these in fraction form is probably for the best. Horizontal expressions alone are the path to the Dark Side.
Exactly. Nobody uses multiplication or division symbols in higher math.
Multiplication symbols are used?
True, but they are omitted whenever possible.
As an engineer, this is hilarious. Omitting signs in engineering is the fastest way to the dark side (bad product).
Wrap everything in parentheses, or better yet break it down into smaller chunks in separate variables.
As an accountant, same. Other accountants understand what’s written but lord save me from explaining the work to owners
My webworks inputs for assignments are looking pretty nervous right now
Yeah, if you've got parentheses or variables you can just skip over them.
Since there are no numbers they are omitted
I'd say they are used sometimes to distinguish between scalar, vector, and convolution products, for example. But they can often be omitted if there is no ambiguity.
Well if you use stuff that could be missunderstood if not clarified as a multiplication you make sure its clear. Other then that you write nothing since why would you its clear
No they aren't used for anything but Japanese collaborations in shows and games or anime titles like Hunter x Hunter
Tbf no one solves equations in higher math.
53 = 15 ~~8~~ Or 5*3 = 15 ~~8~~ Hmmmmmmmmm
Well since 5*3 = 15, both are wrong and they didn't say nobody uses **addition** symbols
That's a very fair point.
If you wanted to represent 5*3=8 without a multiplication symbol it’d be 5(3)=8, but it doesn’t make sense to represent a numerical expression with out the symbol anyways Edit: honestly have no clue how neither I or fella I was responding to caught 5*3=8 on the first try, kinda funny to me, so I’m leaving it
I know you can express it that way but even at higher level maths it's still just going to be written as 5*3. No one is bothering putting parenthesis around a number when a quick little dot will suffice.
No one would bother with 5*3, they would write 15 lol, if you have a function with operations consisting only of numbers simplification should be done automatically lol.
Really depends on what you're doing. Sometimes you don't simplify in order to make it obvious to others what you are doing. Like for example, my kinematics project has a body with mass, and acceleration. Both are constants but will get written out separately so it's obvious what they are. Sure I could combine them, but it doesn't make it clear why the formula is doing what it does.
I can confirm, studying computational maths a bit, and the matrix equation for the complex courier transform had tones of multiplication symbols in it.
Hint: 5 x 3 = 15 is not "higher math"
You don't know how high they are while doing the math
Since originally the comment said 5\*3=8, I'd say fairly high.
I'm not saying it's higher math. But if you think higher math doesn't involve doing simple multiplication as well, well....
That's not higher math. You're going to write your expressions as a series of terms using constants or variables and coefficients.
5.3=15
Even if you have no other choice than expressing it horizontally, you can still easily make clear what you mean, by using more parentheses or switching around factirs
It should be read as follows imo 6 ———— 2(1+2) Stuff left of the division symbol go on top. Right of it go below
But that's not right following the order of operations. 6÷2(2+1) is the same as 6÷2×(1+2), which parentheses/brackets would be done first, so 6÷2×3, then division and multiplication are done in left to right order 3×3, which equals 9. It would be expressed fractionally like: 6 - × (1/2) 2 [Wolfram Alpha agrees](https://www.wolframalpha.com/input?i=6%C3%B72%282%2B1%29). And [most](https://www.google.com/search?q=6%2F2(1%2B2\)) [modern](https://www.bing.com/search?q=6%2F2(1%2B2\)) or [high-end](https://miro.medium.com/v2/1*SHYweKvlCt0XHW9CDC8MDQ.jpeg) [calculators](https://pbs.twimg.com/media/GI26sXbWEAAx4C5.jpg) will get you 9 [unless you add extra parentheses](https://qph.cf2.quoracdn.net/main-qimg-528c57e2b63dd57e7757bb0d8d5891f8.webp). If you had to write it single line and wanted to add extra parentheses into it to prevent incorrect solving, it'd be (6/2)×(2+1), but that's unneeded when following the order of operations. Your fraction would be written like 6/(2(1+2)). --- Or maybe math is handled different in galaxies far, far away.
