Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.
The “order of operations” tells us the sequence in which we should carry out mathematical operations. It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. You could come up with a different convention, but why bother? Let other people waste their time coming up with arbitrary conventions so that you can merely waste time reading posts about them.
The rules to follow are:
1. Things in parenthesis get computed before things not in parenthesis (this is because of the (little known fact) that parentheses kick ass).
2. Then, exponents get dealt with (including roots). Hence a^b*c = (a^b)*c.
3. Next, multiplication and division get carried out from left to right. So a*b/c*d is ((a*b)/c)*d. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another.
4. Finally, additions and subtractions get carried out from left to right. Subtraction can be thought of as being like addition since a – b = a + (-b) so it makes sense that subtraction is treated on an equal footing with addition.
People sometimes use the expressions PEMDAS, “Please Excuse My Dear Aunt Sally” or “Puke Exhaust Mud Dolls Autumn Sasquatch” to remember these rules. These all stand for “Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction”. It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right). That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch”
One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever. But that implies there is something interesting about them. This is a contradiction, and hence the order of operations form a paradox. They are also about as fun as getting to watch a fish crawl up the stairs as a reward for answering math questions.
But, that won’t stop me from giving an example. Consider:
a^b*c/d+e-f.
Without a convention, it has many possible interpretations, such as
a^(b*(c/(d+e)))-f
or
(a^b)*((c/(d+e))-f)
which could be horribly confusing for people trying to communicate with each other. But, the correct interpretation according to the rules mentioned above, is:
(((((a^b)*c)/d)+e)-f).
When in doubt about what the orders of operation are for an expression, there is a simple solution which works even better than tattooing PEMDAS to your face. Just paste the expression into the google search bar and hit “search”. For instance, if I paste in:
4^2*3/6+1-5
it gives back
Not only is this the correct answer, but it shows you explicitely the order of operations that were used, so it’s useful for learning.
Google even gets this one right:
4^2*3/6 + 1 – 5/14*3 + 6/10 – 4*2/14*6^3/18 + 14
which is the kind of question little sisters give you when they are old enough to finally sort of get what a mathematician is.
It’s important to note that not all calculators handle the order of operations in the standard way. Why? Because people who make calculators hate children. The preceding sentence was not intended to be a factual statement.
Also, a lot of programming languages use the standard order of operation rules, but then you have exceptions like Smalltalk.
Oh, and in case you were wondering:
a^b^c = 
Listen closely: 2x = (2x) = (2 * x)
If you think that 2x does equal (2 * x), then I have news for you.
Also, as I have stated: You cannot change the coefficient of (2+1), which is clearly 2, into (1/2).
Just to re-iterate:
2x = (2x)
So if you would like to substitute something different into the original equation (or expression), then you require parentheses.
Like I stated in me previous post:
6 ÷ 6 = 1
Let 6 = 2 * 3
Now I have: 2 * 3 ÷ 2 * 3 = ?
This is basic algebra and mathematics techniques. You are not allowed to substitute “willy-nilly”. 🙂
@mathman
Just a quickie here for you. 6÷6=1 is absolutely not the same as 2*3÷2*3 unless you use parentheses. * and ÷ have the same precedence in Order of Operations. 2*3÷2*3=9. Don’t be “willy nilly”.
@RB: My point exactly. You can’t just insert a multiplication sign between a set of factors, coefficients, and literals without containing it parentheses.
(6y²z⁵ ÷ 2xyz²)² is not the same as (6y²z⁵ ÷ 2 * xyz²)²
@mathman: I think the problem comes from IMPLIED meaning, and since people don’t want to write parenthesis around everything they do, they just assume them and move on. When you write (6y²z⁵ ÷ 2xyz²)² what is usually meant is (((6(y²))(z⁵)) ÷ ((2x)(y)(z²)))², which is a lot of writing for what should be something pretty straightforward. But, written out inline without those parenthesis, it SHOULD be interpreted as ((6)*(y²)*(z⁵) ÷ (2)*(x)*(y)*(z²))². But, because of a convention that teaches us to imagine the parethesis instead of writing them, it creates confusion at the elementary levels with problems like 6/2(2+1). You either follow the order of operations to the letter, from left to right without implied parenthesis: (((6) / (2)) * (2+1)) OR ((6) * (½) * (2+1)), or you treat is as an algebraic coefficient, which would really read as (6 / (2 * (2+1))).
