6÷2(2+1)

There are many problems like this on the internet. I'm going to use this one as an example, but the analyses can be applied to any of the “viral” problems of similar ambiguous notation. I take deep issue with the proclamation that the “modern answer” is 9 and believe that our continued insistence upon its validity as an answer could have negative ramifications on the average person's ability to understand how math really works. There are important reasons that “the general public” chooses to interpret the question the way they do. You do need to satisfy the other end of your division symbol. Everyone is aware that “divided by” is an open question that needs answered. When you input “6÷2(2+1)=” into a scientific calculator, it will, before your very eyes, re-write the problem to validate the answer 9. These are compelling reasons for choosing to put 2 with 6. I intend to walk through several key elements of the controversy which should help guide a clearer perspective into public awareness.

The first question, as with any math problem, has to be: “Is this properly notated?” Both sides can probably agree it is not properly notated. Attempting to solve a problem left to right using PEMDAS/BODMAS/the order of operations (I will simply refer to this method as “PEMDAS” from now on) without accounting for notation is the equivalent of pronouncing the word “boat” like “Bow-At” or “music” like “muss-sick” because of phonetics and the left to right reading rule. So it stands to reason that both sides of the argument, 9 and 1, are changing the notation to answer the problem, at least to some extent. The general rule when/if you run into improper notation “in the field” is to:

1. Make the fewest assumptions as possible and
2. Favor the simpler answer.

“Simpler answer” doesn't mean “1 is smaller than 9, therefore 1 must be the answer.” What it means is, if I did my problem my way and you did your problem your way, and you came up with 9 and I came up with 1.7266666, I wouldn't have a leg to stand on. Clearly in that scenario, the interpretation that you would be using to arrive at 9 would be the simpler and better way to interpret the information. But that isn't the case. 1 and 9 are both equally “simple”, seemingly elegant solutions. So, we seem to be looking at which answer is making the fewest number of assumptions about the notations and the question being asked.

In order to get “6÷2(2+1) to equal 9, with proper notation, the solver or calculator contains 6÷2 into parentheses, rewriting the problem like this:

(6÷2)(2+1)=9

and uses PEMDAS to solve, “parentheses first”

(6÷2) = 3

(2+1)=3

(3)(3)= 9

Except “(3)(3)” is not proper notation either, because that isn't how simple multiplication questions are notated. It should be 3 * 3 or 3 x 3. So in order to fix this, another assumption must be made in the notation from the beginning, and that is the addition of * between the two (3)'s like this:

(3) * (3) = 9

It's true that some scientific calculators will not add the parentheses around 6/2 and let it stand without them making the problem look like this:

6÷2 * (3)

instead, but in 100% of cases, it will add a multiplication symbol between the two 3s. It must assume the multiplication symbol. After all, if “(3)(3)” were the same thing as saying “(3) * (3)” or “3x3”, why would the calculator insist upon adding a seemingly redundant symbol? Don't worry, you should understand this by the end.

First of all, we're going to have to agree that “division symbol” is not really used in higher level math. By the time someone who has done a lot of math sees a “ ÷ ” again, their brain has been automatically rewired to view it as a fraction bar. That is because always, without fail, “ ÷ ” in our human math language means “a fraction bar sits here”. Even the 9-sayers agree that 6 is a numerator of SOME fraction, as in 6 halves, or their precious (6/2). No assumptions are made by turning the division symbol into a fraction bar, and this is agreed upon by both sides. I also think we can all agree that:

(6/2)*(2+1) Does indeed equal 9, while 6 (fraction bar) 2(2+1) does indeed equal 1 (at least I hope we are together on this)

So then, the real discrepancy between getting the answer 1 and the answer 9 has nothing to do with the division symbol. It is whether 2(2+1) is one term, or can be separated from each other. How far should the fraction bar “stretch”? Across just the 2? Or across the entire 2(2+1)? To answer this, lets examine the difference between “2 * 3” or “2 x 3” and “2(3)”.