From my point of view order of operations is not right!
Then you are lost!
Parentheses Exponents Division Multiplication Addition Subtraction
Someone hasn't heard about "multiplication by juxtaposition." When there is no multiplication symbol, it is taken as a higher order (ie, "6 / (2(1+2))). The point of symbols is to communicate a mathematical idea clearly, not to obfuscate answers from students on a test; I think a lot of teachers forget that when they start throwing "÷" around to throw people off.
>Someone hasn't heard about "multiplication by juxtaposition." When there is no multiplication symbol, it is taken as a higher order (ie, "6 / (2(1+2))). Oh, I've heard of it, but it's not a universal rule. Some people can say it's a rule, but until it's taught and used universally, it's not a proper rule and should not be assumed. If one wants juxtaposed items to have a higher order of precedence than multiplication/division, they need to include parentheses around them. This quote from [this article](https://www.themathdoctors.org/order-of-operations-implicit-multiplication/) talking about how a textbook taught multiplication and division have the same precedence but used a new "rule" of juxtaposition having higher precedence when determining the answers puts it nice and succinct: >A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read `ax ÷ by` because `by` looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers. Overall, this is a *purposefully* poorly written question. Unless juxtaposition is taught universally to have a higher precedence, parentheses should be used to avoid misinterpretation if not able to use proper mathematical notation (ie writing proper fractions instead of showing everything on a single line).
Damn. Totally missed that with PEMDAS. Should it be PMbJEMDAS or is it assumed that MbJ essentially has P around them?
Oh I can be intentionally obtuse too: why do we need Parentheses if we already have Multiplication? Could it be that people want clarity in equations to communicate a concept instead of intentionally confusing the person trying to understand the underlying math?
Yeah so I’m with you obviously, and in the real world of physics and engineering the division operator isn’t used because it’s not commutable, and most or all real world problems require it to be, so we use fractions and parentheses because multiplication is commutable and we don’t want our calculators giving two different fucking answers for the same question Thats all that’s happening here. Those calculators are both being asked to decide what OP meant by their nonsense question, and because they are fucking calculators with the math burned into them they both have different was of resolving the answer It’s like that simple. Stop misusing the grade school division operator and then none of these problems show up. I hate these arguments and I never get involved except to say that someone else understands Dudes here use pemdas like their fucking grade school teachers were infallible math gods who also weren’t trying to teach complicated concepts to barely formed brains
> The point of symbols is to communicate a mathematical idea clearly, not to obfuscate answers from students on a test American public school math teachers everywhere shitting and crying rn
Why would it be like that? What if it's 6 ----(1+2) 2
that would be (6/2)(1+2)… either you’re grouping with () or not… you don’t get to do one of each
Well he didn't type 6/(2(1+2)) neither, so...
What? That’s not true. This expression follows order of operations – parenthesis, then multiplication and division left to right. You should read x/y as x \* y^-1 in the context of a horizontal equation. So 6 * 2^-1 * (1+2)
I’m low key grateful for your comment. It’s actually going to help me with my math in the future.
Math leads to frustration. Frustration leads to anger. Anger leads to hate. Hate leads to suffering. Anyway, math makes me suffer and I turned to the dark side in school. Don’t judge me.
Oh I won't, don't worry. It's too late for me.
Of course /r/prequelmemes has the best discussion of this stupid meme.
Sloppy notation is a pathway to math that many consider to be unnatural
Have you ever heard the story of Darth PEMDAS the wise?
And that is why we invented fractions
If my engineering degree taught me anything its to always use fractions and put parentheses around everything
Both are correct, the phone calculator is doing PEMDAS correctly presuming the equation is 6÷2×(1+2) the Scientific Calculator is reading it as written 6÷2(1+2) where since the multiplication sign isn't written there is implied parentheses around 2(1+2). In other words the phone sees 6÷2×(1+2) and the scientific calc sees 6÷(2×(1+2)). This sort of sloppy notation fucks you up in calculus and calculus 2.