Imagined parenthesis also leads to bad math reasoning like:
6 / 6 = 1
6 / 3 * 2 = 1
now, following order of operations
(6 / 3) * 2 = 1
2 * 2 = 1
4 = 1 ??
look, i just broke math!
Moral of the story, parenthesis are probably the most overlooked but important tool you can use to say what you really mean. The reason there is so much confusion is, in my humble opinion, the somewhat necessary teaching (so many parenthesis!) of a convention that tells you to assume.
Thank you for commenting, but it actually should ~not~ be interpreted as you wrote it. An expression as written, should never be presented to an elementary student in the first place. Also, mathematicians, physicists, chemists, etc, do want to write out dozens of parentheses, as you can imagine, because it would like your example. We string together variables and understand that it is one unit to avoid such things. Putting the multiplication symbol between them all accomplishes nothing but confusion, and would require parentheses around the entire group, which makes it look ugly and slows down the ability to read the information quickly. A product of coefficients is the coefficient of the remainder of them, and so on. They are also commutative, as in abc = bca = cab … It isn’t a terminally difficult concept. 2 = (2), right? There are always implied parentheses around numbers. The same holds true for groups of variables and the like. I am not sure if this got omitted in schools in the last 20 years, but it is a very basic algebraic concept.
Ciao
I was speaking in a purely elementary fasion for interpreting the problem. If you are just holding order of operations as the highest authority then that is how it ~should~ be read, but the algebraic convention says otherwise because we assume those parenthesis!
As for the multiplication; it is commutative, but the division has to be handled with much more care. Without those parenthesis, are you multiplying (since division is a form of multiplication and is, in my opinion, a much simpler way to avoid confusion) by 1/2, or is it multiplying by 1/(2xyz²)? in algebra, this is a very easy answer, because we teach the CONVENTION of using coefficients and implying parenthesis (again, for good reason).
But, when you take that same logic and apply it to elementary math, you end up with a disconnect that crops up with cases like 6/2(2+1). without more clarification, it is pretty ambiguous whether the WRITER of the equation meant multiply by 1/2 or 1/(2*(2+1)). If you convert it to an algebraic form, then it is (a/(b(b+c))), which is 1. But if the person is writing from/for an elementry level, the question would instead be written out as (a(1/b)(b+c)), which is 9. Without knowing the context of the question, and with two different conventions (implied vs absolute parenthesis) to chose from, how can you be sure what the author meant?
The solution is either rewrite the original question to include parenthesis and remove any ambiguity, standardize the convention and teach elementary math using the same assumptive parenthesis as algebra, or teach numerators/denominators instead of the division symbol at the elementary level.
or, if we boil the problem down to its simplest form, we have to ask whether the “/” sign means “÷” or “everything following is the denominator”, since unfortunately it seems to be used interchangebly by people to mean either one.
@MK if youre going to start factoring 6 then you need to do it correctly.
6/6=1
6/2*3=1 you went wrong right there
6/(2*3)=1
I believe I found a better way to understand this
for people who always get 9
Lets write this out with variables.
6/2(2x+x)=y
simplify
6/2(3x)=y
is the next step 6 * 3x = 2y or is it 6=2(3x)y or is it 6/3x=2y or is it 6/3x=y/2
the answer is 6=2(3x)y or 6=6xy then divide both sides by 6 and you get xy=1 already knowing that x=1 therefore y=1 and the math checks itself.
do this expecting to get 9 for y and you will not end up with x equalling 1
This whole equation was originally written as 6÷2(1+2) rather than 6/2(1+2), so somewhere in the hundreds of posts, someone could not figure out how to write ÷, so it was transformed. Having said that, order of operations says that we first take care of the (1+2) which gives us a next step of 6÷2(3). Since division is considered multiplication and the rules for Order of Operations indicate that both multiplication and division have the same precedent and should be followed from left to right, the division would be handled next giving an intermediate step of 3(3). The last operation to occur would be the multiplication, giving us an answer of 9.
Again. not a dilemma of order of operations, but rather one of operators and operands. 2(3) is not simply 2 * 3.
When you use that rule, it states that, for example, 2a = the product of 2 * a. That is what everyone is missing. 2(3) is the same as 2a where a = 3, except that we can’t write it the same way, that is, without the parentheses since it is 2 single digits adjacent to each other.
REMEMBER THIS: 2(3) = (2)3 = (2 * 3)
just as abc = bac = cba, etc.
Anyone who has spent any amount of time in some serious education, whether it be calculus, stats, chemistry, physics, or mechanics, would know this. That is the frustrating part of it all: Those who haven’t had the opportunity to be immersed in such depths of any of the degrees listed, yet they consider themselves experts.