The former, “2*3” and “2x3”, are interchangeable. They both simply mean “multiply two times three” and both are properly notated basic math questions. However, 2(3) is not in the same realm. Are we still multiplying two times three? Yes. Are we still coming up with 6? Yes. However, 2(3) doesn't “equal” 6. It “represents” 6.

Two slices of cake times 3 kids equals six slices of cake looks like:

2x3=6 or

2*3=6.

But “2(3)” will always mean “2 sets of 3”. It is 2 litters of 3 kittens, or 2 groups of 3 kids or 2 bouquets of 3 flowers. Parentheses are a “grouping” of things which will give you an answer as to how many of that specific “thing” there are. If I have 2(3) slices of cake, for example, I'm not just stating that I have 6 slices of cake. I am stating that I have 3 slices of cake each, set up on two different tables. 2 x 3 = 6 is simply telling you how many slices there are. But, 2(3) = 6 is giving you more information. It represents splitting the 6 slices into 2 groups, the product of which will tell you how many slices there are total. This is why we don't use parentheses for simple math problems like 3x3=9. Understanding how 2(3) represents 6 but 2x3 equals 6 is a subtle but important difference. A representation of a number that can be further worked out isn't quite “equal” to a number, at least not yet. The rules of parentheses, therefore, are different than the rules of “regular old” multiplication. They are not interchangeable with other multiplication symbols because you can and will end up with estranged units. This explains why you cannot just “grab” the 2 away from 2(2+1). That 2 doesn't quite equal two. It is 2 groups of something.

When you are working with P.E.M.D.A.S., that first P-step (“Parentheses”) doesn't just mean “multiply” or “multiplied by”. This seems to be a common misconception. Because parentheses mean “I am a set of something”, the correct way to deal with them is to solve what is inside (this is undisputed) and then multiply that number by whatever number is OUTSIDE of the parentheses to the left. The “P”-step requires you to resolve the parentheses entirely in this way before moving on with the rest of the question. Someone who learned “The distributive property” instead of PEMDAS already knows this. In most online PEMDAS tutorials, they define the step simply as “Parentheses” before moving on to list “Exponents”, as if just saying “Parentheses” clarifies what is meant in the step. This has been a disservice. Solving inside the parentheses was never meant to be the end of the step. “P for Parentheses” is just a mnemonic to help you remember what to do first in the series. It tragically clarifies nothing about the step or what it fully entails.
If we apply the updated P-step to the viral math problem, we will first solve (2+1) but then you are looking at “2(3)” which is correct notation for “two sets of 3 things”, not “2x3”. When you attempt to solve 2(2+1), and turn that into 2(3), you've solved what is inside the parentheses, but the parentheses are still there! They didn't go away yet.

At this very moment, you can go to online PEMDAS lessons or look at any old textbook where the “P” step is interpreted as “solve inside” instead of clarifying the whole (real) step, and you will not see any practice questions or examples where there are numbers situated outside the parentheses of the problem to the left, such as in these sets. You will not find a 2 just hanging out by itself in front of a (2+1) without a + or – or * in between them. Your PEMDAS homework probably looked like this:

(6÷2) * (2+1) =

But never this:

(6÷2) * 2(2+1)

Many people, it seems, were not exposed to a number existing outside parentheses, or what that means for “reading” the question and how to resolve it. The creators of your textbooks/lessons were very careful not to give you these problems so that you could focus on learning the basic principles of PEMDAS without getting confused by “multiple sets”. This has created a situation where, as the student, you don't know how much you don't know until you move upward in the math lessons. If you never move upward to higher level math, you continue to think that you reached the top of knowledge about PEMDAS when in reality you never learned to resolve parentheses. PEMDAS is still used in higher level math, sometimes multiple times in the same problem, but ALWAYS with the P-step completely resolving parentheses, not just solving what is inside them.

So, why do the parentheses seem to disappear on their own in some problems, but not others?

When you're solving a problem in PEMDAS class, say for example this problem:

(6*3) – (3+2) =

You first solve (6*3) and (3+2). Why do you then get to re-write the problem as “18 – 5” and not “(18) - (5)”? Where did the parentheses go? They ONLY disappeared here because there is no visible number outside them to multiply by. When you have 3+2 inside parentheses like this (3+2), signifying a “set” of (3+2), what you are really saying is:

“Whatever thing is inside the parentheses, we have exactly ONE set/bundle of it.”

or

“1 times three plus two”

or
(3+2) represents ONE “set” of (3+2)

2(3+2) represents TWO “sets” of (3+2).