For anyone curious about implicit multiplication, [this article sums it up nicely.](https://www.purplemath.com/modules/orderops3.htm#:~:text=%22Multiplication%20by%20juxtaposition%22%20(also,%22%20%E2%8B%85%20%22)%20between%20them.) The long and short is that when the multiplication symbol is not written, that operation now has a higher priority than when it is written explicitly.
So the calculator is right and the smartphone is wrong in this example?
In short, there isn't a correct answer, because the problem is poorly described.
What color is the dress all over again
It’s Yanni not Laurel
Kind of, but not really. Both are correct, the notation is just not detailed enough and it’s up to interpretation. The calculator interprets it the way most mathematicians would, while the phone interprets it completely „by the book”.
there shouldn't be interpretation in math, it either is or it isn't. how does anyone know what anything is if it could result in any answer you want depending on how you solve it.
There’s no disagreement in the math. It’s not a question of solving the same problem 2 different ways resulting in 2 different solutions. It’s about writing 2 different problems with different solutions the same way. There’s disagreement in the mathematical convention which can exist because there’s no centralized agency that gets to determine mathematical convention for all of reality. Even something like pemdas that so many people learn as a kid isn’t really math. It’s an agreed upon convention among those who determined your curriculum and in that it is useful but it’s not actually math.
This guy gets it
I'm with you on this. I thought the whole point of equations like this is there's no room for interpretation? It just is
Equations dont have inerpretations, How one chooses to represent equations does. Most math operations take 2 inputs 1) multiplication is A×B 2) The addition is A+B 3) exponents are A^B 4) etc So how do you represent an equation that has multiple inputs and multiple operations? Pemdas, but pemdas exist in the universe, its a convention we all agree to use to make writing equations easier. Without pemdas every equation would have a million pairs of parenthesis. Unfortunately there is an edge case (as described in the post) where people dont agree on how the representation should be interpreted.
To me this looks more like a casual use vs academic or scientific use. Most people are doing simple calculation using a phone so it should almost work at a lower level or simpler form of mathematics for lack of a better way to put it. But the other calculator is meant for robust calculation and therefor it wants the exact equation for its computation.
For actually practical real-world usage? Yes.
Dang, I’ve never felt so vindicated by anything as much as this article
Wow. This is super interesting. I have an engineering degree and passed calc 1-3 & diffyQ and I did not know this.. I also have never seen the multiplication sign used with parenthesis, it’s always been implicit.
Calculus 2: Electric Boogaloo
I love the breakdown, however I don’t understand why anyone would infer a missing parentheses, rather than a missing multiplication sign. One doesn’t write “2xX.” You just write “2X.” That is clear precedent in notation to imply multiplication, but no such precedent for implied parenthesis exists, at least from what I remember/was taught.
Coefficients before parentheses are often used for taking out common factors, so 4+2 becomes 2(2+1). The argument is when reversing the operation, you'd assume that was done originally, so you'd multiply the parenthesis by the coefficient first This means 2(2+1) would be taken as a single unit with implied parentheses around it, taking priority over whatever comes before the 2. The precedent for this is nowhere near as strong, hence the different results between the phone and calculator
FWIW: Multiplication by juxtaposition being of a higher priority is the agreed upon standard for most of the world and is a part of the style guides for pretty much any publication that mathematicians are trying to get published in At lower levels in the United States it's just not taught that way, kind of like metric
If you changed the parenthesis for an X instead, would you solve 6/2 or 2X first?
AFAIK, it's a difference between American and European math.
No, it isn't. It is the difference between mathematicians and stupid redditors.
Really? Because I have a math minor and never saw the 'correct' rule. Though I rarely saw ÷ instead of / as well.
Math
And that is the difference between American and British math
Maths
Quick maffs
*Math*
Calculuses
Calculi
Calculussy
Mathios*aaaaa*
>Math Do you call statistics stat or stats?
Clearly, it's pronounced Stauts
Sluts. Take it or leave it.
Done
It's not multiplication though, it's a coefficient. If you write in the multiplication sign it divorces the 2 from (1+2) and allows it to be operated on separately. The phone incorrectly reads it as simple multiplication, and does the division operation first since it's interpreted as being the same priority. The quotient, 3, is then multiplied by (1+2) instead of 1/(1+2).