I repeat: 2a = the PRODUCT of 2 * a, which means it is (2 * a).
Someone, somewhere, said it was ok to substitute back 2 * a into the original expression, when it is not. But hey, they read it on the internet right? So it must have been correct.
It is not 9, folks.
@Ray Did you get that ^^^^ 2(2+1) is the same as [2*(2+1)] and that is why you are wrong. You do not separate the individual numbers of the expression. That’s a lame excuse to say that the problem is that people confuse the division symbol for a fraction line….a fraction line is division so smack your head for me please. This is why there is something called implied multiplication implicated by juxtaposition. The (2+1) belong to the 2 because the math implied has been simplified removing all extra operands. I’ve been telling my teachers they were wrong about this for years. And the problem is that people dont even care to check the math. Try writing this equation using 1=x and trying to solve for y=9. 6/2(2x+x)=y Solve for x and if you dont get 1 when using 9 for y then your are wrong, and since I’ve already done this I’ll just let you try it out on your own.
@mitchell: i understand that, which is what i was trying to show. The incorrect use (and sometimes assumptions) of parenthesis lead to math errors such as the one i explained, many of which showed up in the comments on this thread.
@mathman: that is essentially my point. with an elementary problem like that and varying math levels, it is difficult to know if the original author meant 6÷2*3 or 6÷(2*(3)). I don’t think either answer is wrong as much as the equation needs to be clarified a little to remove possible ambiguity. I agree that the question, as written, should be interpreted as 6÷(2*(3)), but that doesn’t preclude the possibility that the author intended 2(3) to be shorthand for 2*3 (as it often is in lower level math). This doesn’t make either convention necessarily wrong, it just means you have to know which convention to apply (though i do vote for chosing only one to fix situations like this).
Or, to put it a slightly different way, if i write the date 07-10-13 in america, most people will read it as July 10th, 2013. But for someone from europe who is visiting, they might interpret it as 7th October, 2013. Neither is wrong, you just need to know which was intended.
This intention of the author is neither here nor there. 🙂 The problem is just that: a problem. I have had calculus profs that “intended” a solution, but when I solved otherwise, and was marked wrong, I showed the error and was awarded the marks. For the problem at hand, mathematical techniques must be ignored for the solution to be something other than 1 or 2 (for the corresponding problem).
However, this ‘problem’ should never be presented to an audience that isn’t knowledgeable enough to solve it properly, which happens to be about 2/3’s of the internet by reading most of the polls online. Let me correct myself with (2/3)’s … 😉
Just because most people get it wrong or don’t know any better, doesn’t make this a convention issue, but rather a educational one.
I have never had a problem discerning that when a number is placed immediately outside of parentheses it and the values inside the parentheses are one single expression that must be simplified before the equation can continue its order of operations. As stated above and quite clearly there is no need for clarification on the syntax of this equation, the clarification needs to be taught in school that 2(2+1) is shorthand for [2*(2+1)], it is not all that hard.
6/2(2+1)=1
6/2*(2+1)=9
its that simple
basically you are trying to allow people to be correct in assuming that the equation is equivalent to 6/2*(3) instead of 6/2(3). There is a profound difference and it needs to be taught. Science cannot afford mistakes like this. I hate misinformation and allowing people to believe the answer can be 9 or 1 is WRONG. Theres is a perfectly simple explanation for why but people are willfully ignorant because they were taught the ORDER OF OPERATIONS in school so it must be absolutely true. It is true but the fact is that no matter what level of math you have learned all RULES apply at ALL times. I’m and getting tired of people cherry picking the rules so they can make 9 fit without even rearranging the equation to see if 9 actually works which it doesn’t.
Lets have you work backwards. We’ll start with:
6/2(2x+x)=y solve for X and then insert 9 for y and tell me if you get 1 for x
6/[(4x+2x)]=y
6/6x=y
6=6xy
1=xy
y=x
so how can y be 9 if we know x is 1
@Mitchell: I agree with you that the answer is one. However, the original article we are replying to indicates the answer should be 9, and he is a professional mathematician. So I guess it isn’t cut and dry like we wish. I believe the 2 is a coefficient of the parenthetical expression, and thus is linked to it in priority over PEMDAS. But calculators and sites as he describes in the article don’t agree.