You can resolve (3+2) simply by knowing 3+2, because there is only “one set” of (3+2) and “anything times 1 is itself”. Basic math. Since no number in front of the parentheses actually means “one set of whatever this is”, “one set” of (5), is the same as saying 1 x 5 = 5. You still multiplied by the number outside the parentheses to the left, it's just that the number was a 1, allowing the parentheses to be eliminated fully without changing what was calculated inside of them. This gives the illusion in most carefully constructed problems, that solving what is “inside” the parentheses makes them disappear as if by magic. It's also why you'll never see PEMDAS lessons with incomplete “P-steps” offering problems with multiple “sets” of things. The “number of sets” have always been there, hiding in all of the PEMDAS lessons, and forgotten like the “1” in “1x”. You can think of it as an invisible “times 1 multiplier” outside of every set of parentheses unless otherwise specified with a different number. “2(2+1)” or “2(3)” for example, has specified that there are 2 “sets” of 3 things. However, in “(3)(3)”, the parentheses being “alone” looks, and is, wrong. This is because you need an operator between the two “sets” in order to engage the “x1 multiplier” rule and resolve the parentheses entirely.

So this:

(3) + (3) = 6

Is REALLY this:

1(3) + 1(3) = 6

Without the operator between to engage the “x1 multiplier”, writing “(3)(3)” is like saying “there are 3 in each set and there are 3 in each set” without a way to resolve the questions “how many sets are there, and what am I doing with them?”. As soon as you begin dealing with multiple sets, the “P” in PEMDAS takes on its role more visually instead of doing the work behind the scenes. You don't notice the parentheses disappear when you're multiplying by 1, but as soon as that becomes any other number, you realize that you have to do the multiplication yourself or else the parentheses don't resolve.

If you go back to the viral math problem now, “6÷2(2+1)”, you should be beginning to understand the rather large assumption you are making when you group 6 and 2 together. You are assuming that 2 does not belong as a set with (2+1), even though “2(2+1)” is correct notation for 2 sets of (2+1). To arbitrarily throw the 2 in with 6 goes against what would appear to be a more simple answer, which also makes the least assumptions about the notation. Grouping 6÷2 together just so you can hurry to resolve the uncomfortable division symbol doesn't seem practical or rational. 2(2+1) is one inseparable term. It is notated correctly as such, making no assumptions.

When you input “6÷2(2+1)” into the scientific calculator, it will begin assuming notation almost right away. It comes up with it's own interpretation, it's own notation of what you are asking. It looks, again, like this:

(6÷2) * (2+1) = 9 or

6÷ 2 * (2+1) = 9

Many say that we should stop here. That's the answer. Game over. In my experience, though, calculators are smart but they aren't all knowing. There is a reason it's doing this, and it is not because it has the “correct” or the “simplest” interpretation. It's because the calculator does not recognize that division symbols can be resolved by complex denominators. You might be trying to enter “2(2+1)” as a denominator, but the limited capacity of the calculator won't allow anything to be understood past the number 2. As soon as the “open question” division symbol is satisfied, the calculator is satisfied. It isn't smarter than its operator. It doesn't know how to do a fraction with a denominator that has some nuance and flair. The calculator grabs the 2 as soon as it's available, away from its rightfully notated spot next to the parentheses, in order to satisfy its division symbol. However, to its horror, you continue to type. (Some calculators decide to start their parentheses as soon as you put in the division symbol because they know that whatever is coming after a division symbol is going to be the end of that road. It simply MUST divide right away.) When your next move is to input an open parentheses, this is a big problem for the calculator because parentheses, (the P step in PEMDAS), absolutely must be resolved entirely. The calculator can't just leave your (2+1) hanging off to the right without a way to resolve the parentheses they are in because that leads to “(3)(3)”. Unfortunately, it already impatiently stole the “2” which would have resolved them just fine. Assuming the multiplication symbol and separating the question into two questions is the calculators last-ditch effort to salvage meaning from what you're typing after making a huge error from the beginning. It NEEDS to believe there is a multiplication symbol or SOME symbol there to kick in your invisible x1 multiplier and eliminate your parentheses! Otherwise, it's a joke and won't get invited to parties.