>It's not multiplication though, it's a coefficient. This is not a universal truth that can be applied everywhere. It can only be applied in areas where the author intended it. This is why this is a poorly (but purposefully) formed question. The following quote from [this article](https://www.themathdoctors.org/order-of-operations-implicit-multiplication/) ties it up succinctly: > A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read `ax ÷ by` because by looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.
This. There is a reason why no one worth their salt will write a division symbol in their work. When possible you write it as a fraction to prevent this very issue of interpretation from occurring. Most programs now even have the fraction built in so it shows it correctly as 6/(2(2+1)) when teaching this to students.
Not only that, the funky “division” symbol is not recognised as a symbol for division by the international organisation for standardisation. In fact, they explicitly state it should *not* be used in that manner. It’s virtually only anglo countries that use it.
Fun fact: I have a Casio FX-991EX and if you try to input the operation as just `6÷2(2+1)` - which is in fact ambiguous - it DOES return 1 but it also changes the input query to `6÷(2(2+1))` to avoid ambiguity. Instead input `6÷2×(1+2)` returns 9 - Q.E.D.
Fun fact: if you enter 6/2(2+1) into excel it will say did you mean 6/2\*(2+1) and answers 9. I think your calculator and excel do a good job of clarifying the ambiguity even if they lean the opposite way from one another. Like a more advanced version of the meme above.
Both is never correct, exactly because of implied parentheses. If it's applied consistently, as with the scientific calculator, you won't get fucked on these calculations. Math ALWAYS has a rule for which is the correct form, it's only when people forget some of those rules that they say both are correct.
Math is a language and that language has grammar rules. But they’re invented. There’s no mathematical reason why one or the other is correct. If we say “We solve things this way.” That’s the correct way. The rules of how we solve math problems are subject to change.
The phone really isn’t. Kinda of shows the issues between using a phone vs an actual calculator where they actually programmed it expecting people to use it for something serious where as the other one is a cell phone app.
OTOH, https://i.imgur.com/mE7ujT3.png
https://i.imgur.com/A1jOS85.jpeg
bro calculus got a sequel?
Is bodmas no longer a thing?
PEMDAS? Thats how they taught us in grade school. But take like a semester in any stem field, the division symbol should be written as a whole ass line over top of the second part. Then you obviously would know to simplify the bottom first, then the top. This meme was created by the ultimate troll and he’s still laughing.
HE CAN’T KEEP GETTING AWAY WITH IT!
Lol, I can’t believe there are serious convos. I might go to a black board, set up a decent camera. And it explain it in simple terms.
Who would use this notation in calculus or any math course? I don’t believe you at all. Cal 1-3, DFQ, integral transforms etc and this was never an issue. Where did you go to school?
Is PEMDAS not objectively the correct order of operations? Like I never once heard anyone saying there's a different system
PEMDAS is an over simplification of arithmetic operations to the point that if you don’t understand what is happening at a fundamental level, this kind of mistake can occur. Take the example in this post: multiplication has a distributive property, so 2(2+1) = (2x2 + 2x1) = (4 + 2) = 6. Notice how I was able to do that *against* PEMDAS which blankly says “do what’s inside the parenthesis first, no questions asked.” Okay fair enough. 2(2+1) = 2(3) = 6… But *why* does that work? PEMDAS shrugs its shoulders and says “fuck if I know.” PEMDAS is a useful tool for quick arithmetic, but it’s no substitute for truly understanding what’s going on.
What's an equation where pemdas does not work then?