Calculators are programmed. They are not absolute on correct mathematical procedure. Computers can only do what we tell them to. If the calculator operand precedence is flawed within the programming how could we possibly expect people to know the right answers to questions like this? I do agree that the problem here is the education of the math involved. People think that they only have to apply the rules of arithmetic PEMDAS when it requires higher level math. The people who get 1 have A)Higher IQs
B)Better Education
C)Common Sense that dictates 2*(1+2) is completely different from 2(1+2)
6÷2(1+2) How do you read this? is it (6÷2)(1+2)=(3)(3)=9 which means you have created 2 new parenthesis which means you have changed the equation and decided to include the 6 in the multiplication with the (1+2) which means you have changed the answer and if we were dealing with millions of $$ you would .. well you get the picture I hope. Or do you read it as 6÷[2(1+2)]=6÷[2+4]=6÷[6]=1 and in this case we also added some parenthesis so we could come with the OTHER answer. I have no problem going left to right as long as the problem have been simplified and completely. so lets do this again without changing or adding anything to this simple problem : 6÷2(1+2)=x using simple Algebra I learned in 6th grade: 6÷ (2 times 1+ 2 times 2)=x which means we took the 2(1+2) and simplified it to its least confusing shape. now we have the following: 6÷[(2×1)+(2×2)]=x are you still with me? now we have 6÷[2+4]=x Plz don’t fall asleep on me am almost done: x=6÷6= 1
Left to right, up and down, dogie style, what ever pleases you simplify first
@Raed: Is 2(1+2) the same as 2 x (1+2) (x as in multiply operation)? 2(1+2)=6 and 2 x (1+2)=6 Since they both are 6, we should agree that it is the same. Now try this using Order of Operations: My tablet will not allow me to use ASCII, so I will spell out obelisk. 6 obelisk 2 x (1+2) OofO states division and multiplication have same precedence, so this is 6 divided by 2 times (1 plus 2) and by OofO, next step is 6 divided by 2 times 3. Next step 3 times 3 and final answer is 9.
Back to my laptop. Just to reiterate that last problem done without the assistance of ASCII characters. We will use basic Pre-Algebra for this one. As stated before, we know that 2(1+2) = 6. We also know that, because of the implied multiplication of the (), that this is the same as 2 x (1+2) = 6. Now let us write this with all out in the open, including the multiplication between the 2 and the (1+2). 6 ÷ 2 x (1+2) is still the same problem, is it not? Now let’s read up in our Pre-Algebra book and it says that we first clear the parentheses. This leaves us with 6 ÷ 2 x 3. Now we again look at the Orders of Operation, same book, and it tells us that both division and multiplication have the same precedence. After all, it is the same as 6 x 1/2 x 3. But I digress and going even this far may confuse some people. So, having Orders of Operation in mind, and looking at 6 ÷ 2 x 3, we know that we must now go left to right because of equal precedence and we now have 3 x 3. Last step, 3 x 3 = 9. Or, is it just better to leave this as too ambiguous to please everyone.
Excel gets 9. When I was in school for Engineering x(2)=x*2. Excel does the same thing… So does EES and Mathcad… And my Ti92 plus!
Since when do we simplify parentheses by inserting a multiplication symbol between the factor and parenthetical? Have you ever seen an example or proof of this sort of thing in a published text? I sure haven’t.
The proper way to simplify parentheses is:
a(b+c) = (ab + ac)
We are allowed to replace this entire group of terms, according to the definition of math properties.
6 ÷ a(b+c)
6 ÷ 2(2+1)
or, if a is an expression of more than one term or operators, it is always contained in parentheses itself (a + b)(c+ d) or other variations.
Since
6÷2(1+2) ≠ (6÷2)(1+2)
Because it is NOT implied
Do NOT imply a parenthesis which is NOT there
Since it is NOT shown
AND
Since
(6÷2)(1+2) = 9
Therefore
6÷2(1+2) ≠ 9
Since
6÷2(1+2) ≠ (6÷2)(1+2)
THAN
6÷2(1+2) = 6÷(2+4) = 6÷6 = 1
Where the ‘2’ is part of the parenthesis
The problem is that we are implying that
6÷2(1+2) = (6÷2)(1+2)
However
The equation is NOT written that way!
There is NO parenthesis sign () there!