The scientific calculator, with adjusted settings, can account for complex denominators in its calculations by being instructed not to “rush” to resolve the division symbol. This is like the calculator suddenly being able to “recognize” that there could be more to a denominator IF certain conditions are met. Having no marks or numbers at all between the 2 and the (2+1), again, is proper notation for 2 sets of (2+1), and the calculator can now accept this one single term as a valid way to satisfy the division symbol. Suddenly, the answer, according to the calculator, is 1 instead of 9, and no assumptions have been made in the notation at all. So while the 9-sayers re-write the problem (6÷2) * (2+1) =9 to mimic the calculators assumptions, turning off its ability to add assumptions at all and declaring boldly that “6÷2(2+1) EQUALS something” the calculator will finally reveal to you that this has been an expression of “1” all along.
You didn't just regress your calculator back to 1917 math settings. Calculators are designed to work from left to right which is directly reflected in that “1917” change. When you type the division symbol into your calculator, it KNOWS the rule is to satisfy it with ONE term on the other side. That's precisely why it does just that, regardless of how that might change the meaning of anything else that is thereafter typed in. In fact, the arguments against “9” have nothing to do with choosing to use the whole section on the right as the denominator arbitrarily “because of some old math grammar” involving the division symbol. It has everything to do with recognizing a complex denominator for what it is without the aid of the calculator at all, and recognizing the practicality of seeing 2(2+1) as one term.

When you work out “(6/2)”, you are reducing or evaluating a fraction: Six halves. This is it's own little math problem because it is a fraction that reduces. 6 divided by 2, or “6 halves”, or “one-half, six times”, reduces to 3, and 3 is the only relevant information in your journey to the answer 9. For example:

We're playing a measuring game with some kids. There are 2 boys and 1 girl for a total of 3 children. (That's the undisputed (2+1)). I give each child a scoop that holds ½ cup of water and a bucket and I ask “how much water will be in your bucket if you do 6 scoops?” The children are working out the problem “(6/2)” or “six halves” =3. When they are done measuring out 3 cups of water into each of their buckets, they bring the buckets of water to you. So then I come to you and ask, “how many cups of water do you have across all buckets? This, after all, is the question I have to ask in order to get the answer “9”. The problem you are then working through is: 3 cups of water per bucket x 3 buckets to get your answer 9.

The kids could have measured in whole cups. They could have measured in liters or tablespoons, inches or centimeters, grapes or fingernails. At the end of the day, as long as they got 3 cups of water into the bucket, it won't change your math or your outcome. Even if the ½ cup measurement is seemingly relevant in the situation, it is NOT relevant to the question which leads you to the answer “9”. This type of bizarre phrasing is what your scientific calculator is assuming when it rewrites the problem the way it does. It is understanding the question as “3 x 3” with extra arbitrary steps. Embellishing a simple problem with fancy notation by adding irrelevant information is the math equivalent of asking, “How many hours does it take to get to San Francisco given that it's 6 o'clock?” You started with information that was leading in one direction and then asked something else in the same sentence.

The types of questions that are common with multiple “sets” of things, complex denominators, and fancy notation such as this, you would never see embedded with all whole numbers, no exponents, no variables, and no combined terms. Most often, maybe exclusively, this type of notation is seen in problems that look more like:

6xy^2/ 2xy^2(2+1) * 2(3x+1)

The answer here is not so immediately recognized and the puzzle is not immediately solvable like the viral math problem. Yet, this is the format we are expected to believe is being utilized to express something as simple as “3x3”? If we agree that the best answer to a poorly notated problem is one that makes the fewest assumptions and arrives at the simplest answer, why are we then interpreting the question in such a complicated way? And referring back to that same example: What mathematician will suggest that the “2” in “2(3x+1)” somehow “belongs” with the previous “6xy^2” and its denominator, just because of a “left to right” general rule? By doing that, you would be robbing “3x+1” of an entire set of itself, which will dramatically change your answer. The complexity of the notation being assumed doesn't match with the simplicity of the problem or question at hand. We appear to be changing the question in order to fit an answer we like, rather than seeking out a complex but more realistic truth.