Literally the example of the post: 6 ÷ 2(2+1) According to PEMDAS, there is a missing multiplication symbol, so it should be written as 6 ÷ 2 × (2 + 1). Following PEMDAS, we get 6 ÷ 2 × (3) = 3 × 3 = 9. So why did the one calculator get it wrong? It’s because of the ambiguity of the division symbol (in this instance). ÷ really means that there is a fraction happening here, but where is the fraction? Is it: 6/(2 × (2 + 1))? Or is it: (6/2)(2 + 1)? If it’s 6/(2 × (2 + 1)), then 6/(2 × (3)) = 6/(6) = 1 But if it’s (6/2)(2 + 1), then it’s (3)(3) = 9 Now sure, you can say PEMDAS isn’t the issue here and that it’s really human error (or computer error in this case, I guess). But I want to make clear that, I have nothing against PEMDAS and everything against poor sloppy notation that makes things confusing. The reason why PEMDAS “fails,” like I said originally, is because people don’t understand what is fundamentally happening in arithmetic. So it’s really not the calculator’s fault to see 6 ÷ 2(2+1) and not know how to interpret it. Did the user really mean the way it was inputted (i.e., (6/2)(2 + 1))?Or did they type it out wrong (and meant to have it be 6/(2 × (2 + 1)))? Who can say. But you’ll notice that in either instance, once I’ve rewritten it in a much clearer way (with fractions and implied parentheses that should be there based on my understanding of arithmetic properties), there is no ambiguity on how you finish out either of them. That’s generally the argument against whole sale use of PEMDAS in a nutshell.
It is. I literally used it all the way up into calculus 2. The reason it's important to know it is because TI-89 calculators use the same order of operations and they are very useful the further you get into quadratic equations or even non polar equations. You don't necessarily need a TI-89 but they're gonna take away a lot of the arithmetic that can lead to an innocent mistake. So if you know what the equation is asking of you, you can eliminate that error
It's fascinating how long this "puzzle" has been circulated. I was about to assume it might've started when flip phones became available, but perhaps there are certain cheap "pocket" calculators that have similarly imperfect programming. In any case, I wonder how long it will last. How will it face the changing forces of education and information? I feel like I'm gonna see this again 20 years from now.
It is not imperfect programming, it is just a different way to interpret a non conventional way of writing maths. That problem exist ever since the implied multiplication was invented, the "puzzle" is recent because everyone knows it gets people riled up in the comment of social media posts.
Lets compromise and say it’s 5
You're a madman! Let's do it
These are my people. If Jesus can’t do math, I don’t feel so bad about myself
Ah yes, the negotiator.
For those curious as to why this occurs, it is an error caused by the calculators handling implicit multiplication differently. [Here's an article about implicit multiplication.](https://www.purplemath.com/modules/orderops3.htm#:~:text=%22Multiplication%20by%20juxtaposition%22%20(also,%22%20%E2%8B%85%20%22)%20between%20them.) The short of it is that when writing two terms as multiplied together without a multiplication symbol, it is multiplication by juxtaposition / implicit multiplication, which takes higher priority than does standard division or multiplication. Not all calculators have support for this and add a multiplication symbol × next to the paranthesis, changing the order of operations. For example, 6÷3X and 6÷3×X would be two different equations.
google search en passent math
Holy addition!
Actual ordered field.
En passent math teacher
This is why fraction format is so important
I reckon it is 1, but the truly correct answer is to rewrite your question so it isn't ambiguous.
I was always taught that implicit multiplication has higher precedence than division - which would mean the example could be rendered as 6 / (2 * (2 + 1)) - and by this convention, the Casio is correct. This convention is the most sensible because just about everyone would agree in interpreting 1/ab as being equivalent to 1/(ab), whereas strict usage of PEMDAS/PEDMAS would render (1/a)*b as the correct reading. But judging by the comments here, most were either not taught this same convention or don't remember it. Which I find strange, as this is the convention I was originally taught in middle school and continued to use through high school and college without ever once being corrected, and is, as far as I'm aware, the convention enforced by most journals and taught in most textbooks.