@mathMan is right. Call it Distributive Property of Multiplication (Algebra ). It’s the implied Multiplicaton(aka multiplication by juxtaposition). It’s a matter of logic and understanding that math is not only linear. The ‘2’ in the equation: 48÷2(9+3) alone is not an operand and the writer made it clear by NOT putting a operator after the 2….purposely. The multiplication operator is left out ON PURPOSE. The fact that it is left out, doesn’t mean its not multiplication but that it’s on a different tier so to speak. If it was meant to be handle with the same precedence as the ÷ then it would have been written visually just like the ÷ was. Here is a scenario/example of this exact erquation. 48 apples divided by 2 boxes of 12 apples each. As you can see the 2 means nothing without the 12 apples to dicatate the total quantity of that (denominator). 48 is the numerator. 2(9+3)=2(9)+2(3)=2a+2b
And some of you guys think that pemdas along with the left to right are the only 2 rules in math. There are books apon books in algebra especially with equations written exactly like this. When there is no physical multiplication symbol written you do not ad it cuz that now changes the flow. There ARE rules already in place to handle this equation dont forget multiplication by juxtaposition. The placement of the 2 as it is, with no additional parenthesis already gives enough information to state that it is part of the parenthesis. The mere absence of the multiplication symbol IS AS important as any presence of one. And since it is part of the parenthesis operand. Then it gets simplified before continuing with pemdas. If the equation was longer you would see that pemdas and left to right would still apply but you must first simplify and that is a rule also.
(∂^4 w)/(∂x^4 )+2 (∂^4 w)/(∂x^2 ∂y^2 )+(∂^4 x)/(∂y^4 )=q/D
how can i solve “(∂^4 w)/(∂x^4 )+2 (∂^4 w)/(∂x^2 ∂y^2 )+(∂^4 x)/(∂y^4 )=q/D”?
by the best wishess
put 2x/2x into google. it plots 2x/(2*x)
then put 2x/2*x into google it plots x^2
put 6/2(1+2) google=9
6/2*(1+2) it also =9
only 6/(2(1+2)) gives the 1 using implicit * takes precedence over / rules.
curious inconsistency
This whole thing boils down to “intended interpretation”.
Mathematics is not just about being able to crunch equations, but also being able to convey a message to an audience in a clear and concise manner.
I’d like a mathematician’s view on the following:
Would you ever write 6 / 2a is you intended someone interpret 6a / 2 ?
Would you write 6 / 2(2+1) if you wanted someone to read 6(2+1) / 2 ?
Also see this. Those who solve for 9 (which is ok) are saying “by using standard order of operations, 6 / 2a is 6 / 2 x a because 2a is nothing more than 2 x a.”
Well what happens when you use the same logic for cos2a ?
I have never had anyone try and say cos2a is cos2 x a instead of cos(2a).
So how does the “2a is simply 2 x a” apply here ?? 🙂
You must always simplify problems like this by dividing all sides by the multiplier 2 in this case and u get a much simpler problem to solve whether ir not u use the / or the division symbol used in the problem. Therfore when u divide everything by 2, u get 3/3=1 and thats the only answer
One would assume that 6÷2(1+2) would be one and the same as 6÷(2+4), Correct, then apply the PEDMAS, Et al.
According to Measure and Integral: An Introduction to Real Analysis
By Richard Wheeden, Richard L.
μ/2(k+1) =
μ
———-
2(k+1)
Let μ=6, k=2. What is the result?
No math text I have ever seen writes: a/bc = (a/b)c. That is because there are inherent parentheses around the “bc” that are implied. So directly entering such a thing into a computer (today), may result in an incorrect solution, thus using parentheses might be required when using calculator.
1/2y = (1/2)y is NOT any standard by any math text.
An Introduction to Real Analysis: The Commonwealth and International Library
By Derek G. Bal
ε/2K =
ε
—-
2K
Let’s suppose ε=6 and K=2+1 here. What is the result ?
Let’s not forget Introduction to Real Analysis by Bartle and Sherbert 3rd Edition
where 1/n(n+1) =
1
——–
n(n+1)
Let’s apply this to, for example, : 6/n(n+1) and let n=2. What is the result ?
I think everyone gets the point. What book says 1/n(n+1) = 1 / n x (n+1) = (n+1)/n ? None. That’s how many.
If no books use this notation, would you consider it STANDARD ?
Heck no should be what comes to mind
What books uses 1/2y = (1/2)y instead of 1 / (2y)?
None.
A Radical Approach to Real Analysis By David M. Bressoud
Do they write 1/2(200/199) when they mean (1/2)(200/199) ? NOPE. The write it as:
(1/2)(200/199)
F(x) := (2/3)x^(3/2)
What about (1/2)x ? Just that, and NOT 1/2x.