In reality, the 9-sayers are solving a puzzle (not a real math problem) by doing a series of simple math problems. That series of simple problems is notated like this:

6÷2 = 3

and

2+1=3

so

3x3=9

Using overdressed notation, this mutant is being created:

(6/2) * (2+1)= 9

which is technically notated correctly, so the calculator is able to give an answer. However, there is no compelling reason to use it in real life or assume that this is a legitimate translation. We aren't evaluating all of those factors “at once” so they don't truly belong in the same sentence.

But you can interpret the viral math problem exactly as shown in such a way that renders each term relevant in contributing to a coherent question and answer; by considering it to be evaluating an expression of 1. For example, instead of having a bucket game where ½ cups are measured for seemingly no reason, you have something more like this:

“There are 6 slices per whole cake. 2 boys and 1 girl are coming to the party and I want each child to have 2 slices of cake. How many whole cakes do I need to buy?”

Though this question is represented by a fraction with a “complex denominator”, it is undeniably simple, common and demonstrable in a real-life scenario, and each element is relevant information in answering the one question presented.

Conclusion:

One possible reason for the public's propensity to invoke complex notation and argue that it is actually “more simple or correct” could be the desire to avoid “dealing with” a live fraction. Especially in America, people don't seem to understand fractions well, and actively avoid them when possible. We can fall back on elementary school math if we interpret “divide” as just another simple operator. “6 divided by 2” is a lot easier for us to stomach than “6 divided by 2 sets of some number we've yet to work out.” Thus, it is being treated as “simpler” to rush to resolve the division. I believe this treatment of fractions could have a negative impact on how people understand math and math notation going forward. It may create the false impression with “math newbies” that this complicated notation legitimately reflects how we would go about pursuing and notating basic information in real world math, science, and technology. It creates the impression that we routinely add unnecessary information and complexity to our math problems, and that when we ask (or write) questions, we are not really sure where the answer comes from. Isn't math complicated enough?

Pretend for a moment that an alien species beamed down a mathematical message: 6÷2(2+1). Their symbology is certainly different, but we were able to translate the message down to “six divided by two times two plus one.” Given these options, which is the more “simple” answer that makes the fewest assumptions? :

1. The aliens meant us to work left to right using our childish little division symbol that requires immediate gratification?
2. The aliens were going out of their way to be extra transparent or extra confusing with their run-on question by using irrelevant information? Or
3. The aliens sent a fraction with a complex denominator that reduces to the number 1?

It seems rational that “6÷2(2+1)” is not a “problem” at all in the sense that it requires a solution, but rather it is an “expression” of the number 1. What a human could almost intuitively recognize as a valid representation of “1”, our calculator with it's left/right algorithm and blind loyalty to the division symbol, missed out on the simplicity of this message and instead sent back to the aliens an irrational and presumptive “9”. For some, believing the calculators process is the “correct” way to re-shape the question appears to be (at least in part) persuaded and validated by the calculators decision to do it that way. I find that problematic because they're essentially saying “the calculator is right because it's a calculator and calculators are right.”

Two “sets” of something divided by 6 would be a simpler and more universal interpretation, even if it is more difficult for us to notate in our calculators and, perhaps, articulate with words. To be fair to the 9-sayers, if I were given this problem by a 2^(nd) grader in their adorable scribble-y handwriting and asked if I could figure out their handmade “cool math puzzle”, I would certainly reconsider my position on this whole matter. However, I refuse to bestow the same handicap onto the adult academic world. By coming out more squarely on the side of “9” instead of “1”, we are encouraging bad habits in everyone who learned or understands their P-Steps in PEMDAS incorrectly, and are treating math like a number puzzle rather than something that reflects a real question and answer in our physical world.