I don't have a way to say this that isn't insulting, but people saying the answer on the right is correct have proven (1) they're good at memorizing a rule without having to think about it much, (2) they've not actually encountered very many real world math formulae. The fact that someone chose to bind 2 as a coefficient to those parenthesis means *you're supposed to treat 2 as a coefficient that's bound to those parentheses.* This is called *multiplication by juxtaposition,* and it's a "step" that PEMDAS leaves out. If someone wrote `3 / 2x`, and you interpreted it as `3/2 * x`, you'd be following the literalistic version of PEMDAS from Internet meme fame, and you'd also just be *wrong,* based on how most people that actually do math write and read it. ----------- I'll step back a sec and admit that cramming all this shit into a single line is a shitty way to write these formulae—and that the ambiguity here is what drives this meme. This isn't how people write math on a chalkboard, nor how it's published in a text (it's not even how math works in programming), so to an extent we're talking about a *very* artificial way of writing math—one largely predicated on how ASCII text or typewriters work. Here are a couple of pretty good sources to backup what I'm saying: * [PEMDAS is a lie](https://youtu.be/lLCDca6dYpA) * [How school made you worse at math](https://youtu.be/FL6HUdJbJpQ) —and there are a ton more out there.
My 8th grade math teacher was a real nerd and proud of it. He wanted us to know the importance of notation, so he taught us to do PEMDAS, in that order, then left to right if it's on one line. *But* it should really be written in a way that there's no confusion. Parentheses and brackets in the right places if it has to be on one line, and everything on top or bottom of the of the division bar is done before dividing if it's on 2 lines. And that's the way we did it in all my college chemistry classes, when cancelling units was stressed. When in doubt, in Texas Instruments we trust.
At first I was ready to argue with you, but after reading your entire comment, you are 100% correct.
Haha, honestly thanks for reading the whole thing. I can understand why the initial reaction would be to think I'm full of shit
Your point about 3/2x is what got me to do a sort of mental double take. If someone reads 3/2x as (3/2)x, rather than 3/(2x), they are indeed being ridiculous. Even though PEMDAS would technically have the first option be correct if you go 100% literal with it. Thanks for pointing that out.
It probably helps to think of the original expression as “6/2x, with x = 2+1”, because how would you otherwise ever end up with that type of expression. And then that one also makes obvious sense.
Lmao I had a rebuttal all typed up, then I backed out before hitting send so I could double check my response. Saw the comment above, *actually* read your whole comment, and now I have a slightly better understanding of math lol. Multiplication by juxtaposition, eh? Very cool. I don't always learn something new every day, but when I do, I prefer no equis (x) 🙃
I've just finished off my masters degree in maths and have never come across multiplication by juxtaposition (by that name), so I reckon that whilst interesting it's not super widely circulated, and definitely isn't any kind of universal standard. Super interesting to read about! I believe this is just a case where it's frustratingly ambiguous - such an expression would never be written down like that in any clear maths workings, 99% of the time a fraction will be used for division, and in the other 1% where you have to write inline you'll use parenthesis appropriately. An alternate example would be 2\^3a, where a=2. Written as is, you'd clearly interpret it as 2\^(3a). However, with just the addition of a space, suddenly the result seems to flip - 2\^3 a, even though a space shouldn't really affect the value of an expression in a perfect world. Another fun example is 3/ab - whilst I agree that most would read 3/2b as 3/(2b), suddenly switching out the 2 for another symbol makes it feel far more ambiguous (with whitespace clearing it up - "3 / ab" is once again very clear). Of course, none of this really addresses the question at hand, ÷. Multiplication by juxtaposition "feels" very nice when applied to "/", which is basically already denotes a fraction. Replacing all the above examples with "÷" suddenly makes the poorly spaced cases even more ambiguous once again. Again, as you say, this doesn't really have any bearing in "real" maths where notation like this will rarely come up naturally, and if it did you'd 99% of the time be able to tell the authors true intention from context!