Another oldie:
Bonnycastle’s Introduction to Algebra
By John Bonnycastle, James Ryan, John Francis Jenkins
Division. RULE: Set the dividend OVER the divisor in the MANNER OF A FRACTION, and reduce it to its simplest form, by cancelling letters and figures that are common to each term.
Examples:
6ab ÷ 2a =
6ab
—– = 3b
2a
He is saying it is a mathematical RULE and wikipedia that says 1/2y = 1 / (2y) is exception to the “standard”, yet I can’t find a single text that uses 1/2y = (1/2)y.
I will stick with all the books:
6/2(a+b). Let a=2, b=1
/shrugs.
Equation: 6/2(1+2) is what is given. your normal math rules should get everyone to atleast. —6/2(3) right? which is shorthand for—-6/2*3 as someone pointed out saying that shorthand for the normal equation was —-6/2*(1+2).
Then it’s just really basic understanding. from left to right. you get—-6/2 = 3.
3*3 = 9.
I like the understanding behind getting 1 with the use of going from —6/2(1+2) = 6/(2+4). It makes sense, But I will stick to the basic math rules that are easily straight forward, as well as what every calculator says. (9)
These are all constant so the implied rules of multiplication between a coefficient and a variable do not apply.
Normal Equation: 6/2(1+2)
6/2(3) = 9
if they weren’t all constants…
6/2x = 1, where x = 3
2 different “sets” of rules for digits or letters, do not exist my friend. Numerals, letters, digits, what-have-you, for representing numbers, the same rules for operations exist.
6/a(a+b) =
6
—–
a(a+b)
let a=2, b=1
I think Bob is right. they are all constants. when you have 2x, it is always (2x) because you already have the product. but when you say (2)(x), its another story, those are still factors so it will be changed depending on the operations. when we have 6/2(2+1) (/ is divided by anyway), () is treated as multiplication in PEMDAS so you have to apply division first whether you like it or not. unless you have 6/(2(2+1)). 1/2x is not the same with 1/2(x) by the way.
It still seems to boil down to whether or not you’re comfortable with 2x/2x being x^2 rather than 1
2x / 2x is always 1. Look at my post from 3 April 2015. Every math text says so, therefore it is an agreed upon standard in the mathematical community.
Galois: Answer me this pretty please: If 2x = (2x), then substitute x=2+1 on both sides of that equation and tell us what you have! 🙂
“1/2x is not the same as 1/2(x)”
→ What rule/property/convention did you use to conclude this? Just curious as I’d like to see the book reference.
@ Bob: You say that “they are all constants”. I have stated several times that mathematics doesn’t change with the representation of a number.
Pi is a constant. 1 / 2π = 1 / (2π). Not (1/2)π.
And all calculators don’t give 9. Just recently most calculators have gone to a similar programming standard which are strictly left to right, with no interpretation to any grouping as seen in my post from 3 April 2015.
In order to use a calculator properly, you need to know how it is going to handle your input, and use apporpriate parentheses when required, even if you don’t need them on paper. All computer programs are garbage in → garbage out.
6 ÷ 2a =
6
—
2a
Let a = 2+1:
6 ÷ 2(2+1) =
6
——
2(2+1)
Consistency is key to mathematics.
How to solve 7:48::12:?
I have a doctorate in mathematics and have been teaching college mathematics (everything from arithmetic and pre-algebra up through calculus) for over 20 years now. Allow me to put in my 2 cents, if I may.
The idea that 2x/2x is always 1 is simply not true. Higher mathematical texts (say, those used for courses beyond calculus) may adopt a convention where that is true since doing so alleviates the need to endlessly write parentheses, as in 2x/(2x), which at that level often become more of a hindrance to communication of advanced concepts. There is little danger in that setting of the reader stumbling over the order of operations and missing the point.
However, and this to me is the heart of the matter, when working in the context of basic arithmetic and elementary algebra, the interpretation of a/bc or a/b(c) is exactly the same as a/b*c. Indeed, in the context of basic arithmetic, there is no distinction made between, say, 2*3 and 2(3). Parentheses are being used here to simply indicate multiplication, and that multiplication is no different than that indicated by the dot symbol or the * symbol or the x symbol. It’s all the same thing. There is nothing “stickier” or “stronger” or whatever about 2(3) than there is with 2*3. It’s exactly the same thing.
If you believe that 2(3) does not mean exactly the same thing as 2*3 or 2×3 in the context of basic arithmetic, then show me the beginner’s pre-algebra or elementary algebra text being used in today’s schools that makes a distinction. They don’t. At least none that I’ve ever seen or used over the years. Ever. Don’t quote from a Real Analysis text to make a case for basic arithmetic. The authors of those advanced texts may take the liberty to opt for other conventions. But that’s the point. They’re working with a different convention.