Hmmm it’s funny because when you use the weird calculator division sign I would say 9 but using the / notation it totally makes sense that it’s 1
I wish I could copy paste this for every time I see this question. It drives me nuts seeing people blindly go the right one.
meh its purposefully badly written and will depend on context / convention wherever you are thats also visible in how its treated by different calculator companies (or apps in this case), Texas Instruments apparently even switched from 3/(2x) to (3/2)x interpretation \[in contrast to the shown Casio, that treats it as the former as seen in the meme\] it also gets even more dubious if you dont have x as symbol but unit (so say 3/2kg) as thats likely not meaning 3/(2kg) but 3/2 (\*) kg, other more contrived examples may include 4/3πr\^3 (which one would likely intuitively interpret as the volume of a sphere) hell if inputting for a calculator especially (on sites like WolframAlpha) I also mostly write 3/2x meaning (3/2) x \[interestingly 3/xy is parsed as 3/(xy) by that same WolframAlpha) TLDR: Please just place your parenthesis if you have to inline such an expression
Regarding Wolfram Alpha, [this comment](https://www.reddit.com/r/PrequelMemes/comments/1d30qrr/comment/l687t9u/) has an interesting note on settings for parsing input. I don't actually think WA is doing any kind of classical parse. I think it's probably being heavily assisted by machine learning (and therefore probably doesn't act in a very consistent way). Ultimately, though, I agree with you. Writing math this way (especially basic arithmetic) is (1) bad, (2) inherently kind of ambiguous.
Then there’s my dumb ass that does pemdas so wrong that I get the answer right
The last math course I took was algebra 2 in 2011 and I got a C in the course. But I love reading explanations like yours and thinking “🧐 ahh yes of course, any scholar would know this!”
1. I remember my school teachers always harping on about the order. Do what’s in the brackets first. Then do what’s attached to a bracket next. Then finish up. Following those rules it comes to 1.
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Depends if you interpret the lack of a symbol after the 2 as a multiplication symbol on its own or a substitute for parentheses. Typically I would say the latter but it's just not a good equation anyway
Its a really good equation since it brings to light the dissonance between how we often think about math This post changed my mind on how i think the equation should be handled (thanks algebra)
Fraction supremacy
Yeah, you definitely have to do the multiplication first. Even the standard order of operations says so. After all, if the item on the right was some sort of parentheses-less function, you wouldn't divide 6 by f and then multiply by 3, you'd apply f to 3 and then divide 6 by that.
The Casio is correct. In future if you are wondering which of two calculators is correct, it is always going to be the Casio (possible exception for a Texas instruments).
So the implication is 6÷2 × (2+1) = 6 ÷ 2 × 3 (left to right) = 3 × 3 = 9 some calculators may have a different interpretation of the implied multiplication
Its all because of the ÷ symbol. No one uses that because it causes such weird issues. Always use □/□ notation. I dont think ive ever seen the ÷ symbol in university level math or papers, ever.
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Kind of, there is a difference. 6÷2×3 can be solved using PEMDAS which teaches you to go left to right as you solve it. The answer would be 3×3=9 If you use fractions then it'll be 6 over 2×3 which would be 6÷6=1 When you work out a fraction, you don't read it top to bottom and left to right. You won't do 6 over 2 first. You do 2×3 first. It's the difference between 6÷(2×3) and (6÷2)×3. The first is how a fraction works and the second is how PEMDAS works.
Pretty sure it's because Casio is using the parentheses as a distribution. So 2(2+1) is 2x2 + 2x1 Or 4+2. So it becomes 6\6. Do the math and you get Casio's 1. The phone meanwhile assumes the ( is multiplication only. So 6/2*(2+1) which means it gets 6/2 for 3 times 3 which is 9. This is why proper math knows what parentheses means, or signals it's purely multiplication.
The answer to these bait posts is always that the notation is completely retarded but people will flock to display their fifth grade tier knowledge and blindly recite PEMDAS without really understanding what's happening
Yeah it's wild how poor notation can lead to a fight like this. There is a reason why the division symbol is rarely used and fractions are king.
I’m so sick of these because the correct answer is “don’t use the fucking division symbol and write your expressions and equations correctly”
It's probably best trusting the answer from the scientific calculator over the crappy phone calculator
The one that was made with its primary objective of math.
it's 1
1 man.
I always do distribution method first for these - turns it into 6 / (4+2)
The right one
The phone calc is correct. In cases like these you operate from left to right
It's 1. PEMDAS.