Now, about that sleight of hand in which some of you are engaging. I keep noticing some people making statements like 2a = (2a). Well, sure, in isolation you can put parentheses around the whole of anything, and it will be equivalent. But that doesn’t mean 6/2a is the same 6/(2a). Not in the context of basic arithmetic. In the context of basic arithmetic, 2a is always the same as 2*a. Therefore, in that context, 6/2a = 6/2*a = 3*a = 3a which is not the same as 6/(2a). So, while 2a = (2a) in isolation, taking the expression 6/2a and adding parentheses to get 6/(2a) fundamentally changes the expression.
In other words, you are not allowed in the context of basic arithmetic to simply add those parentheses to 6/2a to create the expression 6/(2a) while citing 2a = (2a) as your justification. Why? Because in the expression 6/2a, the 6 is being divided by the 2 only, not by 2a. How so? Because in the context of basic arithmetic, 2a = 2*a ALWAYS. Thus, 6/2a = 6/2*a = 3*a = 3a which is not equivalent to 6/(2a).
Moreover, in the context of basic arithmetic, 2a = 2(a) = (2)(a) = (2)a = 2*a ALWAYS. There is no distinction made among any of these. They all mean exactly the same thing: “two times a.” Thus, in the context of basic arithmetic, 6/2(a) = 6/2*a = 3*a ALWAYS.
Consequently, in the context of basic arithmetic, 6/2(1+2) = 6/2(3) = 6/2*3 = 3*3 = 9. Period. The issue about using the * is moot. There is no distinction made in the context of basic arithmetic between 2(3) and 2*3.
As someone has pointed out, not every calculator arrives at this same answer for 6/2(1+2). This is truly unfortunate, for surely errors will be made on account of the discrepancy. Because of this as well as other potential pitfalls, I try to teach my students to be a little wary of and a little careful with technology. How important it is to know your technology!
For what it’s worth, from what I can tell the convention that leads to 9 as the answer to 6/2(1+2) seems to have the upper hand among calculators, online computing apps, and such, but it is by no means unanimous. It sure would be nice to see uniformity. Authors of advanced texts would still be free to use whatever conventions they desire, but at the basic level of arithmetic and for the benefit of us all, we really should be on the same page.
To Dr. R! Thank you. You have explained that wonderfully and much better than I have frustratingly done that on this site. I agree with you wholeheartedly, but alas, I can guarantee that you will still receive much resistance from those that make up their own math rules to further their ideas or agendas. Good luck and I hope you have convinced at least one who stands on the other side. I don’t even try anymore. You are awesome.
a/b = a÷b = _a_
b
Where the whole is an expression and where “a” is the numerator and the “b” is the denominator.
If we look upon 2(2+1) as being “b” meaning it’s the denominator than we ought to treat 2(2×1) as a whole expressed vaue being the denominator.
From here it’s simple.
You are treating the “/” as being not the same as the “:” or the horizontal line where they are all having same mening therefore you apply the “order” falsely ignoring the fact that 2(2+1) is a whole expression and must be treated as such.
Good luck to you and to Dr. K
Can you please provide some references. I have provided lots. You are insinuating that “basic arithmetic and algebra” is different than “advanced math and arithmetic”. The rules don’t magically change at some point.
Some of the references are BASIC ALGEBRA!! :
*Order of Operations*
The order in which we carry out the operations that connect our numbers is as follows:
• Parentheses (or any other grouping process, such as brackets or braces)
• Exponents
• Multiplication and Division, as they occur from left to right in the problem
• Addition and Subtraction, as they occur from left to right.
then shows an example: 6x³÷2x = 3x²
The Everything Guide to Algebra By Christopher Monahan
Basic Algebra I: Second Edition
By Nathan Jacobson
“a/b + c/d = (ad +bc)/bd”
Introduction to Algebra
by Peter Jephson Cameron
“Since mx/nx = m/n we can always divide out common factors.”
Of course 2a = 2 * a
But the quantity 2a in 6/2a is
6
—
2a
Now do you substitution:
6
—-
2 * a
It all the context. I can’t find ANY books that do: 6/2a = (6/2)a.
They ALL use fractions OR the intended meaning shown above.
SO now that I have made my case using “Basic” algebra, as well as basic analysis books, what say you now?
If we want to represent this equation by real … Let’s assume that there is a factory produces some pieces … his client’s request to receive two 2 pieces every 6 minutes .. then asked to add one piece (2 +1) … after a period of time and then told him to double as requested by (1 + 2) 2 over the same period of 6 minutes … manufacturer wanted mathematical knowledge process requires factory production of pieces per minute >>>Division 6 minutes to the requirements of the customer of pieces after the last amendment 2 (2 +1), becoming a equation 6/2 (2 + 1)resulting really 6/6 = 1 and is realistic no doubt done to achieve the customer’s requirements one piece per minute … Greetings
I agree with all of mathman’s comments… and more importantly the textbooks he references!
Imagine if the equation was written 6/2(x+y). Using the distributive property couldn’t you solve it by starting with 6/(2x+2y) versus 6x/2 + 6y/2?
MattP: That is correct. I haven’t found a single reference that shows the second interpretation. Every single reference uses the first interpretation, yet we have ‘supposed mathematicians’ who are saying the opposite, which makes me wonder what they learned math from, or maybe some are not who they say they are…we ARE using the internet where nothing can be verified, EXCEPT the book references I posted.
Anything that cannot be verified is heresay. I learned from math books as well as mathematician and engineer professors. We used books which all used:
6 ÷ 2(a+b) = 6 ÷ (2a + 2b)
I have no doctorates in math. I am neither an engineer, nor a physicist, nor a mechanic. I did not get math in high school. I last took math in 8th grade.
Now that we’ve gotten how dumb I am out of the way, I took basic and advanced Algebra in 8th grade. By 10th grade, rather than math, I was receiving various computer design courses like CAD for my math requirements. I am a published electronics engineering expert, a certified biomedical equipment technician, a certified laser repair technician, and I write government policies, regulations, field manuals, etc. pertaining to medical electronic equipment.
I use math every day. I can say that, as an employer, or manager, I would never hire someone who got “9” as the answer to this equation. All these arguments about lower order math vs. higher order math are completely ridiculous. Sure, the conventions might be different, but elementary math conventions don’t have parentheses used in the expression 2(3), or shouldn’t. Any person who uses math, not studies, or teaches (you know what they say about those who teach), but uses math, will see the parentheses around the 3 as an indication that something is yet to be resolved.
Again, I am not a mathematician, so I only know the application- not the pretty language that convinces hard headed math teachers. A parentheses does not equal a simple sign of multiplication. The parentheses is as attached to the 2 as it is to the 3. The best argument to make this as simple as possible for all the hard headed people is the example of Pi- using a formula like (I have to use n for Pi) 2nr^2, how would you justify separating the Pi, or the radius squared (Pi is a constant, and the radius squared is a variable) from the two in this formula??? If we say 1/2nr^2, are we supposed to accept 1/(2n)r^2, (1/2)nr^2, and 1/2(nr^2), and other variations of the interpretation of this equation as equals??? That’s just ridiculous.
Again, I am not a mathematician, and I have not had a day of math class since 8th grade. Are you going to tell me that, because I USE math, I am using some kind of higher order math???
I will say this- if higher order math begins at the 8th grade, and there aren’t children debating this problem, why the heck are we talking about lower order math??? Also, if higher order math begins at the 8th grade, why even consider lower order (children’s) math as a standard for computing ANYTHING?
I believe Regulatory Guy provided a very functional explanation, especially with his comment on how a parenthetical, even a simplistic one, such as (3), presents the idea of something being incomplete, as well as his pi function denominator example. That said, I would like to expand on this, just a tine bit, to clarify on the incompleteness of a simplistic parenthetical, as given in the original question. 6/2(3) is not ambiguous at all. This is 6/6, or 1. There are no two ways around it. The reason being as follows: When you have a simplistic parenthetical, with a preceding multiplier, not separated by a clear product indicator, the multiplier is no longer a lone multiplier. It is a coefficient. Coefficients travel with their counter parts.
Dear readers and those who commented..the discussion is old and goes for years..
But how could the correct answer be 9? Seriously look further than just numbers alone look to what place they stand and what if they resemble something physical. Like Apple’s or cake slices. If I have 6Apple’s which needs to be divided over 2 sets of kids each set contains 1boy + 2girls..how the fuck can each kid get 9 apples? I thought it was mathematic and not magic! You can’t simply shift symbols and brackets if they mean something. As for those who answered 9, really give me an example with some physical product how you can come to 9?
Because it’s not what ur saying. It’s 6 divided by two times three. U divide first and then multiply. That’s how math goes. Ask a calculator, Google or wolphram alpha the answer is 9.