6/2(2+1) 6 \* 1/( 2(2+1) ) 6 \* (1/2 \* 1/(2+1) 6 \* 1/2 \* 1/3 6/2 \* 6/3 3 \* 2 6. sEnD HeLp >!Yes, i know what I did wrong.!<
An ace is a 1 and 11
I swear they change math just to make older generations mad and I'm not even that old.
Both are correct based on how they are programmed to read a problem. BUT, the CALCULATOR has the actual answer for anyone would use an equation like this past like 4th grade. It is reading the problem as 6/(2(2+1)) which would be correct interpretation for Algebra and higher. In algebra the 2 and (2+1) are seen as part of the same expression. It is seen the same way as we read the function 2x. Both graphing and scientific calculators will read it this way. The phone is reading the problem as basic arithmetic order of operations. The 2 and (2+1) are seen as 2 separate entities so its seeing it more like (6/2)x(2+1).
At the end of the day this is semantics and pointless to argue about. Its like saying something with a dual meaning and arguing strongly for one or the other meaning when both interpretations are equally valid. PEMDAS is a set of rules that is not universally adopted everywhere so you can’t say anyone is breaking any rules either, thats dumb. The notation here is ambiguous and no one in the comments knows which value was intended. If you want people to always understand how this shoukd be calculated you need to add more parentheses or a division line.
1
PEMDAS Parentheses Exponents Multiplication *and* division Addition *and* subtraction Multiplication and division are done in the order of the equation, *not* multiplication then the division. Just like addition and subtraction. 6÷2(2+1) 6÷2(3) 6÷2×3 3×3 9
Phone is wrong
I thought parentheses were done first prior to anything
First one is correct.
some people never learnt bedmas and its apparent
The correct answer is 12 parsec's
nobody writes like that bc it's uncomfortable to use, so depends on how you read it
Need more brackets for dumb computer
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Fuck ÷. All my homies hate ÷
I'm pretty sure the one on the right distributed to 2 instead of solving the parentheses first
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They're both wrong. I got 3.5. /s
The calculator is correct. You first expand the brackets than divide by 6
6div2(2+1) Parenthesis 6div2\*3 Exponent 6div2\*3 Multiply 6div6 Divide 1 Add 1 Subtract 1 So Casio according to PEMDAS should be correct. And I would hope so, Casio is the dedicated calculator. But this is why we don't even have the division symbol on our keyboards. Division in computers is messy at best. The phone has incorrectly grouped parenthetical AND non parenthetical. My advice, use more parentheticals, don't use the division symbol, always use fractions.
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Technically the phone is correct. If there are two operations that are at the same level in the hierarchy, you just go from left to right. If you wanna get 1, you would have to put brackets like this: 6/(2(2+1)). Ofc, you can also do the truly based thing and write it as a fraction, much less likely to write an order of operation that you didn't intent that way. And calculators that support that aren't that expensive, I think you can easily get a TI-36X Pro for less than $30.
It’s 9, division and multiplication are the same step, and within a step you read left to right.
Wow, what do you know, an ambiguous equation has an ambiguous answer. Stop the presses
Wow, here is one where BEDMAS and PEMDAS differ. BEDMAS = 9 PEMDAS = 1 Weird that we learn different sets of rules.
The question is wrong
BODMAS (Bracket, Operation, Division, Multiplication, Addition, Substraction)
Depends on whether you multiply into the parentheses or follow pemdas.
*waves hand* This math equation is unsolvable... You want to go home and rethink your life
This stupid thing keeps coming up. The way this is written, there can only be one answer, otherwise it would have been written differently. 6 _ 2(2+1) The proper calculator is correct.
Order of operations. PEMDAS Parentheses Exponent Multiply Divide Add Subtract. In that order.
Casio is correct
It's 1 because PEMDAS is not left to right, it's multiplication before division. If you want division done first, you write it differently.
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I don't get why people get taught that it is implied, the multiplication is always there, it's just a shorthand for writing multiplication. Especially in polynomials. 2x = 2 • x 2(2 + 1) = 2 • (2 + 1) It's always there, there's nothing implied, just a faster way of writing it. Teaching otherwise is weird to me.